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International Mathematics Research Notices
, Volume 2018 (2) – Jan 1, 2018

34 pages

/lp/ou_press/the-closure-of-two-sided-multiplications-on-c-algebras-and-phantom-bhleeDpgzq

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.
- ISSN
- 1073-7928
- eISSN
- 1687-0247
- D.O.I.
- 10.1093/imrn/rnw248
- Publisher site
- See Article on Publisher Site

Abstract For a $$C^*$$-algebra $$A,$$ we consider the problem when the set $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ of all two-sided multiplications $$x \mapsto axb$$$$(a,b \in A)$$ on $$A$$ is norm closed, as a subset of $$\mathcal{B}(A)$$. We first show that $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ is norm closed for all prime $$C^*$$-algebras $$A$$. On the other hand, if $$A\cong \Gamma_0(\mathcal{E} )$$ is an $$n$$-homogeneous $$C^*$$-algebra, where $$\mathcal{E} $$ is the canonical $$\mathbb{M}_n $$-bundle over the primitive spectrum $$X$$ of $$A$$, we show that $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ fails to be norm closed if and only if there exists a $$\sigma$$-compact open subset $$U$$ of $$X$$ and a phantom complex line subbundle $$\mathcal{L} $$ of $$\mathcal{E} $$ over $$U$$ (i.e., $$\mathcal{L} $$ is not globally trivial, but is trivial on all compact subsets of $$U$$). This phenomenon occurs whenever $$n \geq 2$$ and $$X$$ is a CW-complex (or a topological manifold) of dimension $$3 \leq d<\infty$$. 1 Introduction Let $$A$$ be a $$C^*$$-algebra and let $$\mathop{\mathrm{IB}}\nolimits(A)$$ (resp. $$\mathop{\mathrm{ICB}}\nolimits(A)$$) denote the set of all bounded (resp. completely bounded) maps $$\phi : A \to A$$ that preserve (closed two-sided) ideals of $$A$$ (i.e., $$\phi(I)\subseteq I$$ for all ideals $$I$$ of $$A$$). The most prominent class of maps $$\phi\in \mathop{\mathrm{ICB}}\nolimits(A) \subset \mathop{\mathrm{IB}}\nolimits(A)$$ are elementary operators, that is, those that can be expressed as finite sums of two-sided multiplications$$M_{a,b} : x \mapsto axb$$, where $$a$$ and $$b$$ are elements of the multiplier algebra $$M(A)$$. Elementary operators play an important role in modern quantum information and quantum computation theory. In particular, maps $$\phi : \mathbb{M}_n \to \mathbb{M}_n$$ ($$\mathbb{M}_n$$ are $$n \times n$$ matrices over $$\mathbb{C}$$) of the form $$\phi=\sum_{i=1}^\ell M_{a_i^*,a_i}$$ ($$a_i \in \mathbb{M}_n$$ such that $$\sum_{i=1}^\ell a_i^*a_i=1$$) represent the (trace-duals of) quantum channels, which are mathematical models of the evolution of an “open” quantum system (see e.g., [21]). Elementary operators also provide ways to study the structure of $$C^*$$-algebras (see [2]). Let $$\mathcal{E}\ell(A)$$, $$\mathop{\mathrm{TM}}\nolimits(A),$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ denote, respectively, the sets of all elementary operators on $$A$$, two-sided multiplications on $$A$$ and two-sided multiplications on $$A$$ with coefficients in $$A$$ (i.e., $$\mathop{\mathrm{TM}}\nolimits_0(A) = \{ M_{a,b}: a, b \in A\}$$). The elementary operators are always dense in $$\mathop{\mathrm{IB}}\nolimits(A)$$ in the topology of pointwise convergence (by [23, Corollary 2.3]). However, more subtle considerations enter in when one asks if $$\phi \in \mathop{\mathrm{ICB}}\nolimits(A)$$ can be approximated pointwise by elementary operators of cb-norm at most $$\|\phi\|_{cb}$$ ([24] shows that nuclearity of $$A$$ suffices; see also [26]). It is an interesting problem to describe those operators $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ (resp. $$\phi \in \mathop{\mathrm{ICB}}\nolimits(A)$$) that can be approximated in operator norm (resp. cb-norm) by elementary operators. Earlier works, which we cite below, revealed that this is an intricate question in general, and can involve many and varied properties of $$A$$ and $$\phi$$. In this article, we show that the apparently much simpler problems of describing the norm closures of $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ can have complicated answers even for rather well-behaved $$C^*$$-algebras. In some cases, $$\mathcal{E}\ell(A) = \mathop{\mathrm{IB}}\nolimits(A)$$ (which implies $$\mathcal{E}\ell(A) = \mathop{\mathrm{ICB}}\nolimits(A)$$); or $$\mathcal{E}\ell(A)$$ is norm dense in $$\mathop{\mathrm{IB}}\nolimits(A)$$; or $$\mathcal{E}\ell(A) \subset \mathop{\mathrm{ICB}}\nolimits(A)$$ is dense in cb-norm. The conditions just mentioned are in fact all equivalent for separable $$C^*$$-algebras $$A$$. More precisely, Magajna [25] shows that for separable $$C^*$$-algebras $$A$$, the property that $$\mathcal{E}\ell(A)$$ is norm (resp. cb-norm) dense in $$\mathop{\mathrm{IB}}\nolimits(A)$$ (resp. $$\mathop{\mathrm{ICB}}\nolimits(A)$$) characterizes finite direct sums of homogeneous $$C^*$$-algebras with the finite type property. Moreover, in this situation we already have the equality $$\mathop{\mathrm{ICB}}\nolimits(A)=\mathop{\mathrm{IB}}\nolimits(A)=\mathcal{E}\ell(A)$$. It can happen that $$\mathcal{E}\ell(A)$$ is already norm closed (or cb-norm closed). In [13, 14], the first author showed that for a unital separable $$C^*$$-algebra $$A$$, if $$\mathcal{E}\ell(A)$$ is norm (or cb-norm) closed then $$A$$ is necessarily subhomogeneous, the homogeneous sub-quotients of $$A$$ must have the finite type property and established further necessary conditions on $$A$$. In [14, 15] he gave some partial converse results. There is a considerable literature on derivations and inner derivations of $$C^*$$-algebras. Inner derivations $$d_a$$ on a $$C^*$$-algebra $$A$$, (i.e., those of the form $$d_a(x) = ax-xa$$ with $$a \in M(A)$$) are important examples of elementary operators. In [35, Corollary 4.6] Somerset shows that if $$A$$ is unital, $$\{ d_a : a \in A\}$$ is norm closed if and only if $$\mathrm{Orc}(A) < \infty$$, where $$\mathrm{Orc}(A)$$ is a constant defined in terms of a certain graph structure on $$\mathrm{Prim}$$(A) (the primitive spectrum of $$A$$). If $$\mathrm{Orc}(A) = \infty$$, the structure of outer derivations that are norm limits of inner derivations remains undescribed. In addition, if $$A$$ is unital and separable, then by [19, Theorem 5.3] and [35, Corollary 4.6] $$\mathrm{Orc}(A) < \infty$$ if and only if the set $$\{ M_{u,u^*} : u \in A, u \mbox{ unitary}\}$$ of inner automorphisms is norm closed. In [12, 15] the first author considered the problem of which derivations on unital $$C^*$$-algebras $$A$$ can be cb-norm approximated by elementary operators. By [15, Theorem 1.5] every such a derivation is necessarily inner in a case when every Glimm ideal of $$A$$ is prime. When this fails, it is possible to produce examples which have outer derivations that are simultaneously elementary operators ([12, Example 6.1]). While considering derivations $$d$$ that are elementary operators and/or norm limits of inner derivations, we realized that they are sometimes expressible in the form $$d = M_{a,b} - M_{b,a}$$ even though they are not inner. We have not been able to decide when all such $$d$$ are of this form, but this led us to the seemingly simpler question of considering the closures of $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$. In this article, we see that non-trivial considerations enter into these questions about two-sided multiplications. Of course the left multiplications $$\{ M_{a,1} : a \in M(A)\}$$ are already norm closed, as are the right multiplications. So $$\mathop{\mathrm{TM}}\nolimits(A)$$ is a small subclass of $$\mathcal{E}\ell(A)$$, and seems to be the basic case to study. This article is organized as follows. We begin in Section 2 with some generalities and an explanation that the set of elementary operators of length at most $$\ell$$ has the same completion in the operator and cb-norms (for each $$\ell \geq 1$$). In Section 3, we show that for a prime $$C^*$$-algebra $$A$$, we always have $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ both norm closed. In Section 4, we recall the description of ($$n$$-)homogeneous $$C^*$$-algebras $$A$$ as sections $$\Gamma_0(\mathcal{E})$$ of $$\mathbb{M}_n$$-bundles $$\mathcal{E} $$ over $$X= \mathrm{Prim}(A)$$ and some general results about $$\mathop{\mathrm{IB}}\nolimits(A)$$, $$\mathop{\mathrm{ICB}}\nolimits(A)$$ and $$\mathcal{E}\ell(A)$$ for such $$A$$. In Section 5, for homogeneous $$C^*$$-algebras $$A = \Gamma_0(\mathcal{E} )$$, we consider subclasses $$\mathop{\mathrm{IB}}\nolimits_1(A)$$ and $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ of $$\mathop{\mathrm{IB}}\nolimits(A)$$ that seem (respectively) to be the most obvious choices for the norm closure of $$\mathop{\mathrm{TM}}\nolimits(A)$$ and of $$\mathop{\mathrm{TM}}\nolimits_0(A)$$, extrapolating from fibrewise restrictions on $$\phi \in \mathop{\mathrm{TM}}\nolimits(A)$$. For each $$\phi \in \mathop{\mathrm{IB}}\nolimits_1(A),$$ we associate a complex line subbundle $$\mathcal{L} _\phi$$ of the restriction $$\mathcal{E} |_U$$ to an open subset $$U \subseteq X = \mathrm{Prim}(A)$$, where $$U$$ is determined by $$\phi$$ as the cozero set of $$\phi$$ ($$U$$ identifies the fibres of $$\mathcal{E} $$ on which $$\phi$$ acts by a non-zero operator). For separable $$A$$, the main result of this section is Theorem 5.15, where we characterize the condition $$\phi \in \mathop{\mathrm{TM}}\nolimits(A)$$ in terms of triviality of the bundle $$\mathcal{L} _\phi$$. We close Section 5 with Remark 5.21 comparing our bundle considerations to slightly similar results in the literature for innerness of $$C(X)$$-linear automorphisms when $$X$$ is compact, or for some more general unital $$A$$. Our final Section 6 is the main section of this article. For homogeneous $$C^*$$-algebras $$A = \Gamma_0(\mathcal{E} )$$, we characterize operators $$\phi$$ in the norm closure of $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ as those operators in $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ for which the associated complex line bundle $$\mathcal{L} _\phi$$ is trivial on each compact subset of $$U$$, where $$U$$ is as above (Theorem 6.9). As a consequence, we obtain that $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ fails to be norm closed if and only if there exists a $$\sigma$$-compact open subset $$U$$ of $$X$$ and a phantom complex line subbundle $$\mathcal{L} $$ of $$\mathcal{E}|_U$$ (i.e., $$\mathcal{L} $$ is not globally trivial, but is trivial on each compact subset of $$U$$). Using this and some algebraic topological ideas, we show that $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ both fail to be norm closed whenever $$A$$ is $$n$$-homogeneous with $$n\geq 2$$ and $$X$$ contains an open subset homeomorphic to $$\mathbb{R}^d$$ for some $$d \geq 3$$ (Theorem 6.18). 2 Preliminaries Throughout this article, $$A$$ will denote a $$C^*$$-algebra. By an ideal of $$A$$ we always mean a closed two-sided ideal. As usual, by $$Z(A)$$ we denote the centre of $$A$$, by $$M(A)$$ the multiplier algebra of $$A$$, and by $$\mathrm{Prim}(A)$$ the primitive spectrum of $$A$$ (i.e., the set of kernels of all irreducible representations of $$A$$ equipped with the Jacobson topology). Every $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ is linear over $$Z(M(A))$$ and, for any ideal $$I$$ of $$A$$, $$\phi$$ induces a map ϕI:A/I→A/I,which sendsa+I to ϕ(a)+I. (2.1) It is easy to see that the norm (resp. cb-norm) of an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ (resp. $$\phi \in \mathop{\mathrm{ICB}}\nolimits(A)$$) can be computed via the formulae ‖ϕ‖=sup{‖ϕP‖ : P∈Prim(A)}resp.‖ϕ‖cb=sup{‖ϕP‖cb : P∈Prim(A)}. (2.2) The length of a non-zero elementary operator $$\phi\in \mathcal{E}\ell(A)$$ is the smallest positive integer $$\ell=\ell(\phi)$$ such that $$\phi=\sum_{i=1}^\ell M_{a_i,b_i}$$ for some $$a_i,b_i \in M(A)$$. We also define $$\ell(0)=0$$. We write $$\mathcal{E}\ell_\ell(A)$$ for the elementary operators of length at most $$\ell$$. Thus $$\mathcal{E}\ell_1 (A)=\mathop{\mathrm{TM}}\nolimits(A)$$. We will also consider the following subsets of $$\mathop{\mathrm{TM}}\nolimits(A)$$: TMcp(A)={Ma,a∗:a∈M(A)},InnAutalg(A)={Ma,a−1:a∈M(A), a invertible}, andInnAut(A)={Mu,u∗:u∈M(A), u unitary} (2.3) (where cp and alg signify, respectively, completely positive and algebraic). Note that $$\mathrm{InnAut}(A)=\mathop{\mathrm{TM}}\nolimits_{cp}(A)\cap \mathrm{InnAut_{alg}}(A)$$. It is well known that elementary operators are completely bounded with the following estimate for their cb-norm: ‖∑iMai,bi‖cb≤‖∑iai⊗bi‖h, (2.4) where $$\|\cdot\|_h$$ is the Haagerup tensor norm on the algebraic tensor product $$M(A) \otimes M(A)$$, that is, ‖u‖h=inf{‖∑iaiai∗‖12‖∑ibi∗bi‖12 : u=∑iai⊗bi}. By inequality (2.4) the mapping (M(A)⊗M(A),‖⋅‖h)→(Eℓ(A),‖⋅‖cb)given by∑iai⊗bi↦∑iMai,bi. defines a well-defined contraction. Its continuous extension to the completed Haagerup tensor product $$M(A) \otimes_h M(A)$$ is known as a canonical contraction from $$M(A) \otimes_h M(A)$$ to $$\mathrm{ICB}(A)$$ and is denoted by $$\Theta_A$$. We have the following result (see [2, Proposition 5.4.11]): Theorem 2.1. (Mathieu). $$\Theta_A$$ is isometric if and only if $$A$$ is a prime $$C^*$$-algebra. □ The next result is a combination of [36, Corollary 3.8], (2.2), [37, Corollary 2.4], and the facts that for $$\phi = \sum_{i=1}^\ell M_{a_i, b_i}$$, we have $$\|\phi_\pi\|=\|\phi_{\ker \pi}\|$$ and $$\|\phi_\pi\|_{cb}=\|\phi_{\ker \pi}\|_{cb}$$ where for irreducible representation $$\pi \colon A \to \mathcal{B}(H_\pi)$$, $$\phi_\pi = \sum_{i=1}^\ell M_{\pi(a_i), \pi(b_i)} \in \mathcal{E}\ell ( \mathcal{B}(H_\pi))$$ (as in [36, Section 4] or [37, Section 2]). Theorem 2.2. (Timoney). For $$A$$ a $$C^*$$-algebra and arbitrary $$\phi \in \mathcal{E}\ell(A)$$ of length $$\ell$$ we have ‖ϕ‖cb=‖ϕ(ℓ)‖≤ℓ‖ϕ‖, where $$\phi^{(\ell)}$$ denotes the $$\ell$$-th amplification of $$\phi$$ on $$M_\ell(A)$$, $$\phi^{(\ell)} : [x_{i,j}] \mapsto [\phi(x_{i,j})]$$. In particular, on each $$\mathcal{E}\ell_\ell(A)$$ the metric induced by the cb-norm is equivalent to the metric induced by the operator norm. □ 3 Two-sided multiplications on prime $$C^*$$-algebras If $$A$$ is a prime $$C^*$$-algebra, we prove here (Theorem 3.4) that $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ must be closed in $$\mathcal{B}(A)$$. The crucial step is the following lemma. Lemma 3.1. Let $$a,b,c$$ and $$d$$ be norm-one elements of an operator space $$V$$. If ‖a⊗b−c⊗d‖h<ε≤1/3, then there exists a complex number $$\mu$$ of modulus one such that max{‖a−μc‖,‖b−μ¯d‖}<4ε. □ Proof. First we dispose of the simpler cases where $$a$$ and $$c$$ are linearly dependent or where $$b$$ and $$d$$ are linearly dependent. If $$a$$ and $$c$$ are dependent then $$a = \mu c$$ with $$|\mu| = 1$$. So $$a \otimes b - c \otimes d = c \otimes (\mu b-d)$$ and $$\|a \otimes b - c \otimes d\|_h = \|\mu b- d\| < \varepsilon < 4 \varepsilon$$. Similarly if $$b$$ and $$d$$ are dependent, $$b = \bar\mu d$$ with $$|\mu| = 1$$ and $$\|\bar\mu a -c\| < \varepsilon $$. Leaving aside these cases, $$a \otimes b - c \otimes d$$ is a tensor of rank $$2$$. By [5, Lemma 2.3] there is an invertible matrix $$S \in \mathbb{M}_2$$ such that ‖[a−c]S‖<εand‖S−1[bd]‖<ε. Write $$\alpha_{i,j}$$ for the $${i,j}$$ entry of $$S$$ and $$\beta_{i,j}$$ for the $${i,j}$$ entry of $$S^{-1}$$. Since $$\alpha_{1,1} \beta_{1,1} + \alpha_{1,2} \beta_{2,1} =1$$, at least one of the absolute values $$|\alpha_{1,1}|$$, $$| \beta_{1,1}|$$, $$|\alpha_{1,2}|$$ or $$|\beta_{2,1}|$$ must be at least $$1/\sqrt{2}$$. We treat the four cases separately, by very similar arguments. The case$$|\alpha_{1,1}| \geq 1/ \sqrt{2}$$.] From [a−c]S=[α1,1a−α2,1cα1,2a−α2,2c] we have $$\| \alpha_{1,1} a -\alpha_{2,1} c\| < \varepsilon$$, so ‖a−α2,1α1,1c‖<ε|α1,1|≤2ε. (Hence $$\alpha_{2,1} \neq 0$$ as $$\varepsilon \leq 1/3$$ and $$\|a\| =1$$.) Let $$\lambda = \alpha_{2,1}/\alpha_{1,1}$$. Then |1−|λ||=|‖a‖−|λ|‖c‖|≤‖a−λc‖≤2ε, and so $$ |\lambda| \in [ 1 - \sqrt{2} \varepsilon, 1+ \sqrt{2} \varepsilon]$$. Also |λ|λ|−λ|=|1−|λ||≤2ε. So for $$\mu = \frac{\lambda}{|\lambda|}$$ we have $$|\mu| = 1$$ and ‖a−μc‖≤‖a−λc‖+|λ−μ|‖c‖<2ε+2ε<3ε. Then a⊗b−c⊗d=a⊗b−(μc⊗μ¯d)=(a⊗b)−(a⊗μ¯d)+(a⊗μ¯d)−(μc⊗μ¯d)=(a⊗(b−μ¯d))+((a−μc)⊗μ¯d) and thus ‖b−μ¯d‖=‖a⊗(b−μ¯d)‖h≤‖a⊗b−c⊗d‖h+‖(a−μc)⊗μ¯d‖h<ε+3ε=4ε. Case$$|\alpha_{1,2}| \geq 1/ \sqrt{2}$$.] We start now with $$\| \alpha_{1,2} a - \alpha_{2,2} c\| < \varepsilon$$ and proceed in the same way (with $$\lambda = \alpha_{2,2}/\alpha_{1,2}$$). Case$$|\beta_{1,1}| \geq 1/ \sqrt{2}$$.] We use S−1[bd]=[β1,1b+β1,2dβ2,1b+β2,2d] and $$\|\beta_{1,1} b + \beta_{1,2} d\| < \varepsilon$$, leading to a similar argument (with $$\lambda= -\beta_{1,2}/\beta_{1,1}$$ and $$b$$ taking the role of $$a$$). Case$$|\beta_{2,1}| \geq 1/ \sqrt{2}$$.] Use $$\| \beta_{2,1} b + \beta_{2,2} d \| < \varepsilon$$. ■ Corollary 3.2. If $$V$$ is an operator space, the set $$ S_1=\{a \otimes b : a,b \in V\} $$ of all elementary tensors forms a closed subset of $$V \otimes_h V$$. □ Proof. Suppose $$a_n \otimes b_n \to u \in V \otimes_h V$$ (for $$a_n, b_n \in V$$). If $$u = 0$$, certainly $$u \in S_1$$ and otherwise we may assume $$\|u\|_h =1$$ and also that ‖an⊗bn‖h=1=‖an‖=‖bn‖(n≥1). Passing to a subsequence, we may suppose ‖an⊗bn−an+1⊗bn+1‖h≤14⋅2n(n≥1). By Lemma 3.1, we may multiply $$a_n$$ and $$b_n$$ by complex conjugate modulus one scalars chosen inductively to get $$a'_n$$ and $$b'_n$$ such that an⊗bn=an′⊗bn′,‖an′−an+1′‖≤1/2nand‖bn′−bn+1′‖≤1/2n(n≥1). In this way we find $$a = \lim_{n \to \infty} a'_n$$ and $$b = \lim_{n \to \infty} b'_n$$ in $$V$$ with $$u = a \otimes b \in S_1$$. ■ Question 3.3. If $$V$$ is an operator space and $$\ell >1$$, is the set Sℓ={∑i=1ℓai⊗bi:ai,bi∈V} of all tensors of rank at most $$\ell$$ closed in $$V \otimes_h V$$? In particular, can we extend Lemma 3.1 as follows. Let $$V$$ be an operator space and let $$\textbf{a} \odot \textbf{b}$$ and $$\textbf{c} \odot \textbf{d}$$ be two norm-one tensors of the same (finite) rank $$\ell$$ in $$V \otimes_h V$$, where $$\textbf{a},\textbf{c}$$ and $$\textbf{b},\textbf{d}$$ are, respectively, $$1\times \ell$$ and $$\ell \times 1$$ matrices with entries in $$V$$. Suppose that $$\|\textbf{a} \odot \textbf{b}-\textbf{c} \odot \textbf{d}\|_h < \varepsilon$$ for some $$\varepsilon >0$$. Can we find absolute constants $$C$$ and $$\delta$$ (which depend only on $$\ell$$ and $$\varepsilon$$) so that $$\delta \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ with the following property: There exists an invertible matrix $$S\in \mathbb{M}_\ell$$ such that ‖S‖,‖S−1‖≤C,‖aS−1−c‖<δand‖Sb−d‖<δ? □ Theorem 3.4. If $$A$$ is a prime $$C^*$$-algebra, then both $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ are norm closed. □ Proof. By Theorem 2.2 we may work with the cb-norm instead of the (operator) norm. Since $$A$$ is prime, by Mathieu’s theorem (Theorem 2.1) the canonical map $$\Theta : M(A) \otimes_h M(A) \to \mathop{\mathrm{ICB}}\nolimits(A)$$, $$\Theta :a \otimes b \mapsto M_{a,b}$$, is isometric. By Corollary 3.2, the set $$S_1$$ of all elementary tensors in $$M(A) \otimes_h M(A)$$ is closed in the Haagerup norm. Therefore, $$\mathop{\mathrm{TM}}\nolimits(A)=\Theta(S)$$ is closed in the cb-norm. For the case of $$\mathop{\mathrm{TM}}\nolimits_0(A)$$, we use the same argument but work with the restriction of $$\Theta$$ to $$A \otimes_h A$$. ■ Corollary 3.5. If $$A$$ is a prime $$C^*$$-algebra, then the sets $$\mathop{\mathrm{TM}}\nolimits_{cp}(A)$$, $$\mathrm{InnAut_{alg}}(A)$$, and $$\mathrm{InnAut}(A)$$ (see (2.3)) are all norm closed. □ Proof. Suppose that an operator $$\phi$$ in the norm closure of any of these sets. Then, by Theorem 3.4 there are $$b,c \in M(A)$$ such that $$\phi=M_{b,c}$$. Let $$\varepsilon >0$$. Suppose that $$\phi$$ is in the closure of $$\mathop{\mathrm{TM}}\nolimits_{cp}(A)$$. By Theorem 2.2 we may work with the cb-norm instead of the (operator) norm. We may also assume that $$\|\phi\|_{cb}=1=\|b\|=\|c\|$$. Then there is $$a \in M(A)$$ such that ‖Mb,c−Ma,a∗‖cb=‖b⊗c−a⊗a∗‖h<ε (Theorem 2.1). If $$\varepsilon \leq 1/3$$, by Lemma 3.1 we can find a complex number $$\mu$$ of modulus one such that $$\|b-\mu a\|\ < 4 \varepsilon$$ and $$\|c-\overline{\mu}a^*\|<4 \varepsilon$$. Then $$\|b-c^*\|\leq 8 \varepsilon$$. Hence $$c=b^*$$, so $$\phi = M_{b,c} \in \mathop{\mathrm{TM}}\nolimits_{cp}(A)$$. Suppose that $$\phi$$ is in the closure of $$\mathrm{InnAut_{alg}}(A)$$. Then there is an invertible element $$a \in M(A)$$ such that $$\|M_{b,c}-M_{a,a^{-1}}\|< \varepsilon$$. Since $$A$$ is an essential ideal in $$M(A)$$, this implies $$\|bxc-axa^{-1}\|< \varepsilon$$ for all $$x \in M(A)$$, $$\|x\|\leq1$$. Letting $$x=1$$ we obtain $$\|bc-1\|<\varepsilon$$. Hence $$c=b^{-1}$$, so $$\phi = M_{b,c} \in \mathrm{InnAut_{alg}}(A)$$. $$\mathrm{InnAut}(A)$$ is norm closed as an intersection $$\mathop{\mathrm{TM}}\nolimits_{cp}(A)\cap \mathrm{InnAut_{alg}}(A)$$ of two closed sets. ■ 4 On homogeneous $$C^*$$-algebras We recall that a $$C^*$$-algebra $$A$$ is called $$n$$-homogeneous (where $$n$$ is finite) if every irreducible representation of $$A$$ acts on an $$n$$-dimensional Hilbert space. We say that $$A$$ is homogeneous if it is $$n$$-homogeneous for some $$n$$. We will use the following definitions and facts about homogeneous $$C^*$$-algebras: Remark 4.1. Let $$A$$ be an $$n$$-homogeneous $$C^*$$-algebra. By [20, Theorem 4.2] $$\mathrm{Prim}(A)$$ is a (locally compact) Hausdorff space. If there is no danger of confusion, we simply write $$X$$ for $$\mathrm{Prim}(A)$$. (a) A well-known theorem of Fell [10, Theorem 3.2], and Tomiyama-Takesaki [38, Theorem 5] asserts that for any $$n$$-homogeneous $$C^*$$-algebra, $$A$$, there is a locally trivial bundle $$\mathcal{E} $$ over $$X$$ with fibre $$\mathbb{M}_n$$ and structure group $$PU(n)=\mathrm{Aut}(\mathbb{M}_n)$$ such that $$A$$ is isomorphic to the $$C^*$$-algebra $$\Gamma_0(\mathcal{E} )$$ of continuous sections of $$\mathcal{E} $$ which vanish at infinity. Moreover, any two such algebras $$A_i=\Gamma_0(\mathcal{E} _i)$$ with primitive spectra $$X_i$$ ($$i =1,2$$) are isomorphic if and only if there is a homeomorphism $$f : X_1 \to X_2$$ such that $$\mathcal{E} _1 \cong f^*(\mathcal{E} _2)$$ (the pullback bundle) as bundles over $$X_1$$ (see [38, Theorem 6]). Thus, we may identify $$A$$ with $$\Gamma_0(\mathcal{E} )$$. (b) For $$a \in A$$ and $$t \in X$$ we define $$\pi_t(a) = a(t)$$. Then, after identifying the fibre $$\mathcal{E}_t$$ with $$\mathbb{M}_n$$, $$\pi_t: a \mapsto \pi_t(a)$$ (for $$t \in X$$) gives all irreducible representations of $$A$$ (up to the equivalence). For a closed subset $$S \subseteq X$$ we define IS=⋂t∈Skerπt={a∈A : a(t)=0 for all t∈S}. By [11, VII 8.7.] any closed two-sided ideal of $$A$$ is of the form $$I_S$$ for some closed subset $$S \subset X$$. Further, by the the generalized Tietze Extension Theorem we may identify $$A_S=A/I_S$$ with $$\Gamma_0(\mathcal{E} |_S)$$ (see [11, II. 14.8. and VII 8.6.]). If $$S=\{t\}$$ we just write $$A_t$$. (c) If $$\phi\in \mathop{\mathrm{IB}}\nolimits(A)$$ and $$S \subset X$$ closed, we write $$\phi_S$$ for the operator $$\phi_{I_S}$$ on $$A_S$$ (see (2.1)). If $$S=\{t\}$$ we just write $$\phi_t$$. If $$A$$ is trivial (i.e., $$A=C_0(X,\mathbb{M}_n )$$), we will consider $$\phi_t$$ as an operator $$ \colon \mathbb{M}_n \to \mathbb{M}_n $$ (after identifying $$A_t$$ with $$\mathbb{M}_n$$ in the obvious way). If $$U \subset X$$ is open, we can regard $$B= \Gamma_0(\mathcal{E}|_U)$$ as the ideal $$I_{ X \setminus U}$$ of $$A$$ (by extending sections to be zero outside $$U$$) and for $$\phi\in \mathop{\mathrm{IB}}\nolimits(A)$$, we then have a restriction $$\phi|_U \in \mathop{\mathrm{IB}}\nolimits(B)$$ of $$\phi$$ to this ideal (with $$(\phi|_U)_t = \phi_t$$ for $$t \in U$$). (d) $$\mathop{\mathrm{IB}}\nolimits(A)=\mathop{\mathrm{ICB}}\nolimits(A)$$. Indeed, for $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ and $$t \in X$$ we have $$\|\phi_t\|_{cb} \leq n \|\phi_t\|$$ ([29, p. 114]), so by (2.2) we have $$\|\phi\|_{cb} \leq n \|\phi\|$$. Hence $$\phi \in \mathop{\mathrm{ICB}}\nolimits(A)$$. (e) Since each $$a \in Z(A)$$ has $$a(t)$$ a multiple of the identity in the fibre $$\mathcal{E}_t$$ for each $$t \in X$$, we can identify $$Z(A)$$ with $$C_0(X)$$. Observe that $$A$$ is quasi-central (i.e., no primitive ideal of $$A$$ contains $$Z(A)$$). (f) By [25, Lemma 3.2] we can identify $$M(A)$$ with $$\Gamma_b(\mathcal{E} )$$ (the $$C^*$$-algebra of bounded continuous sections of $$\mathcal{E} $$). As usual, we will identify $$Z(M(A))$$ with $$C_b(X)$$ (using the Dauns–Hofmann theorem [33, Theorem A.34]). If $$A=C_0(X, \mathbb{M}_n )$$, it is well known that $$M(A)=C_b(X, \mathbb{M}_n)=C( \beta X, \mathbb{M}_n )$$ [1, Corollary 3.4], where $$\beta X$$ denotes the Stone-Čech compactification. (g) On each fibre $$\mathcal{E}_t$$ we can introduce an inner product $$\langle \cdot,\cdot \rangle_{2}$$ as follows. Choose an open covering $$\{U_\alpha\}$$ of $$X$$ such that each $$\mathcal{E}|_{U_\alpha}$$ is isomorphic to $$U_\alpha \times \mathbb{M}_n$$ (as an $$\mathbb{M}_n$$-bundle), say via isomorphism $$\Phi_\alpha$$. Let ⟨ξ,η⟩2=tr(Φα(ξ)Φα(η)∗)(ξ,η∈Et), (4.1) where $$\alpha$$ is chosen so that $$t \in U_\alpha$$ and $$\mathrm{tr}(\cdot)$$ is the standard trace on $$\mathbb{M}_n$$. This is independent of the choice of $$\alpha$$ since all automorphisms of $$\mathbb{M}_n$$ are inner and $$\mathrm{tr}(\cdot)$$ is invariant under conjugation by unitaries. If $$a, b \in M(A) = \Gamma_b(\mathcal{E})$$ then $$t \mapsto \langle a(t),b(t) \rangle_2$$ is in $$C_b(X)$$. The norm $$\| \cdot \|_2$$ on $$\mathcal{E}_t$$ associated with $$\langle \cdot, \cdot \rangle_2$$ satisfies ‖ξ‖≤‖ξ‖2≤n‖ξ‖(ξ∈Et). (4.2) In the terminology of [8], $$(\mathcal{E}, \langle \cdot, \cdot \rangle_2)$$ is a (complex continuous) Hilbert bundle of rank $$n^2$$ with fibre norms equivalent to the original $$C^*$$-norms (by (4.2)). (h) $$A$$ is said to have the finite type property if $$\mathcal{E} $$ can be trivialized over some finite open cover of $$X$$. By [25, Remark 3.3] $$M(A)$$ is homogeneous if and only if $$A$$ has the finite type property. When this fails, it is possible to have $$\mathrm{Prim}(M(A))$$ non-Hausdorff [4, Theorem 2.1]. On the other hand, $$M(A)$$ is always quasi-standard (see [3, Corollary 4.10]). □ For completeness we include a proof of the following. Proposition 4.2. Let $$X$$ be a locally compact Hausdorff space and $$A=C_0(X, \mathbb{M}_n)$$. (a) $$\mathrm{IB}(A)$$ can be identified with $$C_b(X, \mathcal{B}(\mathbb{M}_n ))$$ by a mapping which sends an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ to the function $$(t \mapsto \phi_t)$$. (b) Any $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ can be written in the form ϕ=∑i,j=1nMei,j,ai,j, (4.3) where $$(e_{i,j})_{i,j=1}^n$$ are standard matrix units of $$\mathbb{M}_n$$ (considered as constant functions in $$C_b(X, \mathbb{M}_n )=M(A)$$) and $$a_{i,j}\in M(A)$$ depend on $$\phi$$. Thus, we have IB(A)=ICB(A)=Cb(X,B(Mn))=Eℓ(A)=Eℓn2(A). □ Proof. Let $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$. (a) Suppose that the function $$t \mapsto \phi_t \colon X \to \mathcal{B}(\mathbb{M}_n )$$ is discontinuous at some point $$t_0 \in X$$. Then there is a net $$(t_\alpha)$$ in $$X$$ converging to $$t_0$$ such that $$\| \phi_{t_\alpha} - \phi_{t_0}\| \geq \delta > 0$$ for all $$\alpha$$. So there is $$u_\alpha \in \mathbb{M}_n $$ of norm at most 1 with $$\| \phi_{t_\alpha}(u_\alpha) - \phi_{t_0}(u_\alpha) \| \geq \delta$$. Passing to a subnet we may suppose $$u_\alpha \to u$$ and then (since $$\|\phi_{t_\alpha}\| \leq \|\phi\|$$ and $$\|\phi_{t_0}\| \leq \|\phi\|$$) we must have ‖ϕtα(u)−ϕt0(u)‖>δ/2 for $$\alpha$$ large enough. Now choose $$f \in C_0(X)$$ equal to 1 on a neighbourhood of $$t_0$$ and put $$a(t) = f(t) u$$. We then have $$a \in A$$ and πtα(ϕ(a))=f(tα)ϕtα(u)=ϕtα(u) for large $$\alpha$$ and this contradicts continuity of $$\phi(a)$$ at $$t_0$$. So $$t \mapsto \phi_t$$ must be continuous (and also bounded by $$\|\phi\|$$). Conversely, assume that the function $$t \mapsto \phi_t$$ is continuous and uniformly bounded by some $$M > 0$$. Then for $$a \in A$$, $$t \mapsto \phi_t(\pi_t(a))$$ is continuous, bounded and vanishes at infinity, hence in $$A$$. So there is an associated mapping $$\phi \colon A \to A$$ which is easily seen to be bounded and linear. Moreover $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ since all ideals of $$A$$ are of the form $$I_S$$ for some closed $$S \subset X$$. (b) First assume that $$A$$ is unital, so that $$X$$ is compact. Then each $$x \in A$$ is a linear combination over $$C(X)=Z(A)$$ of the $${e}_{i,j}$$ and since $$\phi$$ is $$C(X)$$-linear, we have x=∑i,j=1nxi,jei,j ⇒ ϕ(x)=∑i,j=1nxi,jϕ(ei,j). We may write ϕ(ei,j)=∑k,ℓ=1nϕi,j,k,ℓek,ℓ=∑k,r=1nek,rei,j(∑s,ℓ=1nϕr,s,k,ℓes,ℓ) where $$\phi_{i,j, k, \ell} \in C(X)$$. It follows that ϕ(x)=∑i,j=1nxi,j(∑k,r=1nek,rei,j(∑s,ℓ=1nϕr,s,k,ℓes,ℓ))=∑k,r=1nek,r((∑i,j=1nxi,jei,j)∑s,ℓ=1nϕr,s,k,ℓes,ℓ)=∑k,r=1nek,rx(∑s,ℓ=1nϕr,s,k,ℓes,ℓ). Hence, $$\phi$$ is of the form (4.3), where $$a_{i,j}=\sum_{s, \ell=1}^n \phi_{j,s, i, \ell} {e}_{s, \ell} \in M(A)$$. Now suppose that $$A$$ is non-unital (so that $$X$$ is non-compact). By (a) we can identify $$\phi$$ with the function $$t \mapsto \phi_t \colon X \to \mathcal{B}(\mathbb{M}_n )$$, which can be then uniquely extended to a continuous function $$\beta X \to \mathcal{B}(\mathbb{M}_n )$$. This extension defines an operator in $$\mathop{\mathrm{IB}}\nolimits( C( \beta X, \mathbb{M}_n )) = \mathop{\mathrm{IB}}\nolimits(M(A))$$, which we also denote by $$\phi$$. By the first part of the proof, $$\phi$$ can be represented as (4.3). ■ Remark 4.3. In fact, in the case of general separable $$C^*$$-algebras $$A$$, Magajna [25] establishes the equivalence of the following properties: (a) $$\mathop{\mathrm{IB}}\nolimits(A) = \mathcal{E}\ell(A)$$. (b) $$\mathcal{E}\ell(A)$$ is norm dense in $$\mathop{\mathrm{IB}}\nolimits(A)$$. (c) $$A$$ is a finite direct sum of homogeneous $$C^*$$-algebras with the finite type property. Analyzing the arguments in [25], for the implication (c) $$\Rightarrow$$ (a) it is sufficient to assume that $$X$$ is paracompact. □ Since any $$n$$-homogeneous $$C^*$$-algebra is locally of the form $$C(K,\mathbb{M}_n)$$ for some compact subset $$K$$ of $$X$$ with $$K^\circ \neq \emptyset$$, we have the following consequence of Proposition 4.2: Corollary 4.4. If $$A$$ is a homogeneous $$C^*$$-algebra, then for any $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ the function $$t \mapsto \|\phi_t\|$$ is continuous on $$X$$. Hence the cozero set$$\mathrm{coz}(\phi) = \{ t \in X : \phi_t \neq 0\}$$ is open in $$X$$. □ 5 Fibrewise length restrictions Here we consider a homogeneous $$C^*$$-algebra $$A = \Gamma_0(\mathcal E)$$ and operators $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$ such that $$\phi_t$$ is a two-sided multiplication on each fibre $$A_t$$ (with $$t \in X$$, and $$X = \mathrm{Prim} (A)$$ as usual). We will write $$\phi \in \mathop{\mathrm{IB}}\nolimits_1(A)$$ for this hypothesis. For separable $$A$$, the main result in this section (Theorem 5.15) characterizes when all such operators $$\phi$$ are two-sided multiplications, in terms of triviality of complex line subbundles of $$\mathcal E|_U$$ for $$U \subset X$$ open. In addition to $$\mathop{\mathrm{IB}}\nolimits_1(A)$$, we introduce various subsets $$\mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}}(A)$$ and $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ (Notation 5.5) which are designed to facilitate the description of $$\mathop{\mathrm{TM}}\nolimits(A)$$, $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ and both of their norm closures in terms of complex line bundles. The sufficient condition that ensures $$\mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}}(A) \subset \mathop{\mathrm{TM}}\nolimits(A)$$ is that $$X$$ is paracompact with vanishing second integral Čech cohomology group $$\check{H}^2(X;\mathbb{Z})$$ (Corollary 5.11). For $$X$$ compact of finite covering dimension $$d$$ and $$A = C(X, \mathbb{M}_n ) $$ we show that $$\mathop{\mathrm{TM}}\nolimits(A) \subsetneq \mathop{\mathrm{IB}}\nolimits_1(A)$$ provided $$\check{H}^2(X; \mathbb{Z}) \neq 0$$ and $$n^2 \geq (d+1)/2$$ (Proposition 5.12). We get the same conclusion $$\mathop{\mathrm{TM}}\nolimits(A) \subsetneq \mathop{\mathrm{IB}}\nolimits_1(A)$$ for $$\sigma$$-unital $$n$$-homogeneous $$C^*$$-algebras $$A = \Gamma_0(\mathcal{E})$$ with $$n \geq 2$$ provided $$X$$ has a nonempty open subset homeomorphic to (an open set in) $$\mathbb{R}^d$$ with $$d \geq 3$$ (Corollary 5.16). Notation 5.1. Let $$A$$ be an $$n$$-homogeneous $$C^*$$-algebra. For $$\ell \geq 1$$ we write IBℓ(A)={ϕ∈IB(A):ϕt∈Eℓℓ(At) for all t∈X}. □ Lemma 5.2. Let $$A=\Gamma_0(\mathcal{E} )$$ be a homogeneous $$C^*$$-algebra and $$\phi \in \mathop{\mathrm{IB}}\nolimits_\ell(A)$$. If $$t_0 \in X$$ is such that $$\phi_{t_0} \in \mathcal{E}\ell_\ell(A_{t_0}) \setminus \mathcal{E}\ell_{\ell-1}(A_{t_0})$$ (that is, such that $$\phi_{t_0}$$ has length exactly the maximal $$\ell$$), then there are $$a_1, \ldots, a_\ell, b_1, \ldots , b_\ell \in A$$ and a compact neighbourhood $$N$$ of $$t_0$$ such that $$\phi$$ agrees with the elementary operator $$\sum_{i=1}^\ell M_{a_i,b_i}$$ modulo the ideal $$I_N$$, that is ϕ(x)−∑i=1ℓaixbi∈INfor all x∈A. Moreover, we can choose $$N$$ so that $$\phi_t \in \mathcal{E}\ell_\ell (A_{t}) \setminus \mathcal{E}\ell_{\ell-1} (A_t)$$ for all $$t \in N$$ (that is, $$\phi_t$$ is of the maximal length $$\ell$$ for $$t$$ in a neighbourhood of $$t_0$$). □ Proof. Choose a compact neighbourhood $$K$$ of $$t_0$$ such that $$A_K\cong C(K,\mathbb{M}_n )$$ and let $$\phi_K$$ be the induced operator (Remark 4.1 (b), (c)). Then, for $$x \in A_K$$ we have $$\phi_K(x) = \sum_{i=1}^{n^2} c_i x d_i$$ for some $$c_i, d_i \in A_K$$ (by Proposition 4.2 (b)). Moreover, we can assume that $$\{ c_1(t), \ldots, c_{n^2}(t)\}$$ are linearly independent for each $$t \in K$$, and even independent of $$t$$. Since $$(\phi_K)_{t_0} = \phi_{t_0}$$ has length $$\ell$$, we must be able to write (in $$\mathbb{M}_n \otimes \mathbb{M}_n$$) ∑i=1n2ci(t0)⊗di(t0)=∑j=1ℓcj′⊗dj′. We can choose $$d'_1, \ldots, d'_\ell$$ to be a maximal linearly independent subsequence of $$d_1(t_0), \ldots, d_{n^2}(t_0)$$. Then, via elementary linear algebra, there is a matrix $$\alpha$$ of size $$n^2 \times \ell$$ and another matrix $$\beta$$ of size $$\ell \times n^2$$ so that [d1(t0)⋮dn2(t0)]=α[d1′⋮dℓ′],[d1′⋮dℓ′]=β[d1(t0)⋮dn2(t0)] and $$\beta\alpha$$ the identity. We have [c1′⋯cℓ′]=[c1(t0)⋯cn2(t0)]α. If we define [d1′(t)⋮dℓ′(t)][d1(t)⋮dn2(t)] then $$d'_1(t), \ldots, d'_\ell(t)$$ must be linearly independent for all $$t$$ in some compact neighbourhood $$N$$ of $$t_0$$. Thus for $$t \in N$$ we have (in $$\mathbb{M}_n \otimes \mathbb{M}_n$$) ∑i=1n2ci(t)⊗di(t)=∑i=1n2ci(t0)⊗di(t)=∑j=1ℓcj′⊗dj′(t). By Remark 4.1 (b) we can find elements $$a_j,b_j \in A$$ ($$1 \leq j \leq \ell$$) such that $$a_j(t)=c_j'$$ and $$b_j(t)=d_j'(t)$$ for all $$t \in N$$. Since for each $$t \in N$$ both of the sets $$\{a_1(t), \ldots, a_\ell(t)\}$$ and $$\{b_1(t), \ldots, b_\ell(t)\}$$ are linearly independent, we get that $$\phi_t=\sum_{j=1}^\ell M_{a_j(t),b_j(t)}$$ has length exactly $$\ell$$ for all $$t \in N$$ as required. ■ Corollary 5.3. Let $$A$$ be a homogeneous $$C^*$$-algebra and $$\phi \in \mathrm{IB}_1(A)$$. If $$t_0 \in X$$ is such that $$\phi_{t_0} \neq 0$$ then there is a compact neighbourhood $$N$$ of $$t_0$$ and $$a,b \in A$$ such that $$a(t) \neq 0$$ and $$b(t)\neq 0$$ for all $$t \in N$$ and $$\phi$$ agrees with $$M_{a,b}$$ modulo the ideal $$I_N$$. □ Remark 5.4. Let $$A=\Gamma_0(\mathcal{E})$$ be a homogeneous $$C^*$$-algebra, $$a, b \in M(A)=\Gamma_b(\mathcal{E})$$ and $$\phi =M_{a,b}$$. We may replace $$a$$ and $$b$$ by t↦‖b(t)‖‖a(t)‖a(t)andt↦‖a(t)‖‖b(t)‖b(t) without changing $$\phi$$ so as to ensure that $$\|a(t)\| = \|b(t)\|$$ for each $$t \in X$$ and that $$\|\phi_t\| = \|a(t)\|^2 = \|b(t)\|^2$$ for $$t \in X$$. □ Notation 5.5. Let $$A$$ be a homogeneous $$C^*$$-algebra. We write IB1nv(A)={ϕ∈IB(A) : 0≠ϕt∈TM(At) for all t∈X} (where $$\mathrm{nv}$$ signifies nowhere-vanishing). We also use IB0(A)={ϕ∈IB(A) :(t↦‖ϕt‖)∈C0(X)},IB0,1(A)=IB0(A)∩IB1(A),IB0,1nv(A)=IB1nv(A)∩IB0(A), and TMnv(A)=TM(A)∩IB1nv(A). By Remark 5.4, $$\mathop{\mathrm{TM}}\nolimits_0(A) = \mathop{\mathrm{TM}}\nolimits(A) \cap \mathop{\mathrm{IB}}\nolimits_0(A)$$. □ Proposition 5.6. Let $$A=\Gamma_0(\mathcal{E} )$$ be a homogeneous $$C^*$$-algebra and suppose $$\phi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A)$$. Then there is a canonically associated complex line subbundle $$\mathcal{L} _\phi$$ of $$\mathcal{E} $$ with the property that ϕ∈TM(A)⟺Lϕ is a trivial bundle. □ Proof. By Corollary 5.3, locally $$\phi$$ is a two-sided multiplication. That is, given $$t_0 \in X$$ there is a compact neighbourhood $$N$$ of $$t_0$$ and $$a , b \in A$$ such that $$\phi_t = M_{a(t), b(t)}$$ for all $$t \in N$$. We define Lϕ∩(E|N)to be{(t,λa(t)):t∈N,λ∈C}. Then $$\mathcal{L} _\phi $$ is well-defined since if $$N'$$ is another neighbourhood of a possibly different $$t_0' \in X$$ and $$a', b' \in A$$ have $$\phi_t = M_{a'(t), b'(t)}$$ for all $$t \in N'$$, then there is $$\mu(t) \in \mathbb{C} \setminus \{0\}$$ such that $$a'(t) = \mu(t) a(t)$$ for $$t \in N \cap N'$$. The definition we gave of $$\mathcal{L} _\phi \cap (\mathcal{E} |_N)$$ shows that $$\mathcal{L} _\phi $$ is a locally trivial complex line subbundle of $$\mathcal{E} $$. The map :N×C→Lϕ∩(E|N)given by(t,λ)↦(t,λa(t)). provides a local trivialization. If $$\phi \in \mathop{\mathrm{TM}}\nolimits(A)$$, then clearly $$\mathcal{L} _\phi$$ is a trivial bundle. Conversely, If $$\mathcal{L} _\phi$$ is a trivial bundle, choose a continuous nowhere vanishing section $$s : X \to \mathcal{L} _\phi$$. Then for any neighbourhood $$N$$ as above there is a continuous map $$\zeta \colon N \to \mathbb{C} \setminus \{0\}$$ such that $$a(t) = \zeta(t) s(t)$$. If we define $$s' \colon X \to \mathcal{E} $$ by $$s'(t) = (1/\zeta(t)) b(t)$$ for $$t \in N$$, then we have $$s, s' \in \Gamma(\mathcal{E} )$$ well-defined and $$\phi_t(x(t)) = s(t)x(t)s'(t)$$ for all $$x \in A$$. Normalizing $$s$$ and $$s'$$ as in Remark 5.4, we get $$c,d \in \Gamma_b(\mathcal{E} ) = M(A)$$ (Remark 4.1 (f)) with $$\phi = M_{c, d}$$. ■ Notation 5.7. If $$A = \Gamma_0(\mathcal{E})$$ is homogeneous and $$\phi \in \mathop{\mathrm{IB}}\nolimits_1(A)$$, we consider the cozero set $$U = \mathrm{coz}(\phi)$$ (open by Corollary 4.4) and then, for $$B = \Gamma_0(\mathcal{E} |_U)$$, $$\phi|_U \in \mathop{\mathrm{IB}}\nolimits_{1}^{\mathrm{nv}} (B)$$ (see Remark 4.1 (c)). We occasionally use $$\mathcal{L}_\phi$$ for the subbundle $$\mathcal{L}_{\phi|_U}$$ of $$\mathcal{E} |_U$$. □ Proposition 5.8. Let $$A=\Gamma_0(\mathcal{E} )$$ be an $$n$$-homogeneous $$C^*$$-algebra such that $$X$$ is $$\sigma$$-compact. If $$\mathcal{L}$$ is a complex line subbundle of $$\mathcal{E} $$, then there is $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}} (A)$$ with $$\mathcal{L} _\phi = \mathcal{L} $$. □ Proof. Let $$\langle \cdot, \cdot \rangle_2$$ be as in Remark 4.1 (g). With respect to this inner product we have a complementary subbundle $$\mathcal{L} ^\perp$$ of $$\mathcal{E} $$ such that $$\mathcal{L} \oplus \mathcal{L} ^\perp = \mathcal{E} $$. By local compactness, $$X$$ has a base consisting of $$\sigma$$-compact open sets. (If $$t_0 \in U \subset X$$ with $$ U$$ open, choose a compact neighborhood $$N$$ of $$t_0$$ contained in $$U$$ and a function $$f \in C_0(X)$$ supported in $$N$$ with $$f(t_0) = 1$$. Take $$V = \{ t \in X : |f(t)| > 0\}$$.) Since $$X$$ is $$\sigma$$-compact (and since every $$\sigma$$-compact space is Lindelöf), we can find a countable open cover $$\{U_i\}_{i=1}^\infty$$ of $$X$$ such that each restriction $$\mathcal{E} |_{U_i}$$ is trivial and each $$U_i$$ is $$\sigma$$-compact. Then we can find $$n^2$$ norm-one sections $$(e_j^i)_{j=1}^{n^2}$$ of $$\Gamma_0(\mathcal{E} |_{U_i})\cong C_0(U_i, \mathbb{M}_n)$$ such that span{e1i(t),⋯,en2i(t)}=Et≅Mnfor all t∈Ui. By extending outside $$U_i$$ with $$0$$ we may assume that $$e_j^i$$ are globally defined, so that $$e^i_j \in A$$. Define $$f_j^i(t)$$ as the orthogonal projection of $$e_j^i$$ into the fibre $$\mathcal{L}_t$$, so that $$f_j^i \in A$$. We define ϕ:A→Abyϕ=∑i=1∞12i(∑j=1n2Mfji,(fji)∗). Note that $$\phi \in \mathop{\mathrm{IB}}\nolimits_0(A)$$ as a sum of an absolutely convergent series of operators in $$\mathop{\mathrm{IB}}\nolimits_0(A)$$ (and $$\mathop{\mathrm{IB}}\nolimits_0(A)$$ is norm closed). We claim that $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(A)$$ and $$\mathcal{L} _\phi = \mathcal{L} $$. Indeed, for an arbitrary point $$t \in X$$ choose a norm-one (in $$C^*$$-norm) vector $$s\in \mathcal{L} _t$$. Then there are scalars $$\lambda_j^i$$ with $$f_j^i(t) = \lambda_j^i \cdot s$$ and $$|\lambda_j^i| = \|f_j^i(t)\|\leq \sqrt n\|e_j^i(t)\|=\sqrt{n}$$ (by (4.2)). Then ϕt=(∑i=1∞12i(∑j=1n2|λji|2))⋅Ms,s∗. This shows that $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(A)$$ and that for all $$t \in X$$ we have $$\phi_t=M_{a(t),a^*(t)}$$ for some $$a(t)\in \mathcal{L}_t$$. By the proof of Proposition 5.6 we conclude $$\mathcal{L}_\phi=\mathcal{L}$$. ■ In the sequel, by $$\lceil \cdot \rceil$$ we denote the ‘ceiling function’ (i.e., if $$x \in \mathbb{R}$$ then $$\lceil x \rceil$$ is the smallest integer greater or equal to $$x$$). Remark 5.9. Let $$\mathcal{L}$$ be a complex line bundle over a locally compact Hausdorff space $$X$$. (a) $$\mathcal{L}$$ is isomorphic to a subbundle of some $$\mathbb{M}_2$$-bundle $$\mathcal{E}$$. Indeed, let $$\mathcal{F}=\mathcal{L} \oplus (X \times \mathbb{C})$$. Then $$\mathcal{E}=\mathrm{Hom}(\mathcal{F},\mathcal{F})=\mathcal{F} \otimes \mathcal{F}^*$$ is an $$\mathbb{M}_2$$-bundle with the desired property (see [30, Example 3.5]). Further, if $$X$$ is $$\sigma$$-compact, then $$A=\Gamma_0(\mathcal{E})$$ (with $$\mathcal{E}$$ as above) is an example of a $$2$$-homogeneous $$C^*$$-algebra with $$\mathrm{Prim}(A)=X$$ that allows an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(A)$$ such that $$\mathcal{L}_\phi \cong \mathcal{L}$$ (by Proposition 5.8). (b) Suppose that $$\mathcal{L}$$ is a subbundle of a trivial bundle $$X \times \mathbb{C}^m$$. If $$p=\lceil\sqrt{m}\rceil$$, then for each $$n \geq p$$ we can regard $$\mathcal{L}$$ as a subbundle of a trivial matrix bundle $$X \times \mathbb{M}_n$$, using some linear embedding $$\mathbb{C}^m \hookrightarrow \mathbb{M}_n$$. □ Remark 5.10. If the space $$X$$ is paracompact, it is well-known that locally trivial complex line bundles over $$X$$ are classified by the homotopy classes of maps from $$X$$ to $$\mathbb{C} P^\infty$$ and/or by the elements of the second integral \v Cech cohomology $$\check{H}^2(X;\mathbb{Z})$$ (see e.g., [18, Corollary 3.5.6 and Theorem 3.4.7] and [33, Proposition 4.53 and Theorem 4.42].) By [18], we know that complex line bundles over $$X$$ are pullbacks of the canonical bundle over $$\mathbb{C} P^\infty$$ (via a map from $$X$$ to $$\mathbb{C} P^\infty$$). □ In light of Proposition 5.6 and Remark 5.10, for a given a homogeneous $$C^*$$-algebra $$A=\Gamma_0(\mathcal{E} )$$ we define a map θ:IB1nv(A)→Hˇ2(X;Z) (5.1) which sends an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}}(A)$$ to the corresponding class of $$\mathcal{L}_\phi$$ in $$\check{H}^2(X;\mathbb{Z})$$. By Proposition 5.6 we have $$\theta^{-1}(0)=\mathop{\mathrm{TM}}\nolimits^{\mathrm{nv}}(A)$$. As a direct consequence of this observation we have: Corollary 5.11. Let $$A$$ be a homogeneous $$C^*$$-algebra such that $$X$$ is paracompact. If $$\check{H}^2(X;\mathbb{Z})=0$$ then $$\mathrm{IB}_1^{\mathrm{nv}}(A) = \mathop{\mathrm{TM}}\nolimits^{\mathrm{nv}}(A)$$. □ We will now give some sufficient conditions on a trivial homogeneous $$C^*$$-algebra $$A$$ that will ensure the surjectivity of the map $$\theta$$. To do this, first recall that a topological space $$X$$ is said to have the Lebesgue covering dimension$$d<\infty$$ if $$d$$ is the smallest non-negative integer with the property that each finite open cover of $$X$$ has a refinement in which no point of $$X$$ is included in more than $$d+1$$ elements (see e.g., [9]). In this case we write $$d=\dim X$$. Proposition 5.12. Let $$X$$ be a compact Hausdorff space with $$\dim X \leq d<\infty$$. For $$n \geq 1$$ let $$A_n = C(X,\mathbb{M}_n )$$. If $$p= \left\lceil \sqrt{(d+1)/2} \right\rceil$$ then for any $$n \geq p$$ the mapping $$\theta$$ from (5.1) is surjective. In particular, if $$\check{H}^2(X;\mathbb{Z})\neq 0$$, then $$\mathop{\mathrm{TM}}\nolimits^{\mathrm{nv}}(A_n) \varsubsetneq \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}}(A_n) $$ for all $$n \geq p$$. □ To prove this will use the following fact (which may be known): Lemma 5.13. Let $$X$$ be a CW-complex with $$\dim X =d$$. Then each complex line bundle $$\mathcal{L}$$ over $$X$$ is isomorphic to a line subbundle of $$X \times \mathbb{C}^m$$ with $$m = \lceil (d+1)/2 \rceil$$. □ Proof. We consider $$\mathbb{C} P^\infty$$ as a CW-complex in the usual way (see [16, Example 0.6]). Let $$\Psi: X \to \mathbb{C} P^\infty$$ be the classifying map of the bundle $$\mathcal{L} $$ (Remark 5.10). Using the cellular approximation theorem [16, Theorem 4.8] and Remark 5.10 we may assume that the map $$\Psi$$ is cellular, so that $$\Psi$$ takes the $$k$$-skeleton of $$X$$ to the $$k$$-skeleton of $$\mathbb{C} P^\infty$$ for all $$k$$. Since $$\mathbb{C} P^\infty$$ has one cell in each even dimension, $$\Psi(X)$$ is contained in the $$d$$-skeleton of $$\mathbb{C} P^\infty$$, which is the $$(d-1)$$-skeleton if $$d$$ is odd, and is $$\mathbb{C} P^{m-1}$$. Hence $$\mathcal{L} $$ is isomorphic to the pullback $$\Psi^{*}(\gamma)$$ of the canonical line bundle $$\gamma$$ on $$\mathbb{C} P^{m-1}$$ (Remark 5.10), a subbundle of the trivial bundle $$\mathbb{C} P^{m-1} \times \mathbb{C}^m$$. ■ Proof of Proposition 5.12. Let $$\mathcal{L}$$ be any complex line bundle over $$X$$. By the proof of [30, Lemma 2.3] there exists a finite complex $$Y$$ with $$\dim Y \leq d$$, a continuous function $$f: X \to Y$$, and a line bundle $$\mathcal{L}'$$ over $$Y$$ such that $$\mathcal{L} \cong f^*(\mathcal{L}')$$. By Lemma 5.13 we conclude that $$\mathcal{L}'$$ is isomorphic to a line subbundle of $$Y \times \mathbb{C}^m$$, with $$m = \lceil (d+1)/2\rceil$$. Hence, $$\mathcal{L}$$ is isomorphic to a line subbundle of $$X \times \mathbb{C}^m$$. By Remark 5.9 (b) if $$n \geq p= \lceil\sqrt{m}\rceil= \left\lceil \sqrt{ (d+1)/2} \right\rceil$$ ($$\lceil\sqrt{\lceil x\rceil}\rceil=\lceil\sqrt{x}\rceil$$ for all $$x \geq 0$$), we can assume that $$\mathcal{L}$$ is already a subbundle of $$X \times \mathbb{M}_n$$. By the proof of Proposition 5.8 we can find an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}}(A_n)$$ such that $$\mathcal{L}_\phi=\mathcal{L}$$. By Remark 5.10 we conclude that the map $$\theta$$ is surjective. That $$\mathop{\mathrm{TM}}\nolimits^{\mathrm{nv}}(A) \subsetneq \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}}(A_n)$$ ($$n \geq p$$) when $$H^2(X;\mathbb{Z})\neq 0$$ follows directly from previous observations and Proposition 5.6. ■ Example 5.14. If $$X$$ is either the $$2$$-sphere, the $$2$$-torus or the Klein bottle, then it is well-known that $$\check{H}^2(X; \mathbb{Z}) \neq 0$$. In particular, if $$A=C(X, \mathbb{M}_n )$$ ($$n\geq 2$$) then Proposition 5.12 shows that $$\mathop{\mathrm{TM}}\nolimits^{\mathrm{nv}}(A) \subsetneq \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A)$$. □ Theorem 5.15. Let $$A=\Gamma_0(\mathcal{E} )$$ be a homogeneous $$C^*$$-algebra. Consider the following conditions: (a) For every open subset $$U \subset X$$, each complex line subbundle of $$\mathcal{E}|_U$$ is trivial. (b) $$\mathrm{IB}_1(A)=\mathop{\mathrm{TM}}\nolimits(A)$$. (c) $$\mathrm{IB}_{0,1}(A)=\mathop{\mathrm{TM}}\nolimits_0(A)$$. Then (a) $$\Rightarrow$$ (b) $$\Rightarrow$$ (c). If $$A$$ is separable, conditions (a), (b), and (c) are equivalent. □ Proof. (a) $$\Rightarrow$$ (b): Assume (a) holds and $$\phi \in \mathop{\mathrm{IB}}\nolimits_1(A)$$. Let $$U = \mathrm{coz}(\phi)$$ (open by Corollary 4.4). By Proposition 5.6 we may assume that $$U\neq X$$. Let $$B=\Gamma_0(\mathcal{E}|_U)$$ and let $$\phi|_U$$ be the restriction of $$\phi$$ to $$B$$. Then $$\phi|_U\in \mathop{\mathrm{IB}}\nolimits_{1}^{\mathrm{nv}} (B)$$. By (a), $$\mathcal{L} _{\phi}$$ is trivial (on $$U$$) and by Proposition 5.6 we have $$\phi|_U \in \mathop{\mathrm{TM}}\nolimits(B)$$, that is $$\phi|_U = M_{c, d}$$ for some $$c, d \in M(B) = \Gamma_b(\mathcal{E}|_U)$$. By Remark 5.4, we can suppose that $$\|c(t)\|^2 = \|d(t)\|^2 = \|\phi_t\|$$ for $$t \in U$$, so that $$c,d \in B$$. We can then define $$a, b \in A$$ by $$a(t) = b(t) = 0$$ for $$t \in X \setminus U$$ and, for $$t \in U$$, $$a(t) = c(t)$$, $$b(t) = d(t)$$. Then we have $$\phi = M_{a,b} \in \mathop{\mathrm{TM}}\nolimits_0(A)\subseteq \mathop{\mathrm{TM}}\nolimits(A)$$. (b) $$\Rightarrow$$ (c): Take intersections with $$\mathop{\mathrm{IB}}\nolimits_0(A)$$. Now assume that $$A$$ is separable, so that $$X$$ is second-countable. (c) $$\Rightarrow$$ (b): If $$\phi \in \mathop{\mathrm{IB}}\nolimits_1(A)$$, take a strictly positive function $$f \in C_0(X)$$ and define $$\psi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ by $$\psi_t = f(t)^2 \phi_t$$. By (c) and Remark 5.4 we have $$\psi = M_{c,d}$$ for $$c, d \in A$$ with $$\|c(t)\|^2 = \|d(t)\|^2 = \|\psi_t\|$$. We can define $$a, b \in M(A) = \Gamma_b(\mathcal{E})$$ by $$a(t) = c(t)/f(t)$$ and $$b(t) = d(t)/f(t)$$ to get $$\phi = M_{a,b}\in \mathop{\mathrm{TM}}\nolimits(A)$$. (b) $$\Rightarrow$$ (a): Assume (b) holds. Let $$U$$ be an open subset of $$X$$ and $$\mathcal{L} $$ a complex line subbundle of $$\mathcal{E}|_U$$. By Proposition 5.8 applied to $$B = \Gamma_0(\mathcal{E}|_U)$$ ($$U$$ is $$\sigma$$-compact since $$X$$ is second-countable), there is $$\psi \in \mathop{\mathrm{IB}}\nolimits_0(B)$$ with $$\mathcal{L} _\psi = \mathcal{L} $$. Since $$(t \mapsto \|\psi_t\|) \in C_0(U)$$, we can define $$\phi \in \mathop{\mathrm{IB}}\nolimits_0(A)$$ by $$\phi_t = \psi_t$$ for $$t \in U$$ and $$\phi_t = 0 $$ for $$t \in X \setminus U$$. By (b), $$\phi = M_{a,b}$$ for $$a, b \in M(A) = \Gamma_b(\mathcal{E})$$ and then $$a|_U$$ defines a nowhere vanishing section of $$\mathcal{L} $$. ■ Corollary 5.16. Let $$A =\Gamma_0(\mathcal{E})$$ be an $$n$$-homogeneous $$C^*$$-algebra with $$n \geq 2$$. (a) If $$X$$ is second-countable with $$\dim X<2$$, or if $$X$$ is (homeomorphic to) a subset of a non-compact connected $$2$$-manifold, then IB0,1(A)=TM0(A) and IB1(A)=TM(A). (b) If $$X$$ is $$\sigma$$-compact and contains a nonempty open subset homeomorphic to (an open subset of) $$\mathbb{R}^d$$ for some $$d \geq 3$$, then IB0,1(A)∖TM0(A)≠∅ and IB1(A)∖TM(A)≠∅. □ Remark 5.17. By a $$d$$-manifold we always mean a second-countable topological manifold of dimension $$d$$. □ To prove this we will use the following facts (which are well-known to topologists). Remark 5.18. If $$X$$ is a metrizable space with $$\dim X =d <\infty$$, then any locally trivial fibre bundle over $$X$$ can be trivialized over some open cover of $$X$$ consisting of at most $$d+1$$ elements. This follows from Dowker’s and Ostrand’s theorems [9, Theorems 3.2.1 and 3.2.4]. □ Lemma 5.19. Let $$Y$$ be a metrizable space with $$\dim Y =d <\infty$$ and let $$X$$ be a closed subset of $$Y$$. Then any map $$f: X \to \mathbb{C} P^\infty$$ can be, up to homotopy, continuously extended to some open neighbourhood of $$X$$ in $$Y$$. □ Proof. Let $$\mathcal{L}$$ be a complex line bundle over $$X$$ defined by $$f$$ (Remark 5.10). By [9, Theorem 3.1.4], we have $$\dim X \leq \dim Y=d$$. By Remark 5.18 $$\mathcal{L}$$ can be trivialized over some open cover of $$X$$ consisting of (at most) $$d+1$$ elements. In particular, $$\mathcal{L}$$ is determined by some map $$g: X \to \mathbb{C} P^d$$ (see e.g., [18, Section 3.5]) and by Remark 5.10 $$g$$ is homotopic to $$f$$. By [17, Theorem V.7.1], finite dimensional manifolds (in particular $$\mathbb{C} P^d$$) are ANR spaces and so by [17, Theorem III.3.2], $$g$$ extends (continuously) to some open neighbourhood of $$X$$ in $$Y$$. ■ Proposition 5.20. Suppose that $$X$$ is a locally compact subset of a non-compact connected $$2$$-manifold $$M$$. Then $$\check{H}^2(X;\mathbb{Z})=0$$. □ Proof. First assume that $$X=M$$. Then by [28, Theorem 2.2], since every $$2$$-manifold admits a smooth structure (a classical result for which we have failed to find a complete modern reference), $$X$$ is homotopy equivalent to a CW-complex of dimension $$d < 2$$. Using Lemma 5.13 (and Remark 5.10) we conclude that $$\check{H}^2(X;\mathbb{Z})=0$$. Now let $$X$$ be an open subset of $$M$$. Since the previous argument applies to each connected component of $$X$$, we again have $$\check{H}^2(X;\mathbb{Z})=0$$. If $$X$$ is a locally compact subset of $$M$$, then $$X$$ is open in its closure $$\overline{X}$$. Let $$Y=\overline{X} \setminus X$$. Then $$N=M \setminus Y$$ is open in $$M$$ and $$X$$ is closed in $$N$$. Suppose that $$\check{H}^2(X;\mathbb{Z})\neq 0$$ and let $$f : X \to \mathbb{C} P^\infty$$ be any non-null-homotopic map (Remark 5.10). By Lemma 5.19 $$f$$ extends, up to homotopy, to a map defined on some open neighbourhood $$U$$ of $$X$$ in $$M$$. In particular, $$\check{H}^2(U; \mathbb{Z})\neq 0$$ which contradicts the second part of the proof. ■ Proof of Corollary 5.16. For (a) it suffices to show that $$\check{H}^2(U;\mathbb{Z})=0$$ for all open subsets $$U$$ of $$X$$ (by Theorem 5.15 and Remark 5.10). By Proposition 5.20 this is true if $$X$$ is a subset of a non-compact connected $$2$$-manifold. Suppose that $$X$$ is second-countable with $$\dim X<2$$. Then for each open subset $$U$$ of $$X$$ we have $$\dim U\leq \dim X$$ (by the “subset theorem’ [9, Theorem 3.1.19]), so $$\check{H}^2(U;\mathbb{Z})=0$$ (See e.g., [9, p. 94–95]). For (b) we first choose an open subset $$U\subset X$$ for which $$\mathcal{E}|_U \cong U \times \mathbb{M}_n$$ and such that $$U$$ can be considered as an open set in $$\mathbb{R}^d$$$$(d \geq 3)$$. We use the simple fact that $$U$$ contains an open subset that has the homotopy type of the $$2$$-sphere $$\mathbb{S}^2$$. So, replacing $$U$$ by such a subset, we can find a non-trivial line subbundle $$\mathcal{L}$$ of $$U \times \mathbb{C}^2$$. By Remark 5.9 (b) we may assume that $$\mathcal{L}$$ is a subbundle of $$U \times \mathbb{M}_n \cong \mathcal{E}|_U$$. The assertion now follows from the proof of Theorem 5.15. ■ Remark 5.21. In the literature there are somewhat similar phenomena that arise for unital $$C^*$$-algebras $$A$$ of sections of a $$C^*$$-bundle over a (second-countable) compact Hausdorff space $$X$$. The question was to describe when the set $$\mathrm{Aut}_{C(X)} (A)$$ of all $$C(X)$$-linear automorphisms of such $$A$$ coincides with the inner automorphisms of $$A$$ (See e.g., [22, 31, 32, 34]). For example, if $$A$$ is any separable unital continuous trace $$C^*$$-algebra with (primitive) spectrum $$X$$, there always exists an exact sequence 0⟶InnAut(A)⟶AutC(X)(A)⟶ηHˇ2(X;Z) of abelian groups. In general, $$\eta$$ does not need to be surjective unless $$A$$ is stable [31, Theorem 2.1]. If $$A$$ is $$n$$-homogeneous then the image of $$\eta$$ is contained in the torsion subgroup of $$\check{H}^2(X;\mathbb{Z})$$ [31, 2.19]. In particular, $$A=C(\mathbb{S}^2,\mathbb{M}_2 )$$ shows that it can happen that $$\mathop{\mathrm{TM}}\nolimits^{\mathrm{nv}}(A)\subsetneq \mathop{\mathrm{IB}}\nolimits^{\mathrm{nv}}_1(A)$$ even though $$\mathrm{Aut}_{C(\mathbb{S}^2)}(A)=\mathrm{InnAut}(A)$$ (since $$\check{H}^2(\mathbb{S}^2;\mathbb{Z}) \cong \mathbb{Z}$$ is torsion free). Our Proposition 5.12 shows that the map $$\theta$$ from (5.1) is surjective in this case. In contrast to $$\eta$$, there is no obvious group structure on the domain $$\mathop{\mathrm{IB}}\nolimits^{\mathrm{nv}}_1(A)$$ of $$\theta$$. □ 6 Closure of $$\mathop{\mathrm{TM}}\nolimits(A)$$ on homogeneous $$C^*$$-algebras Here we continue to work with $$n$$-homogeneous algebras $$A = \Gamma_0(\mathcal{E})$$. The class $$\mathop{\mathrm{IB}}\nolimits_1(A)$$ considered in Section 5 is rather obviously designed to capture a restriction on the closure of $$\mathop{\mathrm{TM}}\nolimits(A)$$ (and similarly $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ should relate to the closure of $$\mathop{\mathrm{TM}}\nolimits_0(A)$$). We verify right away (Proposition 6.1) that $$\mathop{\mathrm{IB}}\nolimits_1(A)$$ and $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ are indeed closed. However, further restrictions on the operators $$\phi$$ in the closure of $$\mathop{\mathrm{TM}}\nolimits(A)$$ arise because triviality of the line bundles $$\mathcal{L} _\psi$$ associated with $$\psi \in \mathop{\mathrm{TM}}\nolimits(A)$$ is still present for the line bundle $$\mathcal{L} _\phi$$ provided $$U = \mathrm{coz}(\phi)$$ is compact (see Corollary 6.5). If $$U$$ is not compact, this triviality is evident on compact subsets of $$U$$ (see Theorem 6.9, where we characterize the closure of $$\mathop{\mathrm{TM}}\nolimits_0(A)$$). However $$\mathcal{L} _\phi$$ need not be trivial globally on $$U$$ (so that $$\phi \notin \mathop{\mathrm{TM}}\nolimits(A)$$ is possible) and this led us to define the concept of a phantom bundle (Definition 6.11). The terminology is by analogy with the well known concept of a phantom map (see [27]). Thus, in Corollary 6.12, we see that finding $$\phi$$ in the norm closure of $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ with $$\phi \notin \mathop{\mathrm{TM}}\nolimits_0(A)$$ is directly related to finding suitable phantom complex line bundles. For these to exist, we need $$U$$ to have a rather complicated algebraic topological structure, and we find examples with $$\pi_1(U) \cong \mathbb{Q}$$ (Proposition 6.17). In fact, we can also find such examples when $$X$$ contains (a copy of) an open subset of $$\mathbb{R}^d$$ with $$d \geq 3$$ and $$n \geq 2$$ (Theorem 6.18). Proposition 6.1. Let $$A$$ be a homogeneous $$C^*$$-algebra. Then $$\mathop{\mathrm{IB}}\nolimits_1(A)$$ and $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ are norm closed subsets of $$\mathcal{B}(A)$$. □ Proof. If $$(\phi_k)_{k=1}^\infty$$ is a sequence in $$\mathop{\mathrm{IB}}\nolimits_1(A)$$ that converges in operator norm to $$\phi \in \mathcal{B}(A)$$, then it is clear that $$\phi(I) \subset I$$ for each ideal $$I$$ of $$A$$. Thus $$\phi \in \mathop{\mathrm{IB}}\nolimits(A)$$. By (2.2) we have $$\| \phi - \phi_k\| = \sup_{t \in X} \|\phi_t - (\phi_k)_t\|$$ and so $$\lim_{k \to \infty} (\phi_k)_t = \phi_t \in \mathcal{B}(A_t)$$ (for $$t \in X$$). Since $$A_t \cong \mathbb{M}_n$$, invoking Theorem 3.4, we have $$\phi_t \in \mathop{\mathrm{TM}}\nolimits(A_t)$$ (for $$t \in X$$) and hence $$\phi \in \mathop{\mathrm{IB}}\nolimits_1(A)$$. If $$\phi_k \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ for each $$k$$, then $$\|(\phi_k)_t\| \to \|\phi_t\|$$ uniformly for $$t \in X$$. As $$(t \mapsto \|(\phi_k)_t\|) \in C_0(X)$$, it follows that $$(t \mapsto \|\phi_t\|) \in C_0(X)$$ and so $$\phi \in \mathop{\mathrm{IB}}\nolimits_0(A)$$. ■ Lemma 6.2. Let $$A$$ be a homogeneous $$C^*$$-algebra, and let $$\phi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A)$$. Then there is $$\psi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A)$$ with $$\psi_t = \phi_t/\|\phi_t\|$$ for each $$t \in X$$. Moreover $$\phi \in \mathop{\mathrm{TM}}\nolimits(A) \iff \psi \in \mathop{\mathrm{TM}}\nolimits(A)$$. □ Proof. Since $$t \mapsto \| \phi_t\|$$ is continuous by Corollary 4.4, we can define $$\psi_t = \phi_t/\|\phi_t\|$$ and get $$\psi \in \mathop{\mathrm{IB}}\nolimits(A)$$ via local applications of Proposition 4.2. Clearly $$\psi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A)$$. If $$\phi = M_{a,b} \in \mathop{\mathrm{TM}}\nolimits(A)$$ for $$a, b \in M(A)=\Gamma_b(\mathcal{E})$$, then we can normalize $$a$$ and $$b$$ as in Remark 5.4 and then take $$c, d \in A$$ with $$c(t) = a(t)/ \sqrt{ \|\phi_t\| }$$, $$d(t) = b(t)/ \sqrt{ \|\phi_t\| }$$ to get $$\psi = M_{c,d}$$. So $$\psi \in \mathop{\mathrm{TM}}\nolimits(A)$$. We can reverse this argument. ■ Remark 6.3. Let $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}}$$ denote the operator norm closure of $$\mathop{\mathrm{TM}}\nolimits(A)$$, and similarly for $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$. If $$A$$ is homogeneous, then Proposition 6.1 gives $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}} \subset \mathop{\mathrm{IB}}\nolimits_1(A)$$ and $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}} \subset \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. □ Proposition 6.4. Let $$A=\Gamma_0(\mathcal{E} )$$ be an $$n$$-homogeneous $$C^*$$-algebra. Suppose that $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}}$$ such that $$\inf_{t \in X} \|\phi_t\| = \delta > 0$$. Then $$\phi \in \mathop{\mathrm{TM}}\nolimits(A)$$. □ Proof. Let $$(\phi_k)_{k=1}^\infty$$ be a sequence in $$\mathop{\mathrm{TM}}\nolimits(A)$$ with $$\lim_{k \to \infty} \phi_k = \phi \in \mathcal{B}(A)$$. For $$k$$ large enough that $$\|\phi_k - \phi\| < \delta/2$$ we must have $$\|(\phi_k)_t\| > \delta/2$$ for each $$t \in X$$ (and hence $$\phi_k \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A)$$). With no loss of generality we may assume that this holds for all $$k \geq 1$$. Since supt∈X|‖(ϕk)t‖−‖ϕt‖|≤supt∈X‖(ϕk)t−ϕt‖=‖ϕk−ϕ‖, we may use Lemma 6.2 to normalise each $$\phi_k$$ and $$\phi$$ and assume that 1=‖ϕ‖=‖ϕt‖=‖(ϕk)t‖=‖ϕk‖ holds for all $$k \geq 1$$ and $$t \in X$$ (and still $$\lim_{k \to \infty} \phi_k = \phi$$). We now write $$\phi_k = M_{a_k, b_k}$$ for $$a_k, b_k \in M(A)=\Gamma_b(\mathcal{E})$$ such that $$\|a_k(t)\| = \|b_k(t)\| = 1$$ (for all $$t \in X$$ and all $$k$$). We consider the line bundle $$\mathcal{L} _\phi$$ associated with $$\phi$$ according to Proposition 5.6 which is locally expressible as $$\{ (t, \lambda a(t) )\}$$, where $$\phi_t = M_{a(t), b(t)}$$ locally. We assume, as we can, that $$\|a(t)\| = \|b(t)\| = 1$$ (locally). Let $$0<\varepsilon < (18n)^{-1/2}$$. By Remark 4.1 (d), for $$k$$ suitably large (but fixed) and $$t \in X$$ arbitrary, we have $$\|(\phi_k)_t - \phi_t\|_{cb} < \varepsilon$$. Since, by Mathieu’s theorem (Theorem 2.1), we locally have ‖(ϕk)t−ϕt‖cb=‖Mak(t),bk(t)−Ma(t),b(t)‖cb=‖ak(t)⊗bk(t)−a(t)⊗b(t)‖h, by Lemma 3.1, we can locally find a scalar $$\mu_k(t)$$ of modulus $$1$$ such that ‖ak(t)−μk(t)a(t)‖<4ε (note that $$(18n)^{-1/2}<1/3$$ for all $$n \geq 1$$). Consider the inner product $$\langle \cdot , \cdot \rangle_2$$ defined in Remark 4.1 (g). We claim that locally $$\langle a_k(t), a(t) \rangle_2\neq 0$$. Indeed, first (locally) |⟨ak(t),a(t)⟩2|=|⟨ak(t),μk(t)a(t)⟩2| and by (4.2) ‖ak(t)‖2≥1,‖μk(t)a(t)‖2≥1,‖ak(t)−μk(t)a(t)‖2<6nε. Since any two vectors $$v$$ and $$w$$ of norm at least $$1$$ in a Hilbert space satisfy ‖v−w‖22≥‖v‖22+‖w‖22−2|⟨v,w⟩2|≥2(1−|⟨v,w⟩2|), letting $$v=a_k(t)$$ and $$w=\mu_k(t)a(t)$$, we have (locally) |⟨ak(t),μk(t)a(t)⟩2|≥1−12‖ak(t)−μk(t)a(t)‖22>1−18nε2>0. We can therefore define $$a'(t)$$ locally as the normalised (in operator norm) orthogonal projection a′(t)=⟨ak(t),a(t)⟩2|⟨ak(t),a(t)⟩2|⋅a(t). Then $$t \mapsto a'(t)$$ is locally well-defined and continuous (by Remark 4.1 (g)). As $$a'(t)$$ is independent of multiplying $$a(t)$$ by unit scalars, it defines a nowhere vanishing global section of $$\mathcal{L} _\phi$$. By Proposition 5.6, we must have $$\phi \in \mathop{\mathrm{TM}}\nolimits(A)$$, as required. ■ Corollary 6.5. Let $$A$$ be a unital homogeneous $$C^*$$-algebra. Then TM(A)¯¯∩IB1nv(A)⊂TM(A). □ Proof. Let $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}} \cap \mathrm{IB}_1^{\mathrm{nv}}(A)$$. Since $$t \mapsto \| \phi_t\|$$ is continuous (Corollary 4.4) and never vanishing on $$X$$ (which is compact, as $$A$$ is unital), it has a minimum value $$\delta > 0$$. By Proposition 6.4, $$\phi \in \mathop{\mathrm{TM}}\nolimits(A)$$. ■ Example 6.6. Let $$A=C(X,\mathbb{M}_n )$$$$(n \geq 2)$$, where $$X$$ is any compact Hausdorff space with $$\dim X\leq 7$$ and $$\check{H}^2(X;\mathbb{Z})\neq 0$$. Then $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}} \subsetneq \mathrm{IB}_1(A)$$. Indeed, by Proposition 5.12 there exists $$\phi \in \mathop{\mathrm{IB}}\nolimits_1^{\mathrm{nv}} (A) \setminus \mathop{\mathrm{TM}}\nolimits(A)$$. By Corollary 6.5, $$\phi \not \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}}$$. (Since $$A$$ is unital, $$\mathop{\mathrm{TM}}\nolimits_0(A) = \mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathrm{IB}_{0,1}(A)=\mathrm{IB}_1(A)$$.) □ Corollary 6.7. If $$A=\Gamma_0(\mathcal{E})$$ is a homogeneous $$C^*$$-algebra, then both $$\mathrm{InnAut_{alg}}(A)$$ and $$\mathrm{InnAut}(A)$$ (see (2.3)) are norm closed. □ Proof. If $$M_{a, a^{-1}} \in \mathrm{InnAut_{alg}}(A)$$, then for all $$t \in X$$ we have $$\|( M_{a, a^{-1}})_t\| = \|a(t)\| \|a(t)^{-1}\| \,{\geq}\,1$$. Hence if $$\phi$$ is in the norm closure of $$\mathrm{InnAut_{alg}}(A)$$, we have $$\|\phi_t\| \geq 1$$ for each $$t \in X$$. By Proposition 6.4, $$\phi = M_{b,c}$$ for some $$b,c \in M(A)$$. Since $$\phi_t(1) = 1$$, $$c(t) = b(t)^{-1}$$ for each $$t$$ and so $$c = b^{-1} \in M(A) = \Gamma_b(\mathcal{E})$$. The proof for the $$\mathrm{InnAut}(A)$$ is similar. ■ Remark 6.8. The results that $$\mathrm{InnAut}(A)$$ is norm closed if the $$C^*$$-algebra $$A$$ is prime or homogeneous (in Corollaries 3.5 and 6.7) can also be deduced from [3, 19, 35]. To explain the deductions, we first identify $$\mathrm{InnAut}(A)$$ with $$\mathrm{InnAut}(M(A))$$. If $$A$$ is prime, then $$M(A)$$ is also prime (by [2, Lemma 1.1.7]). In particular, $$\mathrm{Orc}(M(A))=1$$ (in the sense of [35, Section 2]), so by [35, Corollary 4.6] inner derivations of $$M(A)$$ are norm closed. Then [19, Theorem 5.3] implies that $$\mathrm{InnAut}(M(A))$$ is also norm closed. If $$A$$ is homogeneous (or more generally quasi-central and quasi-standard in the sense of [3]), then $$M(A)$$ is quasi-standard [3, Corollary 4.10]. Thus we have $$\mathrm{Orc}(M(A))=1$$, and we may conclude as in the prime case. □ Theorem 6.9. Let $$A=\Gamma_0(\mathcal{E} )$$ be a homogeneous $$C^*$$-algebra. For an operator $$\phi \in \mathcal{B}(A)$$, the following two conditions are equivalent: (a) $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$. (b) $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ and for $$U=\mathrm{coz}(\phi)$$ (open by Corollary 4.4) $$\mathcal{L} _{\phi}$$ is trivial on each compact subset of $$U$$. □ Proof. (a) $$\Rightarrow$$ (b): Let $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$, so that $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ (Remark 6.3). For each compact subset $$K \subset U$$, we have $$\phi_K \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits(A_K)}}$$ (recall that $$A_K=\Gamma(\mathcal{E}|_K)$$ by Remark 4.1 (b)). By Corollary 6.5 we have $$\phi_K \in \mathop{\mathrm{TM}}\nolimits(A_K)$$, so that $$\mathcal{L} _{\phi}$$ must be trivial on $$K$$ (by Proposition 5.6). (b) $$\Rightarrow$$ (a): Let $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$, so that $$t \mapsto \|\phi_t\|$$ is in $$C_0(X)$$. For any sequence $$\delta_n>0$$ decreasing strictly to 0 (for instance $$\delta_n = 1/n$$) let Kn={t∈X : ‖ϕt‖≥δn}. Then each $$K_n$$ is compact, $$K_n \subset K_{n+1}^\circ$$ and $$\bigcup_{n=1}^\infty K_n = U$$. By Proposition 5.6, $$\psi_{K_n} \in \mathop{\mathrm{TM}}\nolimits(A_{K_n})=\mathop{\mathrm{TM}}\nolimits(\Gamma(\mathcal{E}|_{K_n}))$$ and so there are $$a_n, b_n \in A_{K_n}$$ with $$\psi_{K_n} = M_{a_n, b_n}$$. Using Remark 5.4, we may assume $$\|a_n(t)\| = \|b_n(t)\| = \sqrt{\|\phi_t\|}$$ for $$t \in K_n$$. By Remark 4.1 (b) we may extend $$a_n$$ to $$c_n \in A$$ with $$c_n(t) = 0$$ for $$t \in X \setminus K_{n+1}^\circ$$ and $$\|c_n(t)\|^2 \leq \delta_n$$ for all $$t \in X \setminus K_n$$. Similarly we extend $$b_n$$ to $$d_n \in A$$ supported in $$K_{n+1}^\circ$$ with $$\|d_n(t)\|^2 \leq \delta_n$$ for $$t \in X \setminus K_n$$. Then $$(M_{c_n, d_n} - \phi)_t$$ has norm at most $$2 \delta_n$$ for all $$t \in X$$ and hence $$\lim_{n \to \infty} M_{c_n, d_n} = \phi$$. Thus $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$. ■ Corollary 6.10. For a homogeneous $$C^*$$-algebra $$A=\Gamma_0(\mathcal{E})$$ the following conditions are equivalent: (a) $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}=\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. (b) For each $$\sigma$$-compact open subset $$U$$ of $$X$$, every complex line subbundle of $$\mathcal{E}|_U$$ is trivial on all compact subsets of $$U$$. □ Proof. (a) $$\Rightarrow$$ (b): Let $$U$$ be a $$\sigma$$-compact open subset of $$X$$, $$B=\Gamma_0(\mathcal{E}|_U)$$ and $$\mathcal{L}$$ a complex line subbundle of $$\mathcal{E}|_U$$. By Proposition 5.8 we can find an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(B)$$ such that $$\mathcal{L}_\phi=\mathcal{L}$$. By extending $$\phi$$ to be zero outside $$U$$, we may assume that $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$, so that $$U=\mathrm{coz}(\phi)$$. By assumption, $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$, so by Theorem 6.9 $$\mathcal{L}$$ is trivial on all compact subsets of $$U$$. (b) $$\Rightarrow$$ (a): If $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ then $$U=\mathrm{coz}(\phi)$$ is an open, necessarily $$\sigma$$-compact subset of $$X$$ (since $$t \mapsto \|\phi_t\|$$ is in $$C_0(X)$$). By assumption, $$\mathcal{L}_\phi$$ is trivial on every compact subset of $$U$$. Hence, $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$ by Theorem 6.9. ■ Definition 6.11. A locally trivial fibre bundle $$\mathcal{F}$$ over a locally compact Hausdorff space $$X$$ is said to be a phantom bundle if $$\mathcal{F}$$ is not globally trivial, but is trivial on each compact subset of $$X$$. □ Corollary 6.12. Let $$A=\Gamma_0(\mathcal{E} )$$ be a homogeneous $$C^*$$-algebra. Then $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ fails to be norm closed in $$\mathcal{B}(A)$$ if and only if there exists a $$\sigma$$-compact open subset $$U$$ of $$X$$ and a phantom complex line subbundle of $$\mathcal{E}|_U$$. If these equivalent conditions hold, then $$\mathop{\mathrm{TM}}\nolimits(A)$$ fails to be norm closed. □ Proof. If $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ fails to be norm closed, there is $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}} \setminus \mathop{\mathrm{TM}}\nolimits_0(A)$$. Note that $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ by Proposition 6.1. By Theorem 6.9, for $$U= \mathrm{coz}(\phi)$$ (open and $$\sigma$$-compact), $$\mathcal{L} _{\phi}$$ is trivial on each compact subset of $$U$$. Moreover $$\phi|_U \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(B)$$ for $$B = \Gamma_0(\mathcal{E}|_U)$$. By Proposition 5.6, if $$\mathcal{L} _{\phi}$$ is globally trivial, then $$\phi|_U \in \mathop{\mathrm{TM}}\nolimits(B) \cap \mathop{\mathrm{IB}}\nolimits_0(B) = \mathop{\mathrm{TM}}\nolimits_0(B)$$. So $$\phi|_U = M_{a,b}$$ for $$a, b \in B$$. Since $$B$$ can be considered as an ideal of $$A$$ (Remark 4.1 (c)), we treat $$a, b \in A$$. Hence $$\phi = M_{a, b} \in \mathop{\mathrm{TM}}\nolimits_0(A)$$, a contradiction. Thus $$\mathcal{L} _{\phi}$$ is a phantom bundle. Conversely, suppose that $$U \subset X$$ is open and $$\sigma$$-compact and that $$\mathcal{L} $$ is a phantom complex line subbundle of $$\mathcal{E}|_U$$. Then, taking $$B =\Gamma_0(\mathcal{E}|_U)$$, Proposition 5.8 provides $$\psi \in \mathop{\mathrm{IB}}\nolimits^{\mathrm{nv}}_{0,1}(B)$$ with $$\mathcal{L} _\psi = \mathcal{L} $$. As $$\mathcal{L} $$ is a phantom bundle, by Proposition 5.6, $$\psi \notin \mathop{\mathrm{TM}}\nolimits(B)$$. We may define $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ by $$\phi_t = \psi_t$$ for $$t \in U$$ and $$\phi_t = 0$$ for $$t \in X \setminus U$$. From $$\psi = \phi|_U \notin \mathop{\mathrm{TM}}\nolimits(B)$$, we have $$\phi \notin \mathop{\mathrm{TM}}\nolimits(A)$$ but $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$ by Theorem 6.9. ■ We now describe below a class of homogeneous $$C^*$$-algebras $$A$$ for which $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ and $$\mathop{\mathrm{TM}}\nolimits(A)$$ both fail to be norm closed. We first explain some preliminaries. Remark 6.13. Let $$G$$ be a group and $$n$$ a positive integer. Recall that a space $$X$$ is called an Eilenberg-MacLane space of type $$K(G, n)$$, if it’s $$n$$-th homotopy group $$\pi_n(X)$$ is isomorphic to $$G$$ and all other homotopy groups trivial. If $$n > 1$$ then $$G$$ must be abelian (since for all $$n>1$$, the homotopy groups $$\pi_n(X)$$ are abelian). We state some basic facts and examples about Eilenberg–MacLane spaces: (a) There exists a CW-complex $$K(G, n)$$ for any group $$G$$ at $$n = 1$$, and abelian group $$G$$ at $$n > 1$$. Moreover, such a CW-complex is unique up to homotopy type. Hence, by abuse of notation, it is common to denote any such space by $$K(G, n)$$ [16, p. 365–366]. (b) Given a CW-complex $$X$$, there is a bijection between its cohomology group $$H^n(X; G)$$ and the homotopy classes $$[X, K(G, n)]$$ of maps from $$X$$ to $$K(G, n)$$ [16, Theorem 4.57]. (c) $$K(\mathbb{Z}, 2)\cong\mathbb{C} P^\infty$$ [16, Example 4.50]. In particular, by (b) and Remark 5.10, for each CW-complex $$X$$ there is a bijection between $$[X, K(\mathbb{Z}, 2)]$$ and isomorphism classes of complex line bundles over $$X$$. □ Proposition 6.14. If $$X$$ is a locally compact CW-complex of type $$K(\mathbb{Q}, 1)$$, then every non-trivial complex line bundle over $$X$$ is a phantom bundle. Moreover, there are uncountably many non-isomorphic such bundles. □ Proof. The standard model of $$K(\mathbb{Q}, 1)$$ is the mapping telescope $$\Delta$$ of the sequence S1⟶f1S1⟶f2S1⟶f3⋯, (6.1) where $$f_n: \mathbb{S}^1 \to \mathbb{S}^1$$ is given by $$z \mapsto z^{n+1}$$ (see e.g., [7, Example 1.9] and [16, Section 3.F]). We first consider the case when $$X=\Delta$$. Applying $$H_1(-; \mathbb{Z})$$ to the levels of the mapping telescope (6.1) gives the system Z⟶(f1)∗Z⟶(f2)∗Z⟶(f3)∗⋯, where $$(f_n)_* : \mathbb{Z} \to \mathbb{Z}$$ is given by $$k \mapsto (n+1)k$$ (see [16, Section 3.F]). The colimit of this system is (by [16, Proposition 3.33]) $$H_1(\Delta; \mathbb{Z})= \mathbb{Q}$$ and all other integral homology groups are trivial. By the universal coefficient theorem for cohomology [16, Theorem 3.2] (See also [16, Section 3.F]) each integral cohomology group of $$\Delta$$ is trivial, except for $$\check{H}^2(\Delta; \mathbb{Z})$$ which is isomorphic to $$\mathop{\mathrm{Ext}}\nolimits(\mathbb{Q};\mathbb{Z})$$. By [39] $$\mathop{\mathrm{Ext}}\nolimits(\mathbb{Q};\mathbb{Z})$$ is isomorphic to the additive group of real numbers. Hence, by Remark 5.10, there exists uncountably many non-isomorphic complex line bundles over $$\Delta$$. We claim that each non-trivial such bundle $$\mathcal{L} $$ is a phantom bundle. Indeed, for $$n\geq 1$$ let $$\Delta_n$$ denote the $$n$$-the level of the mapping telescope (6.1). If $$K$$ be an arbitrary compact subset of $$\Delta$$ then $$K$$ is contained in some $$\Delta_n$$. Since all $$\Delta_n$$’s are homotopy equivalent to $$\mathbb{S}^1$$, and since $$\check{H}^2(\mathbb{S}^1;\mathbb{Z})=0$$, we conclude that $$\mathcal{L} |_{\Delta_n}$$ is trivial. Then $$\mathcal{L} |_K$$ is also trivial, since $$K \subset \Delta_n$$. If $$X$$ is another locally compact CW-complex of type $$K(\mathbb{Q}, 1)$$, then by Remark 6.13 (a), there are maps $$f \colon \Delta \to X$$ and $$g \colon X \to \Delta$$ such that $$g \circ f$$ and $$f \circ g$$ are homotopic (respectively) to the identity maps (on $$\Delta$$ and $$X$$, respectively). If $$\mathcal{L}$$ is a non-trivial complex line bundle over $$\Delta$$, then $$g^*(\mathcal{L})$$ is non-trivial over $$X$$ (Remark 5.10). Moreover $$g^*(\mathcal{L})$$ is a phantom bundle because $$K \subset X$$ compact implies $$g(K) \subset \Delta$$ compact and $$g^*(\mathcal{L})|_K$$ is a restriction of $$g^*(\mathcal{L})|_{g^{-1}(g(K))} = g^*(\mathcal{L}|_{g(K)})$$, which is a trivial bundle. Since $$g$$ is a homotopy equivalence, every non-trivial complex line bundle over $$X$$ must be isomorphic to $$g^*(\mathcal{L})$$ for some $$\mathcal{L}$$. ■ Remark 6.15. With the same notation as in the proof of Proposition 6.14, one can show that for each compact subset $$K$$ of $$\Delta$$ we have $$\check{H}^2(K;\mathbb{Z})=0$$. To sketch the proof, choose an arbitrary complex line bundle $$\mathcal{L}$$ over $$K$$. Then using Lemma 5.19 (and Remark 5.10) $$\mathcal{L}$$ can be extended to an open neighbourhood $$U$$ of $$K$$. The assertion can now be established via an argument with triangulations of $$\Delta$$. There is a triangulation of $$\Delta$$ where $$\Delta_1$$ has $$3$$ triangles and each $$\Delta_{n+1}$$ has $$n+3$$ more triangles than $$\Delta_n$$. We may subdivide the triangles that touch $$K$$ to get finitely many that cover $$K$$ and are all contained in $$U$$. Now consider the union $$T$$ of the triangles that touch $$K$$. It is enough to show $$\mathcal{L}|_T$$ is trivial. We can deformation retract $$T$$ to a union of 1-simplices. To do so, work on one triangle (2-cell) at a time, starting with any 2-cell in $$\Delta_1$$ with a “free” edge not in the boundary of $$\Delta_1$$ relative to $$\Delta_2$$ (where “free” means the edge does not bound a second 2-cell). After each step, consider the remaining 2-cells, edges and vertices. Move on to $$\Delta_2$$ once all 2-cells in $$\Delta_1$$ are exhausted, etc, so as to arrive at a $$1$$-simplex after finitely many steps. As all complex line bundles over 1-simplices are trivial, we have that $$\mathcal{L}|_T$$ is trivial. □ In private correspondence, Mladen Bestvina informed us that we can find phantom bundles even over some open subset of $$\mathbb{R}^3$$, and referred us to [6]. We outline the construction of such a subset. Proposition 6.16. There exists an open subset $$\Omega$$ of $$\mathbb{R}^3$$ of type $$K(\mathbb{Q},1)$$. □ Proof. In [6], a construction is given of dense open sets $$U$$ in the $$3$$-sphere $$\mathbb{S}^3$$ with fundamental groups $$\pi_1(U)$$ that are large subgroups of $$\mathbb{Q}$$. Given a sequence $$n_i$$ of natural numbers $$n_i > 1$$, $$\pi_1(U)$$ can be $$\{ p/q \in \mathbb{Q} : p \in \mathbb{Z}, q = \prod_{i=1}^k n_i \mbox{ for some } k\}$$. In particular we will take $$n_i = i+1$$ and then $$\pi_1(U) = \mathbb{Q}$$. The construction defines $$U$$ as a union of closed solid tori $$U = \bigcup_{i=1}^\infty S_i$$. For each $$i$$, both $$S_i$$ and the complement of its interior $$T_i = \mathbb{S}^3 \setminus S_i^\circ$$ are solid tori with intersection $$S_i \cap T_i$$ a (two-dimensional) torus. At each step, $$T_{i+1}$$ is constructed inside $$T_i$$ as an unknotted solid torus of smaller cross-sectional area that winds $$n_i$$ times around the meridian circle of $$T_i$$. Since $$T_{i+1}$$ can be unfolded to a standard embedding of a torus via an ambient isotopy of $$\mathbb{S}^3$$, $$S_{i+1}$$ must be a solid torus. Let $$f: \mathbb{S}^n\to U$$ be an arbitrary map. Then $$f$$ maps $$\mathbb{S}^n$$ into one of the solid tori $$S_i$$ and these are homotopic to their meridian circle. In particular $$\pi_n(U)=0$$ for all $$n>1$$. By Remark 6.13 $$U$$ has the type $$K(\mathbb{Q}, 1)$$. Choose any point $$t \in \mathbb{S}^3\setminus U$$. Since $$\mathbb{S}^3\setminus \{t\}$$ is homeomorphic to $$\mathbb{R}^3$$, say via the homeomorphism $$F$$, then $$\Omega=F(U)$$ is an open subset of $$\mathbb{R}^3$$ of the type $$K(\mathbb{Q}, 1)$$. ■ Proposition 6.17. Let $$X$$ be any locally compact $$\sigma$$-compact CW-complex of type $$K(\mathbb{Q},1)$$ (e.g., $$X=\Delta$$). Then the $$C^*$$-algebra $$A=C_0(X,\mathbb{M}_n)$$ ($$n \geq 2$$) has the following property: There exists an operator $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(A) \setminus \mathop{\mathrm{TM}}\nolimits(A)$$ such that $$\phi$$ is in the norm closure of $$\mathop{\mathrm{TM}}\nolimits_{cp}(A) \cap \mathop{\mathrm{TM}}\nolimits_0(A)=\{M_{a,a^*} : a \in A\}$$. In particular, $$\mathop{\mathrm{TM}}\nolimits_0(A)$$, $$\mathop{\mathrm{TM}}\nolimits(A)$$ and $$\mathop{\mathrm{TM}}\nolimits_{cp}(A)$$ all fail to be norm closed. Further, if $$X=\Delta$$, we have $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}=\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. □ Proof. Choose any phantom complex line bundle $$\mathcal{L}$$ over $$X$$ (Proposition 6.14). Since, by Remark 6.13 (a), $$X$$ has the same homotopy type as the space $$\Delta$$ of Proposition 6.14 (which is a two-dimensional complex), using Remark 5.10 and Lemma 5.13 we may assume that $$\mathcal{L} $$ is a subbundle of the trivial bundle $$X \times \mathbb{C}^2$$. We also realise $$\mathbb{C}^2$$ as a subset of $$\mathbb{M}_n$$ as $$\{z_1 e_{1,1} + z_2e_{1,2}: z_1,z_2 \in \mathbb{C}\}$$, and in this way consider $$\mathcal{L} $$ a subbundle of $$X \times \mathbb{M}_n$$. By the proof of Proposition 5.8 we can find two sections $$a,b$$ of $$\mathcal{L} $$ vanishing at infinity (so that $$a,b \in A$$) such that $$\mathrm{span}\{a(t), b(t)\} = \mathcal{L} _t$$ for each $$t \in X$$. We define a map ϕ:A→Abyϕ=Ma,a∗+Mb,b∗. Then $$\phi$$ defines a completely positive elementary operator on $$A$$ of length at most $$2$$. Clearly, $$\phi_t \neq 0$$ for all $$t \in X$$, so $$\phi \in \mathop{\mathrm{IB}}\nolimits_{0,1}^{\mathrm{nv}}(A)$$. Also, $$\mathcal{L} _\phi =\mathcal{L} $$. Since the bundle $$\mathcal{L} $$ is non-trivial, by Proposition 5.6 we have $$\phi \not \in \mathop{\mathrm{TM}}\nolimits(A)$$. On the other hand, since $$\mathcal{L} $$ is a phantom bundle, Theorem 6.9 implies $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$. Thus $$\phi \in \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}} \setminus \mathop{\mathrm{TM}}\nolimits(A)$$ and consequently $$\phi$$ has length $$2$$. We have $$\phi_K$$ completely positive on $$A_K = \Gamma(\mathcal{E}|_K)$$ for each compact $$K \subset X$$. Since $$\mathcal{L}|_K$$ is a trivial bundle, $$\phi_K = M_{a,b}$$ for some $$a, b \in A_K$$ and we may suppose $$\|a(t)\| = \|b(t)\|$$ holds for all $$t \in K$$. It follows from positivity of $$\phi_t$$ that $$b(t) = a(t)^*$$ (for $$t \in K$$). By the proof of Theorem 6.9, $$\phi$$ is in the norm closure of $$\mathop{\mathrm{TM}}\nolimits_{cp}(A) \cap \mathop{\mathrm{TM}}\nolimits_0(A)$$. Now suppose that $$X=\Delta$$. Then by Remark 6.15 $$\check{H}^2(K;\mathbb{Z})=0$$ for all compact subsets $$K$$ of $$\Delta$$. By Corollary 6.10 (and Remark 5.10) we conclude that $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}= \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. ■ Recalling that Corollary 5.16 (a) and Proposition 6.1 deal with the cases where $$X$$ is second-countable with $$\dim X<2$$ or if $$X$$ is (homeomorphic to) a subset of a non-compact connected $$2$$-manifold, we now add the opposite conclusion for higher dimensions. Theorem 6.18. Let $$A =\Gamma_0(\mathcal{E})$$ be an $$n$$-homogeneous $$C^*$$-algebra with $$n \geq 2$$. If there is a nonempty open subset of $$X$$ homeomorphic to (an open subset of) $$\mathbb{R}^d$$ for some $$d \geq 3$$, then $$\mathop{\mathrm{TM}}\nolimits_0(A)$$ and $$\mathop{\mathrm{TM}}\nolimits(A)$$ both fail to be norm closed. □ Proof. We first choose an open subset $$U\subset X$$ for which $$\mathcal{E}|_U$$ is trivial and such that $$U$$ can be considered as an open set in $$\mathbb{R}^d$$$$(d \geq 3)$$. Choose any open subset $$V$$ of $$U$$ that has the homotopy type of the set $$\Omega$$ of Proposition 6.16. In particular, $$V$$ is of type $$K(\mathbb{Q},1)$$, so it allows a phantom complex line bundle (Proposition 6.14). Now apply Corollary 6.12. ■ Remark 6.19. Suppose that $$A=\Gamma_0(\mathcal{E})$$ is a separable $$n$$-homogeneous $$C^*$$-algebra with $$n \geq 2$$ such that $$\dim X =d<\infty$$. By Remark 5.18 (applied to an $$\mathbb{M}_n$$-bundle $$\mathcal{E}$$) $$A$$ has the finite type property. Hence, by [25, Theorem 1.1], we have $$\mathop{\mathrm{IB}}\nolimits(A)=\mathcal{E}\ell(A)$$. If $$X$$ is either a CW-complex or a subset of a $$d$$-manifold, the following relations between $$\mathop{\mathrm{TM}}\nolimits_0(A)$$, $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}$$ and $$\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ occur: (a) If $$d<2$$ we always have $$\mathop{\mathrm{TM}}\nolimits_0(A)=\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}=\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ (Corollary 5.16 (a)). (b) If $$d=2$$ we have four possibilities: (i) $$\mathop{\mathrm{TM}}\nolimits_0(A)=\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}=\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. This happens for example, whenever $$X$$ is a subset of a non-compact connected $$2$$-manifold (Corollary 5.16 (a)). (ii) $$\mathop{\mathrm{TM}}\nolimits_0(A)=\overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}\subsetneq \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. This happens for example, for $$A=C(X, \mathbb{M}_n )$$, where $$X=\mathbb{S}^2$$ (by Example 5.14 and since any proper open subset $$U$$ of $$\mathbb{S}^2$$ is homeomorphic to an open subset of $$\mathbb{R}^2$$, so $$\check{H}^2(U;\mathbb{Z})=0$$ by Proposition 5.20). (iii) $$\mathop{\mathrm{TM}}\nolimits_0(A)\subsetneq \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}=\mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. This happens for example, for $$A=C_0(X, \mathbb{M}_n )$$, where $$X=\Delta$$ is the standard model of $$K(\mathbb{Q},1)$$ (Proposition 6.17). (iv) $$\mathop{\mathrm{TM}}\nolimits_0(A)\subsetneq \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}\subsetneq \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$. This happens e.g. for $$A=C_0(X, \mathbb{M}_n )$$, where $$X$$ is the topological disjoint union $$\mathbb{S}^2 \sqcup \Delta$$ (by Proposition 6.17, Corollary 6.10 and Example 5.14). (c) If $$d>2$$ we always have $$\mathop{\mathrm{TM}}\nolimits_0(A)\subsetneq \overline{\overline{\mathop{\mathrm{TM}}\nolimits_0(A)}}\subsetneq \mathop{\mathrm{IB}}\nolimits_{0,1}(A)$$ (by Theorem 6.18 and the fact that $$X$$ must contain an open subset homeomorphic to $$\mathbb{R}^d$$ — if $$X$$ is a subset of a $$d$$-manifold, this follows from [9, Theorems 1.7.7, 1.8.9, and 4.1.9]). Similar relations occur between $$\mathop{\mathrm{TM}}\nolimits(A)$$, $$\overline{\overline{\mathop{\mathrm{TM}}\nolimits(A)}}$$ and $$\mathop{\mathrm{IB}}\nolimits_{1}(A)$$ in parts (a) and (c) of the above cases. □ Funding This work was supported by Irish Research Council [GOIPD/2014/7 to I.G.]; and Science Foundation Ireland [11/RFP/MTH3187 to R.M.T.] Acknowledgments We are very grateful to Mladen Bestvina for his extensive and generous help with many of the topological aspects of the article. References [1] Akemann C. A. Pedersen G. K. and Tomiyama. 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International Mathematics Research Notices – Oxford University Press

**Published: ** Jan 1, 2018

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