Positivity 11 (2007), 497–510
2007 Birkh¨auser Verlag Basel/Switzerland
Ordered Involutive Operator Spaces
David P. Blecher, Kay Kirkpatrick, Matthew Neal and Wend Werner
Abstract. This is a companion to recent papers of the authors; here we
construct the ‘noncommutative Shilov boundary’ of a (possibly nonunital)
selfadjoint ordered space of Hilbert space operators. The morphisms in the
universal property of the boundary preserve order. As an application, we con-
sider ‘maximal’ and ‘minimal’ unitizations of such ordered operator spaces.
Mathematics Subject Classiﬁcation (2000) . Primary 47L07, 47L05; Secondary
46B40, 46L07, 46L08, 47B60, 47B65.
Keywords. Positive operator, Loewner order, operator spaces, operator system,
unitization, noncommutative Shilov boundary, C*- envelope.
An operator space is a closed linear space of bounded operators between Hilbert
spaces, or, equivalently, a subspace of a C*-algebra. Although ordered operator
spaces containing the identity operator are well understood and important (these
are known as operator systems [7, 14]), nonunital ordered operator spaces have
barely been studied at all. This is despite the fact that they occur very naturally;
for example, consider the linear span of three generic positive matrices in the 5× 5
. Indeed, the only theory addressing such spaces which we are aware of
is contained in [15, 16, 17, 6, 5] and a series of papers by Karn (see e.g. [10, 11] and
references therein). In view of the importance of the notion of operator positivity,
we oﬀer, in this companion paper to [6, 5], some results on this topic. In particular,
we construct the ‘noncommutative Shilov boundary’ of such a space X, and use
this to construct a ‘maximal unitization’ of X. At the end of the paper we will
illustrate these two main concepts in the simple case of spaces of ﬁnite matrices.
Blecher was partially supported by grant DMS 0400731 from the National Science Foundation.
Kirkpatrick was partially supported by an NSF REU grant. Neal was supported by Denison
University. Werner was supported by the SFB 487 Geometrische Strukturen in der Mathematik,
at the Westf¨alische Wilhelms-Universit¨at, supported by the Deutsche Forschungsgemeinschaft.