Positivity 6: 59–73, 2002.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
The Daugavet Property of C
Department of Mathematics, University of Texas at Austin, Austin TX 78712, USA
(Accepted 11 May 2000)
Abstract. We prove that a C
-algebra A or a predual N
of a von Neumann algebra N has the
Daugavet property if and only if A (or N ) is non-atomic. We also prove a similar (although somewhat
weaker) result for non-commutative L
-spaces corresponding to non-atomic von Neumann algebras.
We say that a Banach space X has the Daugavet property (DP , in short) if every
rank 1 operator T : X → X satisﬁes the Daugavet equation
+ T=1 +T, (1.1)
is the identity map on X. The study of the Daugavet property was inaug-
urated by Daugavet  in 1961; he proved that every compact operator on C([0.1])
satisﬁes (1.1). Consequently, C([0, 1]) has the DP. Investigation of the Daugavet
properrty continued in the seventies and eighties. In particular, Holub proved in 
and  (see also ) that for any Hausdorff compact topological space K, C(K)
has the DP if and only if K has no isolated points, and, similarly, L
(µ) has the
DP if and only if µ is non-atomic. In the early nineties, the work of Abramovich,
Aliprantis and Burkinshaw (see  and ) infused new ideas into the ﬁeld. The
state-of-the-art information on the Daugavet property can be found, for instance,
in  and .
In this paper we generalize the above results to the non-commutative setting.
We say that a non-zero projection p is a C
-algebra A is atomic if for every a ∈ A,
pap = λp for some λ ∈ C. We say that a C
-algebra A is non-atomic if it has no
atomic projections. if A is a von Neumann algebra, then p atomic if and only if it is
abelian (i.e. pAp is commutative) and minimal (if q is a projection in A such that
q ≺ p,thenq ∼ p). In Section 2 we prove that a C
-algebra A or the predual of a
von Neumann algebra N has the Daugavet property if and only if A (respectively,
N) is non-atomic (Theorem 2.1).
-algebra K of compact operators on
has atomic projections, hence it
fails the DP. However, K has a similar property: If T : K → K is strictly singular,
The author was supported by the NSF grant DMS-9970369.