Property (T) for C*-Algebras

Abstract

A C *-algebra A is said to have the FS-property if the set of all self-adjoint elements in A has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal C *-tensor products even in case of separable nuclear C *-algebras. §1. Introduction A C *-algebra A is said to have the FS-property if the set of all self-adjoint elements in A (= A sa) has a dense subset of elements with finite spectrum. In [3], Brown and Pedersen provided the non-commutative real rank for A (= RR(A)) and showed that the FS-property is equivalent to real rank zero, i.e. A sa has a dense subset of invertible elements. RR(A) is the least integer n such that {(a 0 , a 1 , · · · , a n) ∈ A n+1 sa : n k=0 Aa k = A} is dense in A n+1 sa which is the analogue of the dimension of a topological space X(= dim X). If A is non-unital, its real rank is defined by RR(˜ A), where˜Awhere˜ where˜A is the C *-algebra obtained from A by adjoining a unit. From this definition it is obvious that dim X = RR(C(X)) for a compact Hausdorff space X. In the previous note [9], we showed that there are unital C *-algebras A and C with RR(A) = RR(C) = 0 such that RR(A ⊗ C) = 0. To specify let B be one of the Bunce-Deddens algebras and B(H) the C *-algebra of bounded operators on a countably infinite dimensional Hilbert space H. Then by Blackadar and Kumjian [1], RR(B) = 0 and it is well-known that RR(B(H)) = 0. In [9], we showed that RR(B ⊗ B(H)) = 0. This means that the FS-property for C *-algebras is not stable under taking the minimal C *-tensor products. But B(H) is not separable. In this note, we shall give two examples: Example 1. There are separable unital C *-algebras A and C with RR(A) = RR(C) = 0 such that A and C are nuclear and that RR(A ⊗ C) = 0. Example 2. There are separable unital C *-algebras A and C with RR(A) = RR(C) = 0 such that A is not nuclear but exact, C is nuclear, and that RR(A⊗C) = 0.

Property (T) and strong property (T) for unital C∗-algebras

“…Let us first recall Bekka's notion of property (T ) in [3]. Note that Bekka's definition comes from the original definition of property (T ) for groups (see e.g.…”

“…Bekka introduced in [3] the interesting notion of property (T ) for a pair consisting of a unital C * -algebra and a unital C * -subalgebra. He showed that a countable discrete group G has property (T ) if and only if its full (or equivalently reduced) group C * -algebra has property (T ).…”

Kazhdan's property T andC∗-algebras

“…Kazhdan's revolutionary concept of property T has recently been translated into C * -language in [1]. One of the questions raised by Bekka's paper is whether or not one can generalize to the C * -context the classical fact that a discrete group which is both amenable and has property T must be finite (cf.…”

“…E-mail address: nbrown@math.psu.edu. 1 Partially supported by DMS-0244807. The irritating C-summand cannot be avoided; if B is any C * -algebra with property T and C is any algebra without tracial states then B ⊕ C also has property T. Hence any finite-dimensional C * -algebra plus a Cuntz algebra (for example) will have property T and be nuclear. On the other hand, the theorem above does imply that if A is nuclear, has property T and has a faithful trace then it must be finite-dimensional-this is an honest generalization of the discrete group case since reduced group C * -algebras always have a faithful trace.…”

“…One of the questions raised by Bekka's paper is whether or not one can generalize to the C * -context the classical fact that a discrete group which is both amenable and has property T must be finite (cf. [1,Proposition 11]). Unfortunately, the C * -situation is not quite as simple, but a satisfactory result can be obtained.…”