Elementary operators and subhomogeneous C*-algebras

Abstract

Let A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

Topologically finitely generated Hilbert C(X)-modules

“…Without loss of generality, we may assume that E | U is not of finite type. Using [22,Lemma 2.10] and [22,Lemma 2.11], one can find a compact subset K ⊆ U with the following property: if s 1 , . .…”

“…Remark 3.13. Using [22,Theorem 2.4] together with results of [23], one can show that the answer to Problem 3.12 is positive for (continuous) C * -bundles. Moreover, if the base space X is connected, then all fibres of this C * -bundle must be pairwise * -isomorphic, in particular isometrically isomorphic.…”

The closure of two-sided multiplications on C*-algebras and phantom line bundles

“…It can happen that Eℓ(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 Eℓ(A) is norm (or cb-norm) closed then A is necessarily subhomogeneous, the homogeneous subquotients of A must have the finite type property and established further necessary conditions on A. In [14,15] he gave some partial converse results.…”

On derivations and elementary operators onC*-algebras

“…It is well known that each derivation δ on A is completely bounded with δ cb = δ . In our previous papers [18][19][20] we considered variants of the following two problems. The motivation for considering these problems comes from understanding the operator (or completely bounded) norm closure of E (A).…”

“…On the other hand, in [19,Theorem 2.6] we showed that if a unital separable C * -algebra A satisfies (1.2), then A is necessarily subhomogeneous of finite type. This means that A is subhomogeneous and the C * -bundles corresponding to the homogeneous subquotients of A must be of finite type (see [19] for a detailed explanation). By [20,…”