Elementary operators and subhomogeneous C*-algebras II

Proceedings of the Edinburgh Mathematical Society

2013

Let A be a unital C * -algebra with the canonical (H) C * -bundle A over the maximal ideal space of the centre of A, and let E (A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E (A), and show that such derivations are necessarily inner in the case when each fibre of A is a prime C * -algebra. We also consider separable C * -algebras A for which A is an (F) bundle. For these C * -algebras we show that the following conditions are equivalent: E (A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of A have uniformly finite dimensions, and each restriction bundle of A over a set where its fibres are of constant dimension is of finite type as a vector bundle.

Section: Introductionmentioning

confidence: 81%

Section: Elementary Operators and (F) C * -Bundlesmentioning

confidence: 93%

“…Moreover, if A satisfies (1.2), then by [20,Theorem 2.3], Prim(A) in (1.5) can be replaced by the larger set Primal 2 (A) of 2-primal ideals of A, so (1.2) also implies…”

Section: 5)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On derivations and elementary operators onC*-algebras

Proceedings of the Edinburgh Mathematical Society

2013

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Timoney

2016

Int Math Res Notices

Abstract. For a C * -algebra A we consider the problem of when the set TM 0 (A) of all two-sided multiplications x → axb (a, b ∈ A) on A is norm closed, as a subset of B(A). We first show that TM 0 (A) is norm closed for all prime C * -algebras A. On the other hand, if A ∼ = Γ 0 (E) is an n-homogeneous C * -algebra, where E is the canonical Mn-bundle over the primitive spectrum X of A, we show that TM 0 (A) fails to be norm closed if and only if there exists a σ-compact open subset U of X and a phantom complex line subbundle L of E over U (i.e. L is not globally trivial, but is trivial on all compact subsets of U ). This phenomenon occurs whenever n ≥ 2 and X is a CW-complex (or a topological manifold) of dimension 3 ≤ d < ∞.

Topologically finitely generated Hilbert C(X)-modules

Journal of Mathematical Analysis and Applications

2012

Keywords:Hilbert C (X)-module (F) Hilbert bundle Subhomogeneous Finite type property C (X)-projective tensor product a b s t r a c t For a Hilbert C (X)-module V , where X is a compact metrizable space, we show that the following conditions are equivalent: (i) V is topologically finitely generated, (ii) there exists K ∈ N such that every algebraically finitely generated submodule of V can be generated with k ≤ K generators, (iii) V is canonically isomorphic to the Hilbert C (X)-module Γ (E) of all continuous sections of an (F) Hilbert bundle E = (p, E, X ) over X , whose fibres E x have uniformly finite dimensions, and each restriction bundle of E over a set where dim E x is constant is of finite type, (iv) there exists N ∈ N such that for every Banach C (X)-module W , each tensor in the C (X)-projective tensor product V π ⊗ C (X) W is of (finite) rank at most N.