On derivations and elementary operators on<i>C</i>*-algebras

Gogić, Ilja

doi:10.1017/s0013091512000302

Cited by 5 publications

(6 citation statements)

References 26 publications

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“…More precisely, we showed that all such derivations of A are inner in a case when A is prime [12,Theorem 4.3] or central [12,Theorem 5.6]. This result was further extended in [14,Theorem 1.5] for unital C * -algebras whose every Glimm ideal is prime. The latter result in particular applies to derivations of local multiplier algebras (see e.g.…”

Section: Introductionmentioning

confidence: 72%

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The cb-norm approximation of generalized skew derivations by elementary operators

Gogić

2019

Preprint

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Let A be a ring and σ : A → A a ring endomorphism. A generalized skew (or σ-)derivation of A is an additive map d : A → A for which there exists a map δ :algebra and σ is surjective, we determine the structure of generalized σ-derivations of A that belong to the cb-norm closure of elementary operators Eℓ(A) on A; all such maps are of the form d(x) = bx + axc for suitable elements a, b, c of the multiplier algebra M (A). As a consequence, if an epimorphism σ : A → A lies in the cb-norm closure of Eℓ(A), then σ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime C * -algebras like C(X, M 2 ).

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Section: Introductionmentioning

confidence: 72%

“…In [12,14] we considered derivations of unital C * -algebras A that lie in Eℓ(A) cb . We showed that all such derivations are inner in a case when A is prime [12,Theorem 4.3] or central [12,Theorem 5.6], or more generally, when A is a unital C * -algebra whose every Glimm ideal is prime [14,Theorem 1.5].…”

Section: Counterexamples and Further Remarksmentioning

confidence: 99%

The cb-norm approximation of generalized skew derivations by elementary operators

Gogić

2019

Preprint

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“…Proof The construction of such a CE E : A → C(X ) can be deduced from the proof of [22,Lemma 4.6]. But we include here the main steps of the proof for completeness.…”

Section: Proposition 34 Every Continuous Homogeneous Unital C(x )-Almentioning

confidence: 99%

On unitalC(X)-algebras andC(X)-valued conditional expectations of finite index

Blanchard

Gogić

2016

Linear and Multilinear Algebra

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Abstract. Let X be a compact Hausdorff space and let A be a unital C(X)-algebra, where C(X) is embedded as a unital C * -subalgebra of the centre of A. We consider the problem of characterizing the existence of a conditional expectation E : A → C(X) of finite index in terms of the underlying C * -bundle of A over X. More precisely, we show that if A admits a C(X)-valued conditional expectation of finite index, then A is necessarily a continuous C(X)-algebra, and there exists a positive integer N such that every fibre Ax of A is finitedimensional, with dim Ax ≤ N . We also give some sufficient conditions on A that ensure the existence of a C(X)-valued conditional expectation of finite index. introductionLet B ⊆ A be two unital C * -algebras with the same unit element. A conditional expectation (abbreviated by C.E.) from A to B is a completely positive contraction E : A → B such that E(b) = b for all b ∈ B, and which is B-bilinear, i.e. If E(a * a) = 0 (a ∈ A) implies a = 0, E is said to be faithful. Every faithful conditional expectation E : A → B introduces a pre-Hilbert B-module structure on A, whose inner product is defined by (1.1) a 1 , a 2 E := E(a *

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“…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%

“…In [12,15] the first author considered the problem of which derivations on unital C * -algebras A can be cb-norm approximated by elementary operators. By [15,Theorem 1.5] every such a derivation is necessarily inner in a case when every Glimm ideal of A is prime.…”

Section: Introductionmentioning

confidence: 99%

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

Gogić

Timoney

2016

Int Math Res Notices

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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 < ∞.

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On derivations and elementary operators onC*-algebras

Cited by 5 publications

References 26 publications

The cb-norm approximation of generalized skew derivations by elementary operators

The cb-norm approximation of generalized skew derivations by elementary operators

On unitalC(X)-algebras andC(X)-valued conditional expectations of finite index

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

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