Primitive Limit Algebras and C*-Envelopes

Davidson, Kenneth R.; Katsoulis, Elias G.

doi:10.1006/aima.2001.2068

Cited by 6 publications

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“…Since X ∩ Y ∩ Z is empty, there exists h ∈ A/ ker(X ∩ (Y ∪ Z)) such that h ker(X∩Y ) = λ 4 1 and h ker(X∩Z) = λ 5 …”

Section: Construction Of a V Applying [3 Lemma 33] To Arbitrary Lmentioning

confidence: 98%

“…One way is to calculate the distance d(a, Z(A)) from a to Z(A), the centre of A; the other way is to compute the operator norm of ad(a), the inner derivation of A induced by a. For von Neumann algebras and some closely related algebras one has the interesting result that these two methods yield essentially the same answer: for all elements a in the algebra, ad(a) = 2d(a, Z(A)) [5,7,8,[10][11][12]16,20,21]. The unital C * -algebras for which this equality holds have been characterized in [19] as those C * -algebras A for which the ideal P ∩ Q ∩ R is primal whenever P , Q, and R are primitive ideals of A such that P ∩ Z(A) = Q ∩ Z(A) = R ∩ Z(A).…”

Section: Introductionmentioning

confidence: 97%

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Measuring noncommutativity in C∗-algebras

Archbold

Somerset

2007

Journal of Functional Analysis

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It is well known that if A is a von Neumann algebra then the norm of any inner derivation ad(a) is equal to twice the distance from the element a to the centre Z(A) of the algebra. More generally, this property holds in a unital C * -algebra if and only if the ideal P ∩ Q ∩ R is primal whenever P , Q, and R are primitiveIn this paper we give a characterization, in terms of ideal structure, of those unital C * -algebras A for which the norm of any inner derivation ad(a) at least dominates the distance from a to the centre Z(A). In doing so, we show that if A does not have this property then it necessarily contains an element a, with ad(a) = 1, whose distance from Z(A) is greater than or equal to 3+8 √ 2 14 . We also show how this number is related to the numbers 4 √ 15 and 1 2 + 1 √ 3 which have previously arisen in the study of norms of inner derivations. The techniques used in this work include spectral theory, the theory of primitive and primal ideals, and constrained geometrical optimisation.

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“…Since X ∩ Y ∩ Z is empty, there exists h ∈ A/ ker(X ∩ (Y ∪ Z)) such that h ker(X∩Y ) = λ 4 1 and h ker(X∩Z) = λ 5 …”

Section: Construction Of a V Applying [3 Lemma 33] To Arbitrary Lmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 97%

Measuring noncommutativity in C∗-algebras

Archbold

Somerset

2007

Journal of Functional Analysis

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The C*-envelope of the Tensor Algebra of a Directed Graph

Katsoulis

Kribs

2006

Integr. equ. oper. theory

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Abstract. Given an arbitrary countable directed graph G we prove the C * -envelope of the tensor algebra T + (G) coincides with the universal Cuntz-Krieger algebra associated with G. Our approach is concrete in nature and does not rely on Hilbert module machinery. We show how our results extend to the case of higher rank graphs, where an analogous result is obtained for the tensor algebra of a row-finite k-graph with no sources. IntroductionThe fundamental nonselfadjoint operator algebra associated with a countable directed graph is its tensor algebra T + (G) [6,19,11,14]. Fowler, Muhly and Raeburn have recently characterized [8, Theorem 5.3.] the C * -envelope of the tensor algebra of a faithful strict Hilbert bimodule, as the associated universal Cuntz-Pimsner algebra. When applied to tensor algebras of graphs, their result shows that the C * -envelope of T + (G) where G is a graph with no sources, coincides with the universal Cuntz-Krieger algebra associated with G. This generalizes the well-known fact that, for a single vertex graph with n loop edges, the C * -envelope of the corresponding disc algebra is the algebra of continuous functions on the unit circle (n = 1) and the Cuntz algebra (n ≥ 2) [22]. On the other hand, tensor algebras of graphs that do have sources do not come from injective Hilbert bimodules and therefore

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Crossed Products of Operator Algebras

2019

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Chapter 1. Introduction Chapter 2. Preliminaries 2.1. Generalities 2.2. C ˚-correspondences and tensor algebras 2.3. Crossed products of C ˚-algebras Chapter 3. Definitions and Fundamental Results Chapter 4. Maximal C ˚-covers, Iterated Crossed Products and Takai Duality Chapter 5. Crossed Products and the Dirichlet Property Chapter 6. Crossed Products and Semisimplicity 6.1. Crossed products by discrete abelian groups 6.2. Crossed products by compact abelian groups 6.3. More examples of crossed product Dirichlet algebras 6.4. Semicrossed products and semisimplicity Chapter 7. The Crossed Product as the Tensor Algebra of a C ˚correspondence. 7.1. Discrete groups 7.2. The general case of a locally compact group. 7.3. Hilbert C ˚-bimodules Chapter 8. Concluding Remarks and Open Problems Bibliography

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Primitive Limit Algebras and C*-Envelopes

Cited by 6 publications

References 21 publications

Measuring noncommutativity in C∗-algebras

Measuring noncommutativity in C∗-algebras

The C*-envelope of the Tensor Algebra of a Directed Graph

Crossed Products of Operator Algebras

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