A C ∗ -algebra  for which Ext() is not a Group

Anderson, Joel

doi:10.2307/1971124

Cited by 28 publications

(55 citation statements)

References 7 publications

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“…We use the result to show that the Ext-invariant for the reduced C * -algebra of the free group on 2 generators is not a group but only a semi-group. This problem has been open since Anderson in 1978 found the first example of a C * -algebra A for which Ext(A) is not a group.…”

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confidence: 99%

A new application of random matrices: Ext(C_red^∗(F₂)) is not a group

2005

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In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the following extension of Voiculescu's random matrix result: Let (Xr ) be a system of r stochastically independent n × n Gaussian self-adjoint random matrices as in Voiculescu's random matrix paper [V4], and let (x 1 , . . . , x r ) be a semi-circular system in a C * -probability space. Then for every polynomial p in r noncommuting variablesfor almost all ω in the underlying probability space. We use the result to show that the Ext-invariant for the reduced C * -algebra of the free group on 2 generators is not a group but only a semi-group. This problem has been open since Anderson in 1978 found the first example of a C * -algebra A for which Ext(A) is not a group.

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A new application of random matrices: Ext(C_red^∗(F₂)) is not a group

2005

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“…(See Arveson [3] where all of this is put together nicely.) When A is not nuclear, Anderson [1] showed that Ext(A) is generally not a group. Recently Haagerup and Thorbjornsen [24] have shown that Ext of the reduced C*-algebra of the free group F 2 is not a group.…”

Section: Essentially Normal Operatorsmentioning

confidence: 99%

Essentially Normal Operators

Davidson

2010

A Glimpse at Hilbert Space Operators

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Abstract. This is a survey of essentially normal operators and related developments. There is an overview of Weyl-von Neumann theorems about expressing normal operators as diagonal plus compact operators. Then we consider the Brown-Douglas-Fillmore theorem classifying essentially normal operators. Finally we discuss almost commuting matrices, and how they were used to obtain two other proofs of the BDF theorem. Mathematics Subject Classification (2000). 47-02,47B15,46L80.

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“…(1), suppose that there exists a C * -algebra D and an element x ∈ A D such that x E > x λ . Then the extension is not semi-invertible.…”

Section: Be Asymptotic Homomorphisms Then Their Tensor Product φ T ⊗mentioning

confidence: 99%

“…Beyond the nuclear case not much is known about the general classification of C * -extensions, but more and more examples of non-invertible extensions are coming up [1], [6], [13], [12], [5]. In [9], [10] we suggested weakening the notion of triviality for extensions so that more extensions would become invertible in this new sense.…”

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On the asymptotic tensor C*-norm

Manuilov

Thomsen

2006

Arch. Math.

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We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C * -algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C * -extensions.One of the reasons that the Brown-Douglas-Fillmore theory [2] gives so nice a classification for extensions of nuclear C * -algebras is their invertibility. Beyond the nuclear case not much is known about the general classification of C * -extensions, but more and more examples of non-invertible extensions are coming up [1], [6], [13], [12], [5]. In [9],[10] we suggested weakening the notion of triviality for extensions so that more extensions would become invertible in this new sense. As was shown by S. Wassermann, one of the reasons for non-invertibility is non-exactness and his idea relates many examples of noninvertible extensions to the problem of coincidence for certain tensor product norms. In the present paper we develop this idea to define the asymptotic tensor norm and study this norm in hope to learn more about extensions. Letbe an extension of A by B, i.e., a short exact sequence of C * -algebras. We always assume that B is stable, i.e., B ⊗ K ∼ = B, where K is the C * -algebra of compact operators. We also assume all C * -algebras to be separable with some obvious exceptions like the algebra L(H ) of bounded operators on a Hilbert space H . The extension (1) is called split if there is a * -homomorphism s : A → E that is a right inverse for the surjection p, i.e., p • s = id A . Due to the stability of B, one can fix an isomorphism M 2 (B) ∼ = B, which makes it possible to define a direct sum of two extensions of A by B. An extension of A by B is called Mathematics Subject Classification (2000): Primary 46L06; Secondary 46L05, 46L80.

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A C ∗ -algebra  for which Ext() is not a Group

Cited by 28 publications

References 7 publications

A new application of random matrices: Ext(C_red^∗(F₂)) is not a group

A new application of random matrices: Ext(C_red^∗(F₂)) is not a group

Essentially Normal Operators

On the asymptotic tensor C*-norm

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A C ∗ -algebra  for which Ext() is not a Group

Cited by 28 publications

References 7 publications

A new application of random matrices: Ext(Cred∗(F2)) is not a group

A new application of random matrices: Ext(Cred∗(F2)) is not a group

Essentially Normal Operators

On the asymptotic tensor C*-norm

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A new application of random matrices: Ext(C_red^∗(F₂)) is not a group

A new application of random matrices: Ext(C_red^∗(F₂)) is not a group