Lectures on Amenability

Runde, Volker

doi:10.1007/b82937

Cited by 324 publications

(361 citation statements)

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“…In Banach algebra theory, this led to the discovery and analysis [12] of amenable Banach algebras, which have trivial cohomology with coefficients in dual modules. While the trivial cohomology groups do not provide a useful classification tool for amenable Banach algebras, it follows from the trivial cohomology that many other equations over the Banach algebra can be solved and this leads to other interesting properties [15].…”

Section: Introductionmentioning

confidence: 99%

The Cyclic and Simplicial Cohomology of the Bicyclic Semigroup Algebra

Gourdeau

White

2010

The Quarterly Journal of Mathematics

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Abstract. Let A = 1 (B) be the semigroup algebra of B, the bicyclic semigroup. We give a resolution of ∞ (B) which simplifies the computation of the cohomology of 1 (B) dual bimodules. We apply this to the dual module ∞ (B) and show that the simplicial cohomology groups H n (A, A ) vanish for n ≥ 2. Using the Connes-Tzygan exact sequence, these results are used to show that the cyclic cohomology groups HC n (A, A ) vanish when n is odd and are one-dimensional when n is even (n ≥ 2).

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Section: Introductionmentioning

confidence: 99%

The Cyclic and Simplicial Cohomology of the Bicyclic Semigroup Algebra

Gourdeau

White

2010

The Quarterly Journal of Mathematics

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“…A contractible Banach algebra is unital, and it is conjectured that a Banach algebra A is contractible only when A is finite-dimensional. This is true for C * -algebras (indeed, for closed subalgebras of B(H) for a Hilbert space H, see [25]) and for B(E) when E has, for example, the approximation property (see [30,Section 4…”

Section: Ultra-amenabilitymentioning

confidence: 97%

“…By [30,Corollary 6.4.28] and [30,Corollary 6.4.29], we have that when a C * -algebra is amenable, it has the (metric) approximation property. Hence, if (1) holds, then for any ultrafilter U, we have that (A) U has the approximation property.…”

mentioning

confidence: 99%

Amenability of ultrapowers of Banach algebras

Daws

2009

Proceedings of the Edinburgh Mathematical Society

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We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of A is Arens regular, and give some evidence that this is so if and only if A is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebra. We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of A is amenable. We provide an abstract characterisation in terms of something like an approximate diagonal, and consider when every ultrapower of a C * -algebra, or a group L 1 -convolution algebra, is amenable.

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“…Undefined notation can be found in the above references and [24]. For definitions of the variants of amenability discussed in this paper we refer to [22] and [23].…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, from equation (2.1) T f is non-negative and norm one in L 1 (G/N ) whenever f is non-negative and norm one in L 1 (G). A locally compact group, H, is inner amenable exactly when there is a net f α ∈ L 1 (H) of norm one non-negative elements satisfying [22,Ex. 4.4.4], so the lemma follows.…”

Section: Lemma 35 If G Is Inner Amenable Then So Is G/z(g)mentioning

confidence: 99%