Amenability of weighted discrete convolution algebras on cancellative semigroups

Grønbæk, Niels

doi:10.1017/s0308210500022344

Cited by 24 publications

(10 citation statements)

References 6 publications

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“…We shall use the following notation introduced by Grønbaek in [9]. For s, t ∈ S we define the sets st −1 = {u ∈ S: ut = s} and t −1 s = {u ∈ S: tu = s}.…”

Section: Cancellativitymentioning

confidence: 99%

Homological properties of modules over semigroup algebras

Ramsden

2010

Journal of Functional Analysis

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Let S be a semigroup. In this paper we investigate the injectivity of 1 (S) as a Banach right module over 1 (S). For weakly cancellative S this is the same as studying the flatness of the predual left module c 0 (S). For such semigroups S, we also investigate the projectivity of c 0 (S). We prove that for many semigroups S for which the Banach algebra 1 (S) is non-amenable, the 1 (S)-module 1 (S) is not injective. The main result about the projectivity of c 0 (S) states that for a weakly cancellative inverse semigroup S, c 0 (S) is projective if and only if S is finite.

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“…We shall use the following notation introduced by Grønbaek in [9]. For s, t ∈ S we define the sets st −1 = {u ∈ S: ut = s} and t −1 s = {u ∈ S: tu = s}.…”

Section: Cancellativitymentioning

confidence: 99%

Homological properties of modules over semigroup algebras

Ramsden

2010

Journal of Functional Analysis

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“…We shall thus show that (1) implies (3). That (3) implies (2) follows from the equivalence of (3) and (2) established in [6,Theorem 3.2].…”

Section: Furthermore If a Is Arens Regular Then These Conditions Armentioning

confidence: 81%

Connes-amenability of bidual and weighted semigroup algebras

Daws¹

2006

MATH. SCAND.

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We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σ W C-virtual diagonal, as introduced in [13], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C * -algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras l 1 (S, ω), we have that l 1 (S, ω) is Connes-amenable (with respect to the canonical predual c 0 (S)) if and only if l 1 (S, ω) is amenable, which is in turn equivalent to S being an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Grønbaek in [6], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like C * -algebras.

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“…It is easy to see that −1 ). By w * -continuity of the module actions of M(G, w) on L ∞ (G, w −1 ) * , we have the following lemma, its proof is similar to [Grø1,Lemma 3.1] and is omitted.…”

Section: Normal Virtual Diagonals For Weighted Measure Algebrasmentioning

confidence: 98%

Connes-amenability, super-amenability and normal, virtual diagonal for some Banach algebras

Mahmoodi

2007

imf

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We prove that L p (G) is super-amenable if and only if G is finite. We denote the space of all weakly almost periodic functions on a locally compact group G, by W AP (G). We show that the dual Banach algebra W AP (G) * is Connes-amenable if and only if G is amenable. However, it is proved in [ Run 5], but our proof is different. If G is amenable and ω is a diagonally bounded weight function on G, then M(G, Ω) has a normal, virtual diagonal.

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Amenability of weighted discrete convolution algebras on cancellative semigroups

Cited by 24 publications

References 6 publications

Homological properties of modules over semigroup algebras

Homological properties of modules over semigroup algebras

Connes-amenability of bidual and weighted semigroup algebras

Connes-amenability, super-amenability and normal, virtual diagonal for some Banach algebras

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