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Search: MSC category 43A20 ( $L^1$-algebras on groups, semigroups, etc. )

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1. CMB Online first

Alaghmandan, Mahmood; Choi, Yemon; Samei, Ebrahim
 ZL-amenability Constants of Finite Groups with Two Character Degrees We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009). Keywords:center of group algebras, characters, character degrees, amenability constant, Frobenius group, extraspecial groupsCategories:43A20, 20C15

2. CMB 2010 (vol 53 pp. 447)

Choi, Yemon
 Injective Convolution Operators on l∞(Γ) are Surjective Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results. Categories:43A20, 46L05, 43A22

3. CMB 2008 (vol 51 pp. 60)

Janzen, David
 F{\o}lner Nets for Semidirect Products of Amenable Groups For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where $(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and $(B_\beta)$ is any F{\o}lner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided. Categories:22D05, 43A07, 22D15, 43A20

4. CMB 2007 (vol 50 pp. 56)

Gourdeau, F.; Pourabbas, A.; White, M. C.
 Simplicial Cohomology of Some Semigroup Algebras In this paper, we investigate the higher simplicial cohomology groups of the convolution algebra $\ell^1(S)$ for various semigroups $S$. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than $a$ for some integer $a$. Our results are of two types: in some cases, we show that some cohomology groups are $0$, while in some other cases, we show that some cohomology groups are Banach spaces. Keywords:simplicial cohomology, semigroup algebraCategory:43A20

5. CMB 2004 (vol 47 pp. 445)

Pirkovskii, A. Yu.
 Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators For a locally compact group $G$, the convolution product on the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang \cite{Neuf_PhD}. We study homological properties of the convolution algebra $\nN(L^p(G))$ and relate them to some properties of the group $G$, such as compactness, finiteness, discreteness, and amenability. Categories:46M10, 46H25, 43A20, 16E65

6. CMB 1997 (vol 40 pp. 133)

Blackmore, T. D.
 Derivations from totally ordered semigroup algebras into their duals For a well-behaved measure $\mu$, on a locally compact totally ordered set $X$, with continuous part $\mu_c$, we make $L^p(X,\mu_c)$ into a commutative Banach bimodule over the totally ordered semigroup algebra $L^p(X,\mu)$, in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from $L^p(X,\mu)$ into its dual module and in particular shows that if $\mu_c$ is not identically zero then $L^p(X,\mu)$ is not weakly amenable. We show that all bounded derivations from $L^1(X,\mu)$ into its dual module arise in this way and also describe all bounded derivations from $L^p(X,\mu)$ into its dual for \$1 Categories:43A20, 46M20