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Search: MSC category 20E05 ( Free nonabelian groups )

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1. CMB 2011 (vol 54 pp. 654)

Forrest, Brian E.; Runde, Volker
Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

Keywords:amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotent
Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25

2. CMB 2010 (vol 54 pp. 3)

Bakonyi, M.; Timotin, D.
Extensions of Positive Definite Functions on Amenable Groups
Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented.

Categories:43A35, 47A57, 20E05

3. CMB 2003 (vol 46 pp. 299)

Tomaszewski, Witold
A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group
We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

Categories:20E05, 20F12, 20F05

4. CMB 1997 (vol 40 pp. 352)

Liriano, Sal
A New Proof of a Theorem of Magnus
Using naive algebraic geometric methods a new proof of the following celebrated theorem of Magnus is given: Let $G$ be a group with a presentation having $n$ generators and $m$ relations. If $G$ also has a presentation on $n-m$ generators, then $G$ is free of rank $n-m$.

Categories:20E05, 20C99, 14Q99

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