Gromov’s translation algebras, growth and amenability of operator algebras. (English) Zbl 1157.37004
The authors survey various relations between translation algebras, growth and amenability of finitely generated algebras and operator algebras. The relation between translation algebras of groups and amenability was discovered by G. Elek, who proved [Proc. Am. Math. Soc. 125, No. 9, 2551–2553 (1997; Zbl 0890.20027)] that the group \(G\) is amenable if and only if \(1\neq 0\) in \(K_0(T(G))\), the Grothendieck group of the translation algebra of \(G\). The arguments of Elek are reviewed in the present paper, in particular the existence of a trace in \(T(G)\) for an amenable group \(G\).
The classical notion of amenability has been extended to finitely generated algebras by Elek and to finitely generated dense \(*\)-subalgebras of C\(^*\)-algebras by the second author. The authors call a C\(^*\)-algebra \(\mathcal A\) finitely generated if it admits a dense unital finitely generated \(*\)-subalgebra, and \(\mathcal A \) is Følner if such a \(*\)-subalgebra is amenable. The authors show that Følner C\(^*\)-algebras are weakly filterable and weakly hypertracial, and they also prove that a countably infinite group \(G\) is amenable if and only if the group algebra \(\mathbb C [G]\) is amenable. They remark that this result has been extended to any group algebra \(\mathbb F [G]\) (where \(\mathbb F\) is any field), by L. Bartholdi [Isr. J. Math. 168, 153–165 (2008; Zbl 1167.43001)]. They close the paper with a list of open problems on Følner sequences and growth of algebras.
The classical notion of amenability has been extended to finitely generated algebras by Elek and to finitely generated dense \(*\)-subalgebras of C\(^*\)-algebras by the second author. The authors call a C\(^*\)-algebra \(\mathcal A\) finitely generated if it admits a dense unital finitely generated \(*\)-subalgebra, and \(\mathcal A \) is Følner if such a \(*\)-subalgebra is amenable. The authors show that Følner C\(^*\)-algebras are weakly filterable and weakly hypertracial, and they also prove that a countably infinite group \(G\) is amenable if and only if the group algebra \(\mathbb C [G]\) is amenable. They remark that this result has been extended to any group algebra \(\mathbb F [G]\) (where \(\mathbb F\) is any field), by L. Bartholdi [Isr. J. Math. 168, 153–165 (2008; Zbl 1167.43001)]. They close the paper with a list of open problems on Følner sequences and growth of algebras.
Reviewer: Pere Ara (Bellaterra)
MSC:
| 37B15 | Dynamical aspects of cellular automata |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 20F65 | Geometric group theory |
| 46L05 | General theory of \(C^*\)-algebras |
Keywords:
translation algebra; Følner sequence; Følner condition; amenability; finitely generated *-algebras; C\(^*\)-algebrasReferences:
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