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On the spectrum of the sum of generators of a finitely generated group. II. (English) Zbl 0846.46036

We consider a finitely generated group \(\Gamma\), and the two usual \(C^*\)-algebras coming along with \(\Gamma\): the full \(C^*\)-algebra \(C^* (\Gamma)\) associated with the universal representation \(\pi_{\text{un}}\) of \(\Gamma\) on \({\mathcal H}_{\text{un}}\); and the reduced \(C^*\)-algebra \(C^*_r (\Gamma)\) associated with left regular representation \(\lambda\) of \(\Gamma\) on \(\ell^2 (\Gamma)\).
Given a finite set \(S\) of generators of \(\Gamma\) (here we mainly deal with the non-symmetric case \(S\neq S^{-1}\)), we set: \[ h= {1\over {|S|}} \sum_{s\in S} s\in C^* (\Gamma). \] In part I [Isr. J. Math. 81, No. 1-2, 65-69 (1993; Zbl 0791.43008)], we initiated a study of the spectral properties of \(h\). For example, we proved that the intersection of the spectrum \(\text{Sp } h\) with the unit circle \(\mathbb{T}\) equals either \(\mathbb{T}\) or the group \(C_n\) of \(n\)-th roots of 1, for some \(n\geq 1\). Concerning \(\lambda (h)\), there is the archetypal result of M. M. Day [Ill. J. Math. 8, 100-111 (1964)]: \(\Gamma\) is amenable if and only if 1 is in the spectrum of \(\lambda (h)\), if and only if the spectral radius of \(\lambda (h)\) is 1. In the present paper, we examine more closely how properties of \(\Gamma\) and its representation theory are reflected in properties of \(\text{Sp }h\) and \(\text{Sp } \lambda (h)\).

MSC:

46L05 General theory of \(C^*\)-algebras
43A55 Summability methods on groups, semigroups, etc.

Citations:

Zbl 0791.43008
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