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On a method of determining supports of Thoma’s characters of discrete groups. (English) Zbl 0882.43007

Let \(G\) be a discrete group, and let \(K(G)\) denote the convex set of all normalized, positive definite, conjugation invariant functions on \(G\). The elements in \(E(G)\), the set of extreme points of \(K(G)\), are called characters of \(G\). The GNS-representation \(\pi_\alpha\) associated to \(\alpha\in E(G)\) is a factorial representation of \(G\) and hence of \(C^*(G)\), the group \(C^*\)-algebra. At least when \(G\) is countable, the kernel of \(\pi_\alpha\) is a primitive ideal of \(C^*(G)\), and the resulting map from \(E(G)\) into the primitive ideal space of \(C^*(G)\) is to a large extent responsible for the current interest in \(E(G)\).
The question of where a character vanishes has been studied, in particular for nilpotent groups, in various papers (the author’s reference list is by no means complete). In the paper under review, a classical result of Umegaki on the uniqueness of conditional expectations in von Neumann algebras with finite normal trace is applied to prove that \(\alpha(x)= 0\) for \(x\in G\) and \(\alpha\in E(G)\), under suitable assumptions on both \(\alpha\) and \(x\). It seems that this result is closely related to an earlier result due to the author [Math. Ann. 240, 97-102 (1979; Zbl 0405.43004)].

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
43A40 Character groups and dual objects

Citations:

Zbl 0405.43004
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