Centralization of positive definite functions, weak containment of representations and Thoma characters for the infinite symmetric group
For a discrete group \(G\) with identity \(e\), let \(P(G)\) denote the convex set of all positive definite functions \(f\) on \(G\) with \(f(e)=1\) and \(\text{Tr}(G)\) the subset of all traces on \(G\), that is, the conjugation invariant functions in \(P(G)\). The Gelfand-Naimark-Segal construction associates to each \(f\in P(G)\) a cyclic unitary representation \(\pi_f\) of \(G\), and \(\pi_f\) is irreducible if \(f\) is indecomposable (equivalently, an extreme point of \(P(G))\) and \(\pi_f\) is a factor representation if \(f\) is an extreme point of \(\text{Tr}(G)\) (a so-called character of \(G)\). In many cases, the characters can be described explicitly, whereas there is no way to determine the irreducible representations (up to unitary equivalence). For instance, the set of characters of the infinite symmetric group \(S_\infty= \cup^\infty_{n=1}S_n\) was determined by \textit{E. Thoma} [Math. Z. 85, 40--61 (1964; Zbl 0192.12404)] and has since attained a lot of attention. In the paper under review the author continues his studies of positive definite functions and unitary representations of \(S_\infty\). In order to provide a link between \(P(S_\infty)\) and \(\text{Tr}(S_\infty)\), he introduces the following centralization process. Let \((G_n)_n\) be an increasing sequence of finite subgroups of a group \(G\) with \(\bigcup^\infty_{n=1}G_n=G\). To any \(f\in P(G)\), associate the sequence \((f_n)_n\subseteq P(G)\) defined by \(f_n(g)=\frac {1}{|G_n|}\sum_{a \in G_n}f(a^{-1}ga)\), \(g\in G\), \(n\in\mathbb{N}\). If this sequence converges pointwise, then the limit is a trace. The main body of results of the paper consists of, partly fairly technical and intricate, exploitation of this centralization process. \textit{M. Bożejko} [Studia Math. 95, 107--118 (1989; Zbl 0714.43007)] and \textit{M. Bożejko} and \textit{R. Speicher} [Math. Ann. 300, 97--120 (1994; Zbl 0819.20043)] have shown that the following families of functions on \(S_\infty\) are contained in \(P(S_\infty)\): (1) \(f(g)=r^{|g|}\), \(-1\leq r\leq 1\), where \(|g|\) denotes the usual length of the permutation \(g\); (2) \(f'(g)= q^{\|g\|}\), \(0\leq q\leq 1\), where \(\|g\|\) is the block length of \(g\); (3) \(f''(g)=\text{sgn}(g)q^{\|g\|}\). Theorems 1 and 2 of the paper state that, taking \(G_n=S_n\), in all three cases the centralizing sequence converges pointwise to the Dirac function \(\delta_e\), which is the character associated to the left regular representation \(\lambda_{S_\infty}\) of \(S_\infty\). Consequently, all of the representations \(\pi=\pi_f\), \(\pi_{f'}\), \(\pi_{f''}\) weakly contain \(\lambda_{S_\infty}\). It is an open problem whether the positive definite functions \(f\), \(f'\) and \(f''\) are indecomposable. If they are, then since \(S_\infty\) is amenable, the results mean that each such \(\pi\) forms a dense point in the dual space \(\widehat S_\infty\) of \(S_\infty\). In a previous paper [J. Math. Kyoto Univ. 31, 495--541 (1991; Zbl 0752.22001)], the author has constructed a large family of irreducible unitary representations of \(S_\infty\), obtained by inducing irreducible representations of a wreath product type subgroup of \(S_\infty\) up to \(S_\infty\). Utilizing the centralization process, it is proved that any of these irreducible induced representations weakly contains either \(1_G,\) the trivial representation of \(S_\infty\), or \(\text{sgn}_{S_\infty}\) or \(\lambda_{S_\infty}\) (Theorem 7). It is also shown that the Thoma characters can be obtained as centralizations of coordinate functions of these irreducible representations.
scientific article; zbMATH DE number 5010140