Induced factor representations of discrete groups and their types (Q1316372)

This paper describes, under certain technical conditions, for an induced representation \(\pi = \text{ind}^ G_ K \sigma\) of a discrete group \(G\), the structure of the center \(\pi(G)' \cap \pi(G)''\). This description gives good criteria for \(\text{ind}^ G_ K \sigma\) being a factor representation, especially in the case where \(\sigma\) is irreducible and \(H\) is normal. Let us just present one of the results of the paper. Let \({\mathcal V}_ H = \{g\) in the normalizer \({\mathcal N}(H)\) of \(H\), such that \(\sigma^ g \simeq \sigma\}\) and let for any \(g \in G\), \({\mathcal R}(g) = \{V \in {\mathcal L}({\mathcal H}^ \sigma)\), such that \(\sigma(g^{-1} hg)V = V \sigma(h)\), \(\forall h \in H \cap g^{-1} Hg\), \(\sum_{h \in \Theta_{g^{-1}}} \| V \sigma(h^{-1})v \|^ 2 < \infty\), \(\sum_{h \in \Theta_ g} \| V * \sigma(h^{-1}) v\|^ 2 < \infty\), \(\forall v \in {\mathcal H}^ \sigma\}\). Here \(\Theta_ g := H/(H\cap gH g^{-1})\), \(\Theta_{g^{-1}} := H/(H\cap g^{-1}Hg)\). We say that the conjugacy class \([kH] = \{m^{-1} kmH : m \in {\mathcal V}_ H\}\) (for \(k \in {\mathcal V}_ H\)), satisfies property (\^R), if \(\sigma(k^{-1} m^{-1} km) = B^*_ k B^*_ m B_ k B_ m\) holds for every \(m \in {\mathcal V}_ H\) with \(m^{-1} kmH = kH\) and every unitary \(B_ k \in {\mathcal R}(k)\), \(B_ m \in {\mathcal R}(m)\). Let \(\Omega\) denote a system of representatives of the conjugacy classes of \({\mathcal V}_ H\) that are finite and that satisfy condition (\^R). If \({\mathcal V}_ H \subset {\mathcal N}(H)\) and \(\sigma\) is irreducible, the linear space spanned by the operators \(T^{[g]}\), \(g \in \Omega\), is weakly dense in the center of \(\text{ind } \sigma(G)''\).