A remark on mutually disjoint irreducible decompositions of the regular representation of a group (Q1323257)

The main result of the author's previous work [ibid. 31, No. 4, 1105-1114 (1991; Zbl 0798.22006)] is generalized to the case of more than countably infinite index set \(A\) of nonabelian \(G_ \alpha\), by using direct integral of non-separable Hilbert spaces. In particular, let \(G = \prod_{\alpha \in A}' G_ \alpha\) be the restricted direct product of finite groups \(G_ \alpha\). Let \(d\) be the cardinality of \(A^{non} = \{\alpha \in A \mid G_ \alpha \text{ is non-commutative}\}\). Then there exist \(2^ d\) mutually disjoint irreducible decompositions of the left regular representation \(\ell\) of \(G\) of type \[ \{\ell, L^ 2(G)\} = {^{\{\Phi_ h\}}} \int^ \oplus_ \Xi \{{^{\{F_ h(\xi)\}}}\int^ \oplus_{K(\xi)} \{\rho_{\xi,k}^{{\mathcal O}(\xi)}, {\mathbf V}(\xi,{\mathcal O} (\xi);k)\} d \nu_ \xi(k)\} d\mu (\xi). \]