The coshape category as an imprimitivity algebra (Q1082565)

It was shown by the second author [Cah. Topologie Géom. Différ. 22, 105-110 (1981; Zbl 0453.18006)] how to obtain a special case of Mackey's induced representation theorem by means of an easy categorical argument for the case of a subgroup of finite index in a discrete group G. The purpose of the paper under review is to give such a version in the case where H is a closed subgroup of a separable locally compact group G. The authors consider the representation of G as an enriched functor from the measure algebra M(G) to the enriched base category (satisfying the additional condition \(L^ 1(G) \otimes_{L^ 1(G)} V\simeq V)\) of non- degenerate representations of the algebra \(L^ 1(G)\). The \(M(G)-M(H)^ o\)-bimodule \(\tilde K=M(G) \otimes_{L^ 1(H)} L^ 1(H)\) is a Banach algebra with approximative right unity and can be considered as a distributor \(M(G)\nrightarrow M(H)\), the coshape category of which is the endomorphism algebra \(S=End_{M(H)^ 0}(\tilde K)\) operating on \(\tilde K\) by right centralizers. In a natural way one constructs a canonical coshape functor M(G)\(\to S\) associating to each element of M(G) a left multiplication by such an element. If V is an \(L^ 1(G)\)-essential functor, the authors construct its system of imprimitivity \(\bar V\) in form of a functor \(\bar V:\) \(S\to {\mathcal B}an\) satisfying \(V\simeq L^ 1(G) \otimes_{L^ 1(G)} (\bar VD)\) and \(\tilde K\otimes_{\tilde K} (\bar VQ)\simeq \bar V\). The main result of the paper says that V is induced for some \(L^ 1(H)\)-essential functor U: M(H)\(\to {\mathcal B}an\) if and only if V admits a system of imprimitivity, and this system determines U uniquely up to equivalence.