Group representations in non-archimedean Banach spaces
摘要
This paper deals "with continuous representations of locally compact groups G into non-arc hi me dean Banach spaces E. In order that G has sufficiently many of such representations G must "be totally disconnected, which we assume from nov on. If G carries a K-valued Haar measure (where K is the (non-archime dean valued) scalar field) we have a 1-1 correspondence between the continuous representations of G and those of the group algebra L(G). If G is compact, then L(G) can be decomposed as a direct sum of full matrix algebras over skew fields (Theorem 2.5), which yields as a corollary that every irreductible continuous representation of G is equivalent to a minimal left ideal of L(G). Further, all continuous representations of G can be classified (Theorem 2.8). The theory for compact groups as it is given here is a generalization of the results of [2]. 'If G is locally compact and torsional (i.e., every compact set is contained in a compact subgroup) the results are satisfactory : G then has sufficiently many continuous irreductible representations ; every twosided closed ideal in L(G) is the. intersection of maximal left ideals (Theorem 3.1 and corollaries). About non-torsional G little is known. 1. The Banach algebra L(G). K is a field with a (possibly trivial) non-Archimedean valuation | | such that K is complete relative to the metric induced by | | . The residue class field of K is k. If A. C K, | X [ ^ 1 then X. denotes the corresponding element of k. The characteristic of k is p (which may be 0). G is a totally disconnected locally compact group, Jl the collection of all open compact subgroups of G, ^ the ring of sets generated by the left cosets of the elements of Jf . It is known that ^> consists of the compact open subsets of G