Remark on the Duality for Noncommutative Compact Groups

The duality theorem for noncommutative compact groups was established by Tannaka,1 and later, but independently, by Krein.2 Different proofs have been given by Bochner and Yosida.3 The purpose of the present note is to supplement the theorem by a duality between (closed) subgroups and certain representation-theoretically defined systems. It provides also an extension of Tannaka's theorem itself. As a matter of fact, it is more or less known that the subgroups are in 1-1 correspondence with, say, left-invariant subrings with conjugation of the ring of coefficients of (irreducible) representations.4 But we propose to make the dual objects to be associated with subgroups free from the properties concerned with the group operation, such as left-translations, and obtain purely representationtheoretical dual systems.6 Our intention is also to have a formulation which gives directly the subgroups of a character group in the abelian case. Thus our work has no novelty in analytical respects, and our concern lies mainly in its formulation. 1. Let G be a topological group. We take a representation 25 K from each equivalence class of irreducible bounded representations of G. Denote the totality of (finite) linear combinations of the elements of all the £)« over the field Í2 of complex numbers by 3Î, and that of the elements in a fixed 3), by 9Ϋ. They are rings, here with respect to convolution, and indeed two-sided ideals in the ring R of all almost periodic functions. They are also two-sided G-moduli, operations of zEG on a function f(x) on G being defined by z-f(x) =f(z~1x),f(x) ■z=f(xz~1). Let s, be the degree of ÜD«. 9Ϋ is directly de-