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The duality theorem for inoncommutative 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.5 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 <, from each equivalence class of irreducible bounded representations of G. Denote the totality of (finite) linear combinations of the elements of all the Z2K over the field Q of complex numbers by 9T, and that of the elements in a fixed Z)K by SK,. 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 zCG on a function f(x) on G being defined by z .f(x) =f(z'x), f(x) z =f(xz-1). Let sK be the degree of Z)K. S, is directly de-

Remark on the duality for noncommutative compact groups