# On topological groups

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Journal of <-he Institute of Polytechnics, Osaka C ity University, V o l. 3, No 1 -2 , Series A On topological groups. By Shin-ichi M a t s u s h it a (received September 15, 1952) 0. Introdnction. In this paper, we shall deal with arbitrary topological groups by means of their Marko f f -extensions: the definition of an Markoff- extension is given in Section I. Generally speaking, though the representation theory in matric-aIgebras plays an important role in studying topological groups,i> it becames occationally meaning-less for some type of groups, which have no usual (n®n-trivial) representations; as well known, minimally almost periodic groups are those. However, the Markoff-extension seems to be useful for any topological groups. Section 2 is devoted to an exposition of the relation between the represen tation of a topological group and those of its Markoff-extension. In Section 3, we shall concern the duality theorem of any topological groups, which we would rather call the co-duality theorem- Our theorem coincides with the famous one of Tannaka and Krein^ ^ in maximally almost periodic cases at all, but even if a group is minimally almost periodic, ours may remain still useful. This duality theorem is, on the other hand, considered as the representation theorem in B-algebra, and the process from Theorem 4 to Theorem 5 gives one proof for the Tannaka-Krein’s duality theorem. The space of almost periodic functions is considerd as a commutative B"^ - algebra. This investigation is done in Section 4. Finally, in Section 5, we shall try the theorem of K. Iwasawa^ ^ concerning the group-rings as an interesting application of Markoff-extensions. 1. Preliminary theorem. We begin with the noted theorem of A. Markoff and S. Kakutani on free topological groups.^ ^ That is stated as follows: For any completely regular topological space r» there exists a free topological group F with the following properties ; 1) Recently, Banach representation theo

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