On the maximal Hilbert algebras
Osamu Takenouchi · Tohoku Mathematical Journal · 1951
H.Nakano [42° has extended the results of W. Ambrose concerning his "proper H-algebras" (see [1]), by introducing the notion of "Hubert algebras".In his paper, he showed, among others, that, to each Hubert algebra, there exists a distinguished extension of it (maximal extension) which cannot be extended properly in any way ([4; Theorem 2.2]).After he told to the author this result, W. Ambrose's second paper [2] concering "H-systems" has appeared.Considering the inner relations of these notions, the author was able to show that every Hubert algebra can be extended uniquely to a maximal one, and the considerations of maximal Hubert algebras and H-systems are the same thing, /. e. the "bounded algebra" of an H-system is no other than our maximal Hubert algebra.The structure of this algebra was also determined completely in some extent (i.e., except that we have to introduce the separability assumption at a certain point) by the use of the F. J. Murray and J. von Neumann's theory on rings of operators.In this paper we shall concern with the existence and unicity of a given Hubert algebra and also some principal properties of the maximal Hubert algebras deduced from it.As to the structure, we shall only give the results, as the proof is considerably long though the method is not so new.The fundamentals for the proof will be mentioned.The notions and notations in [4; § 1] will be used freely.The author expresses here his grateful thanks to Prof. H. Nakano for his kind guide and advice to make him obtain these results.