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On the structure of maximal Hilbert algebras

Takenouchi Osamu

In our previous paper [12] , we considered the unicity problem of the maximal extension of a given Hilbert algebra, and established the: most fundamental property of a maximal Hilbert algebra ([12; Theorem 2J). We argued also the decomposition of maximal Hilbert algebras with respect to their centres, and, on doing it, we noticed that there exist two different types of them, i.e., the simple ones and the purely non-simple ones. The decomposition theorem to these types was given in [12; Theorem
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... J with a sketch of the proof, and we announced that. further arguments concerning the decomposition would be given in some other paper. The chief aim of this paper is to give it. In §1 a short cut of the known results is given, and §2 is devoted tothe more detailed exposition of the decomposition of a given Hilbert. algebra into the simple components and the purely non-simple component. A simple Hilbert algebra is one for which the algebras of left and right multiplication constitute a couple of factors in the sense of F. J. Murray and J. von Neumann ([4J), and we are led naturally to make use of their theory. The main problem here is how the dimensionality functional can be expressed by means of the terms of the Hilbert algebra. These are discussed in §3. The reduction theory of a. purely non-simple Hilbert algebra into simple ones is given in §4. This idea, though here only applied to the separable case, can be applied in the non-separable case. But in the most general case we do not yet succeed in proving simplicity character and that will be a future problem. (for any x E m, r f :> 0 is fixed), (for any x E ~r, r, >-0 is fixed) is satisfied ([12, Theorem 1J). The operator, which assignes to every x E '2l the element S:r-f, will be denoted as TJ, and similarly SJ is defined. Thus (1.1) and (1.1') can be restated that TJ or SJ is bounded. Thus to give the complete definition of a maximal Hilbert algebra, we -have to add one more axiom: " (vi) If TJ or SJ is bounded for an! E ~, then! must be in ~," to the axioms (i)•••(v) listed above ([12; Theorem 2J). We shall use later the following 2

doi:10.18926/mjou/33718
fatcat:noetklrifbcppikpnmsrnit724