Complementation for Right Ideals in Generalized Hilbert Algebras

Let 51 be a generalized Hilbert algebra and let 5 be a closed right ideal of S. Let J-1 denote the pre-Hilbert space orthogonal complement of 3 inï. The problem investigated in this paper is: for which algebras % is it true that SI = J ffi ¡f^for every closed right ideal 3"of 8? In the case that % is achieved, a slightly stronger property is characterized and these characterizations are then used to investigate some interesting examples. Introduction. It is known that for full Hilbert algebras, the complementation property holds for both closed left and right ideals. This fact is due to Yood [7]. For achieved generalized Hilbert algebras the problem is more subtle although not difficult. A number of characterizations of a strong complementation property are given below. Perhaps the most useful of these is Theorem 1.11. Several cases are analyzed using this theorem. I would like to thank T. W. Palmer for suggesting this problem to me. Definitions and notation. The definitions and notation of M. Takesaki's printed notes [6] will usually be used without reference. In particular, if 21 is a generalized Hilbert algebra, its involution will be noted by # and for £ £ 21, ni¿f) will denote the unique continuous extension to the completion of 21 of the operator "left multiplication by £". lí b is the conjugate linear adjoint of # and if % is in the domain of la then one can define an operator on 21 by: 7t'(JI)£ = ni^K for all £ £ 21. If 7r'(')l) is bounded on 21 then the unique bounded extension of tt'ÖZ) to the completion of 21 is denoted by tt 01) also. The set of all ÜI in the domain of \> such that 7r'0I) is bounded will be denoted by 21'. By Lemma 3.3 of [6], 21' is an algebra with involution ¡> such that (2I')2 is dense in the domain of !>. In particular, 21 is dense in the completion, K(2I) of 21. If t; is any element of K(2I) one can define an operator tA.0 with domain 21' by: tritffñ. = 7r'0í)f for all V. £ 21'. If tt(£) is bounded on 21' then the unique bounded extension of ni£) to K(2I) is denoted by trig) also. The set of all elements f of K(2I) such that nig) is bounded will be denoted by K... Received by the editors January 4, 1973. AMS (MOS) subject classifications (1970). Primary 46K15, 46L10; Secondary 46K05, 46H10. Copyright © 1973, American Mathematical Society 409 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use