Relation between right and left involutions of a Hilbert algebra
In: Proceedings of the American Mathematical Society, Jg. 102 (1988), S. 57-58
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Zugriff:
Existence of a densely defined right involution in a Hilbert algebra implies existence of a left involution. H*-algebras were introduced by W. Ambrose [1] to characterize Hilbert-Schmidt operators. This notion was generalized by M. F. Smiley [7], who showed that the structure theorems are valid also for a right H*-algebra (a Hilbert algebra whose involution x -> xr satisfies the condition "(yx, z) = (y, zxr)" but not the condition "(xy, z) = (y, XrZ),,). (Hilbert algebra here is a Banach algebra with a Hilbert space norm.) The author showed in [5, Theorem 2] that a proper right H*-algebra is also a left H*-algebra, i.e. it also has another involution x -xl (a left involution) which satisfied the condition "(xy, z) = (y, x'z)."1 In this paper we shall show that the same is true also for the case when the involution x __ Xr (the right involution) is defined on a dense subset only. DEFINITION. Let A be a Hilbert algebra (A is a Banach algebra with a scalar product ( , ) such that (x, x) = IZXH12). We shall say that A is a weak right H*algebra if there is a dense subset Dr of A with the property that for each x E Dr there is some Xr such that (yx, z) = (y, ZXr) for all y, z E A. We define weak left H*-algebra in a similar fashion. The algebra A is said to be proper if each Xr is unique, i.e. A has a right involution x __ Xr, defined on a dense subset. Note that A is proper if r(A) = {u E A: Au = (0)} consists of zero alone. Algebra A in Example 2, p. 54, of [5] is an example of weak right (as well as left) H*-algebra. Also it is easy to show that each weak right H*-algebra is a right complemented algebra. THEOREM. Each proper weak right H*-algebra A is a proper weak left H*algebra. PROOF. Note that the involution x , xr of A, defined on a dense subset Dr, is closable, i.e. the closure of its graph is a graph of some mapping: it is easy to show that if xZ E Dr, xZ -0 and Xr -n y E A, then y = 0 (see end of section 8 in ?5, Chapter I of [3]). Now replace the scalar product (, ) of A and the multiplication Ax of members x of A and complex number A by [, ] and A o x respectively, where these new Received by the editors November 3, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46K15; Secondary 46H20, 47C10.
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Relation between right and left involutions of a Hilbert algebra
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Autor/in / Beteiligte Person: | Saworotnow, P. P. |
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Zeitschrift: | Proceedings of the American Mathematical Society, Jg. 102 (1988), S. 57-58 |
Veröffentlichung: | American Mathematical Society (AMS), 1988 |
Medientyp: | unknown |
ISSN: | 1088-6826 (print) ; 0002-9939 (print) |
DOI: | 10.1090/s0002-9939-1988-0915715-0 |
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