The traceclass of a full Hilbert algebra
HTML articles powered by AMS MathViewer
 by Michael R. W. Kervin PDF
 Trans. Amer. Math. Soc. 178 (1973), 259270 Request permission
Abstract:
The traceclass of a full Hilbert algebra A is the set $\tau (A) = \{ xyx \in A,y \in A\}$. This set is shown to be a $\ast$ideal of A, and possesses a norm $\tau$ defined in terms of a positive hermitian linear functional on $\tau (A)$. The norm $\tau$ is in general both incomplete and not an algebra norm, and is also not comparable with the Hilbert space norm $\left \\right \$ on $\tau (A)$. However, a onesided ideal of $\tau (A)$ is closed with respect to one norm if and only if it is closed with respect to the other. The topological dual of $\tau (A)$ with respect to the norm $\tau$ is isometrically isomorphic to the set of left centralizers on A.References

J. Dixmier, Les algèbres d’opérateurs dans l’espace Hilbertien (Algèbres de von Neumann), GauthierVillars, Paris, 1969.
 Roger Godement, Théorie des caractères. I. Algèbres unitaires, Ann. of Math. (2) 59 (1954), 47–62 (French). MR 58879, DOI 10.2307/1969832
 Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, SpringerVerlag, New York, 1965. MR 0188387
 B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc. (3) 14 (1964), 299–320. MR 159233, DOI 10.1112/plms/s314.2.299
 C. N. Kellogg, Centralizers and $H^{\ast }$algebras, Pacific J. Math. 17 (1966), 121–129. MR 193529, DOI 10.2140/pjm.1966.17.121
 Hidegorô Nakano, Hilbert algebras, Tohoku Math. J. (2) 2 (1950), 4–23. MR 41362, DOI 10.2748/tmj/1178245666
 Marc A. Rieffel, Squareintegrable representations of Hilbert algebras, J. Functional Analysis 3 (1969), 265–300. MR 0244780, DOI 10.1016/00221236(69)900433
 Parfeny P. Saworotnow and John C. Friedell, Traceclass for an arbitrary $H^{\ast }$algebra, Proc. Amer. Math. Soc. 26 (1970), 95–100. MR 267402, DOI 10.1090/S00029939197002674029
 Parfeny P. Saworotnow, Traceclass and centralizers of an $H^{\ast }$algebra, Proc. Amer. Math. Soc. 26 (1970), 101–104. MR 267403, DOI 10.1090/S00029939197002674030
 Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, SpringerVerlag, BerlinGöttingenHeidelberg, 1960. MR 0119112, DOI 10.1007/9783642876523
 James F. Smith, The $p$classes of an $H^{\ast }$algebra, Pacific J. Math. 42 (1972), 777–793. MR 322517, DOI 10.2140/pjm.1972.42.777
 Osamu Takenouchi, On the maximal Hilbert algebras, Tohoku Math. J. (2) 3 (1951), 123–131. MR 50178, DOI 10.2748/tmj/1178245512
 J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911, DOI 10.2140/pjm.1952.2.251
 Bertram Yood, Hilbert algebras as topological algebras, Ark. Mat. 12 (1974), 131–151. MR 380429, DOI 10.1007/BF02384750
Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 178 (1973), 259270
 MSC: Primary 46K15
 DOI: https://doi.org/10.1090/S00029947197303189008
 MathSciNet review: 0318900