Abstract
To each Banach algebra A we associate a (generally) larger Banach algebra A+ which is a quotient of its bidual A″. It can be constructed using the strict topology on A and the Arens product on A″. A+ has certain more pleasant properties than A″, e.g. if A has a bounded right approximate identity, then A+ has a two-sided unit. In the special case A=L1(G) (G a locally compact abelian group) one gets A+=Cu(G)′, the dual of the space of bounded, uniformly continuous functions on G, and we show that the center of the convolution algebra Cu(G)′ is precisely the space M(G) of finite measures on G.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ARENS, R.F.: Operations induced in functions classes, Monatsh. Math.55, 1–19 (1951)
ARENS, R.F.: The adjoint of a bilinear operation, Proc. Amer. Math. Soc.2, 839–848 (1951)
CIGLER, J., LOSERT, V., MICHOR, P.: Banach modules and functors on categories of Banach spaces, Lecture Notes in Pure and Applied Mathematics, vol. 46, New York: Marcel Dekker Inc. 1979
CIVIN, P., YOOD, B.: The second conjugate space of a Banach algebra as an algebra, Pacific J. Math.11, 847–870 (1961)
DAVENPORT, J.W.: Multipliers on a Banach algebra with a bounded approximate identity, Pacific J. Math.63, 131–135 (1976)
GROSSER, M.: Bidualräume und Vervollständigungen von Banachmoduln, Universität Wien, Dissertation 1976; Lecture Notes in Mathematics, vol. 717, Berlin-Heidelberg-New York: Springer 1979
GROSSER, M.: L1(G) as an ideal in its second dual space, Proc. Amer. Math. Soc.73, 363–364 (1979)
GROSSER, M.: Module tensor products of Ko (X,X) with its dual, to appear in: Colloquia Mathematica Societatis Janos Bolyai, vol. 35, Functions, Series, Operators, Budapest (Hungary), 1980, Amsterdam-Oxford-New York: North Holland
GULICK, S.L., LIU, T.S., ROOIJ, A.C.M. van,: Group algebra modules I, Canad. J. Math.19, 133–150 (1967)
HEWITT, E., ROSS, K.A.: Abstract harmonic analysis, Part I, Berlin-Heidelberg-New York: Springer 1963
SENTILLES, F.D., TAYLOR, D.C.: Factorization in Banach algebras and the general strict topology, Trans. Amer. Math. Soc.142, 141–152 (1969)
TAKAHASI, SIN-EI: Dixmier's representation theorem of central double centraliziers on Banach algebras, Trans. Amer. Math. Soc.253, 229–236 (1979)
TAYLOR, D.C.: The strict topology for double centralizer algebras. Trans. Amer. Math. Soc.150, 633–643 (1970)
TOMIUK, B.J.: Multipliers on Banach algebras, Studia Math.54, 267–283 (1975)
WENDEL, J.G.: Left centralizers and isomorphisms of group algebras, Pacific J. Math.2, 251–261 (1952)
YOUNG, N.J.: The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. 224, 59–62 (1973)
ZAPPA, A.; The center of the convolution algebra Cu(G)*, Rend. Sem. Mat. Univ. Padova,52, 71–83 (1974)
ZAPPA, A.: The center of an algebra of operators, Coll. Math.39, 343–349 (1978)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grosser, M., Losert, V. The norm-strict bidual of a Banach algebra and the dual of Cu(G). Manuscripta Math 45, 127–146 (1984). https://doi.org/10.1007/BF01169770
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01169770