On the structure of Banach algebras associated with automorphisms

Lebedev, A.

doi:10.1090/S0077-1554-05-00153-6

On the structure of Banach algebras associated with automorphisms
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by A. V. Lebedev
Translated by: O. Khleborodova

Trans. Moscow Math. Soc. 2005, 105-142

DOI: https://doi.org/10.1090/S0077-1554-05-00153-6

Published electronically: November 18, 2005

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Abstract:

We study the structure of the Banach algebra $B(A, T_g )$ generated by a Banach algebra $A$ of operators in a Banach space $D$ and a group $\{ T_g \}_{g \in G}$ of isometries of $D$ such that $T_g A T^{-1}_g = A$. We analyze relations between the existence of an expectation from $B(A, T_g )$ to $A$, topological (metric) freeness of the action by automorphisms of the algebra $A$ induced by $T_g$, and the dual action of the group $G$ on $B(A, T_g )$. The obtained results are applied to the description of the structure of Banach algebras generated by “weighted composition operators” in various spaces.

References

Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
Magnus B. Landstad, Duality theory for covariant systems, Trans. Amer. Math. Soc. 248 (1979), no. 2, 223–267. MR 522262, DOI 10.1090/S0002-9947-1979-0522262-6
Anatolij Antonevich and Andrei Lebedev, Functional-differential equations. I. $C^*$-theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 70, Longman Scientific & Technical, Harlow, 1994. MR 1413495
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. MR 0251549
N. Bourbaki, Eléments de mathématique. XIII. Première partie: Les structures fondamentales de l’analyse. Livre VI: Intégration. Chapitre I: Inégalités de convexité. Chapitre II: Espaces de Riesz. Chapitre III: Mesures sur les espaces localement compacts. Chapitre IV: Prolongement d’une mesure; espaces $L^p$, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175, Hermann & Cie, Paris, 1952 (French). MR 0054691
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
Paul R. Halmos, The decomposition of measures, Duke Math. J. 8 (1941), 386–392. MR 4718
Jöran Bergh and Jörgen Löfström, Interpolyatsionnye prostranstva, “Mir”, Moscow, 1980 (Russian). Vvedenie. [An introduction]; Translated from the English by V. S. Krjučkov and P. I. Lizorkin. MR 608963