Una nota sobre cocientes finito-dimensionales y el problema de continuidad automática para álgebras de convolución torcida:  A note on finite-dimensional quotients and the problem of automatic continuity for twisted convolution algebras

Felipe I. Flores

doi:10.56754/0719-0646.2702.329

DOI:

https://doi.org/10.56754/0719-0646.2702.329

Abstract

In this note, we will show that the twisted convolution algebra \(L^{1}_{\alpha,\omega}(\mathbb{G},\mathcal{A})\) associated with a twisted action of a locally compact group \(\mathbb{G}\) on a \(C^{*}\)-algebra \(\mathcal{A}\) has the following property: Every quotient by a closed two-sided ideal of finite codimension produces a semisimple algebra. Afterward, we use this property, together with results by H. Dales and G. Willis, to extend previous results by the author and to produce large classes of examples of algebras with automatic continuity properties.

Resumen

En esta nota probaremos que el álgebra de convolución torcida \(L^{1}_{\alpha,\omega}(\mathbb{G},\mathcal{A})\) asociada a una acción torcida de un grupo localmente compacto \(\mathbb{G}\) en una \(C^{*}\)-algebra \(\mathcal{A}\) tiene la siguiente propiedad: Todo cociente por un ideal cerrado, bilátero y de codimensión finita produce un álgebra semisimple. Luego utilizamos esta propiedad, junto con resultados de H. Dales y G. Willis, para extender resultados previos del autor y producir grandes clases de ejemplos de álgebras con propiedades de continuidad automática.

Keywords

Automatic continuity , semisimplicity , cofinite ideal , bimodule , twisted action , convolution algebra

Mathematics Subject Classification:

43A20 , 47L65 , 46H40

Felipe I. Flores Department of Mathematics, University of Virginia, 114 Kerchof Hall. 141 Cabell Dr, Charlottesville, Virginia, United States. https://orcid.org/0000-0003-1441-8136

Pages: 329–341
Date Published: 2025-10-14
Vol. 27 No. 2 (2025): Spanish Edition (40th Anniversary)

F. F. Bonsall y J. Duncan, Complete normed algebras, ser. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York-Heidelberg, 1973, vol. 80.

H. G. Dales, “Automatic continuity: a survey,” Bull. London Math. Soc., vol. 10, no. 2, pp. 129–183, 1978, doi: 10.1112/blms/10.2.129.

H. G. Dales, Banach algebras and automatic continuity, ser. London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 2000, vol. 24, Oxford Science Publications.

H. G. Dales y G. A. Willis, “Cofinite ideals in Banach algebras, and finite-dimensional representations of group algebras,” in Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), ser. Lecture Notes in Math. Springer, Berlin-New York, 1983, vol. 975, pp. 397–407.

F. I. Flores, “On the continuity of intertwining operators over generalized convolution algebras,” J. Math. Anal. Appl., vol. 542, no. 1, 2025, Art. ID 128753, doi: 10.1016/j.jmaa.2024.128753.

A. Y. Helemskii, The homology of Banach and topological algebras, ser. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1989, vol. 41, doi: 10.1007/978-94-009-2354-6.

K. K. Jensen, “Foundations of an equivariant cohomology theory for Banach algebras. II,” Adv. Math., vol. 147, no. 2, pp. 173–259, 1999, doi: 10.1006/aima.1999.1838.

N. P. Jewell, “Continuity of module and higher derivations,” Pacific J. Math., vol. 68, no. 1, pp. 91–98, 1977.

G. Pisier, “Are unitarizable groups amenable?” in Infinite groups: geometric, combinatorial and dynamical aspects, ser. Progr. Math. Birkhäuser, Basel, 2005, vol. 248, pp. 323–362, doi: 10.1007/3-7643-7447-0_8.

V. Runde, “Homomorphisms from L1(G) for G∈[FIA]−∪[Moore],” J. Funct. Anal., vol. 122, no. 1, pp. 25–51, 1994, doi: 10.1006/jfan.1994.1060.

V. Runde, “Intertwining operators over L1(G) for G∈[PG] ∩[SIN],” Math. Z., vol. 221, no. 3, pp. 495–506, 1996, doi: 10.1007/PL00004255.

G. A. Willis, “Derivations from group algebras, factorization in cofinite ideals and topologies on B(X),” Ph.D. dissertation, Newcastle upon Tyne, 1980.

G. Willis, “The continuity of derivations from group algebras: factorizable and connected groups,” J. Austral. Math. Soc. Ser. A, vol. 52, no. 2, pp. 185–204, 1992.