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DIRECTLY FINITE ALGEBRAS OF PSEUDOFUNCTIONS ON LOCALLY COMPACT GROUPS

Part of: Locally compact groups and their algebras Abstract harmonic analysis Selfadjoint operator algebras

Published online by Cambridge University Press:  17 December 2014

YEMON CHOI
YEMON CHOI*
Affiliation:
Department of Mathematics and Statistics, Fylde College, Lancaster University, Bailrigg, Lancaster, Lancashire LA1 4YF, United Kingdom e-mail: [email protected]
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Abstract

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An algebra A is said to be directly finite if each left-invertible element in the (conditional) unitization of A is right invertible. We show that the reduced group C*-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras of p-pseudofunctions, showing that these algebras are directly finite if G is amenable and unimodular, or unimodular with the Kunze–Stein property. An exposition is also given of how existing results from the literature imply that L1(G) is not directly finite when G is the affine group of either the real or complex line.

MSC classification

Primary: 22D15: Group algebras of locally compact groups 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Secondary: 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 693 - 707
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Barnes, B. A., When is the spectrum of a convolution operator on Lp independent of p? Proc. Edinburgh Math. Soc. (Ser. 2) 33 (1990), 327–332.CrossRefGoogle Scholar
2.Bonsall, F. F. and Duncan, J., Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, vol. 80 (Springer-Verlag, New York, 1973).CrossRefGoogle Scholar
3.Choi, Y., Group representations with empty residual spectrum Int. Eq. Op. Th. 67 (2010), 95–107. Erratum: Int. Eq. Op. Th. 69(1) (2011), 149–150.CrossRefGoogle Scholar
4.Cowling, M., The Kunze–Stein phenomenon Ann. Math. (Ser. 2) 107 (1978), 209–234.CrossRefGoogle Scholar
5.Derighetti, A., Convolution operators on groups, Lecture Notes of the Unione Matematica Italiana, vol. 11 (Springer, Heidelberg, 2011).CrossRefGoogle Scholar
6.Diep, D. N., Methods of noncommutative geometry for group C*-algebras, Chapman & Hall/CRC Research Notes in Mathematics, vol. 416 (Chapman & Hall/CRC, Boca Raton, FL, 2000).Google Scholar
7.Dixmier, J., Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann), Gauthier-Villars Éditeur, Paris, 1969. Deuxième édition, revue et augmentée, Cahiers Scientifiques, Fasc. XXV.Google Scholar
8.Dixmier, J., C*-algebras, Translated from the French by Francis Jellett, North-Holland Mathematical Library, vol. 15 (North-Holland Publishing Co., Amsterdam, 1977).Google Scholar
9.Herz, C., The theory of p-spaces with an application to convolution operators Trans. Am. Math. Soc. 154 (1971), 69–82.Google Scholar
10.Herz, C., Harmonic synthesis for subgroups Ann. Inst. Fourier (Grenoble) 23 (1973), 91–123.CrossRefGoogle Scholar
11.Kaplansky, I., Modules over operator algebras Am. J. Math. 75 (1953), 839–858.CrossRefGoogle Scholar
12.Kunze, R. A. and Stein, E. M., Uniformly bounded representations and harmonic analysis of the 2×2 real unimodular group Am. J. Math. 82 (1960), 1–62.CrossRefGoogle Scholar
13.Leptin, H., Lokal kompakte Gruppen mit symmetrischen Algebren, in Symposia Mathematica, vol. 22 (Convegno sull'Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) (Academic Press, London, 1977), 267–281.Google Scholar
14.Lohoué, N., Estimations Lp des coefficients de représentation et opérateurs de convolution, Adv. Math. 38 (1980), 178–221.CrossRefGoogle Scholar
15.Meyer, J., personal communication. MathOverflow. http://mathoverflow.net/questions/16944 (version: 2010-03-09).Google Scholar
16.Montgomery, M. S., Left and right inverses in group algebras Bull. Am. Math. Soc. 75 (1969), 539–540.CrossRefGoogle Scholar
17.Munn, W. D., Direct finiteness of certain monoid algebras. I, Proc. Edinburgh Math. Soc. 39 (2) (1996), 365–369.CrossRefGoogle Scholar
18.Nebbia, C., Groups of isometries of a tree and the Kunze-Stein phenomenon Pacific J. Math. 133 (1988), 141–149.CrossRefGoogle Scholar
19.Palmer, T. W., Banach algebras and the general theory of *-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79 (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
20.Pedersen, G. K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14 (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1979).Google Scholar
21.Rosenberg, J., The C*-algebras of some real and p-adic solvable groups Pacific J. Math. 65 (1976), 175–192.CrossRefGoogle Scholar
22.Wang, X., The C*-algebras of a class of solvable Lie groups, Pitman Research Notes in Mathematics Series, vol. 199 (Longman Scientific & Technical, Harlow, 1989).Google Scholar
23.Z'ep, D. N., The structure of the group C*-algebra of the group of affine transformations of the line, Funkcional. Anal. i Priložen. 9 (1974), 63–64.Google Scholar