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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Rapidly decreasing functions in reduced $C^ *$-algebras of groups
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by Paul Jolissaint PDF
Trans. Amer. Math. Soc. 317 (1990), 167-196 Request permission

Abstract:

Let $\Gamma$ be a group. We associate to any length-function $L$ on $\Gamma$ the space $H_L^\infty (\Gamma )$ of rapidly decreasing functions on $\Gamma$ (with respect to $L$), which coincides with the space of smooth functions on the $k$-dimensional torus when $\Gamma = {{\bf {Z}}^k}$. We say that $\Gamma$ has property (RD) if there exists a length-function $L$ on $\Gamma$ such that $H_L^\infty (\Gamma )$ is contained in the reduced ${C^*}$-algebra $C_r^*(\Gamma )$ of $\Gamma$. We study the stability of property (RD) with respect to some constructions of groups such as subgroups, over-groups of finite index, semidirect and amalgamated products. Finally, we show that the following groups have property (RD): (1) Finitely generated groups of polynomial growth; (2) Discrete cocompact subgroups of the group of all isometries of any hyperbolic space.
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 317 (1990), 167-196
  • MSC: Primary 22D25; Secondary 43A15, 46L99
  • DOI: https://doi.org/10.1090/S0002-9947-1990-0943303-2
  • MathSciNet review: 943303