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Transactions of the American Mathematical Society

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Rapidly decreasing functions in reduced $ C\sp *$-algebras of groups


Author: Paul Jolissaint
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1990-0943303-2
Article copyright: © Copyright 1990 American Mathematical Society