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Rapidly decreasing functions in reduced $$C^*$$-algebras of groups. (English) Zbl 0711.46054
Subalgebras of a $$C^*$$-algebra that consist of $$C^{\infty}$$-elements and are stable under holomorphic functional calculus are an important device in Connes’ noncommutative geometry. The author defines the “Schwartz space” $$H^{\infty}_ L(\Gamma)$$ of rapidly decreasing functions on a discrete group $$\Gamma$$ associated with a length function L. He gives a list of examples of groups for which $$H^{\infty}_ L(\Gamma)$$ is contained in $$C^*_{red}(\Gamma)$$ (called property (RD)) or not.
For example if $$\Gamma_ 1$$, $$\Gamma_ 2$$ have property (RD), so does $$\Gamma_{1^*_ A}\Gamma_ 2$$ for A finite; moreover, property (RD) is inherited by subgroups. If $$\Gamma$$ is finitely generated with word length-functions L then $$H^{\infty}_ L(\Gamma)\subset C^*_{red}(\Gamma)$$ if $$\Gamma$$ is of polynomial growth, and if $$\Gamma$$ is amenable this is also necessary. Note that Pierre de la Harpe has shown [C. R. Acad. Sci., Paris, Sér. I Math. 307, No.14, 771-774 (1988; Zbl 0653.46059)] that the techniques of the present paper apply to establish property (RD) for any hyperbolic group. We also want to direct attention to the author’s accompanying paper [K-Theory 2, No.6, 723-735 (1989; Zbl 0692.46062)] where he proves stability under holomorphic functional calculus for $$H^{\infty}_ L(\Gamma)$$ if $$\Gamma$$ has property (RD) which in particular implies that $$H^{\infty}_ L(\Gamma)$$ and $$C^*_{red}(\Gamma)$$ have isomorphic K-groups.
Reviewer: H.Schröder

##### MSC:
 46L87 Noncommutative differential geometry 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc.
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