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


46L87 Noncommutative differential geometry
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
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