## K-theory of reduced $$C^*$$-algebras and rapidly decreasing functions on groups.(English)Zbl 0692.46062

Let $$\Gamma$$ be a group. A function L of $$\Gamma$$ into $${\mathbb{R}}_+$$ is said to be a length-function of $$\Gamma$$ if (i) $$L(gh)=L(g)+L(h)$$, (ii) $$L(g^{-1})=L(g)$$ and (iii) $$L(1)=0$$ for all g and h in $$\Gamma$$. If there is a length function on $$\Gamma$$, then the space of rapidly decreasing functions $$H^{\infty}_ L(\Gamma)$$ on $$\Gamma$$ with respect to L is the set of all complex valued functions $$\phi$$ on $$\Gamma$$ such that $$\sum_{\Gamma}| \phi (g)|^ 2\cdot L(g)^{2s}<\infty$$ for every $$s\in {\mathbb{R}}$$. The group $$\Gamma$$ is said to have property (RD) if there is a length-function L on $$\Gamma$$ such that $$H^{\infty}_ L(\Gamma)$$ is contained in the reduced $$C^*$$-algebra $$C^*_ r(\Gamma)$$ of $$\Gamma$$. In this case $$H^{\infty}_ L(\Gamma)$$ is a dense *-subalgebra of $$C^*_ r(\Gamma).$$
One defines the technical algebra $$T^{\infty}_ r(\Gamma)$$ to be the set of elements of $$C^*_ r(\Gamma)$$ such that, for every $$\alpha\in (0,1)$$ and every $$q\in {\mathbb{N}}$$, $$\| (1-p_ N)ap_{N- N^{\alpha}}\| +\| p_{N-N^{\alpha}}a(1-p_ N)\| =O(N^{- q})\quad as\quad N\to \infty.$$ Here $$p_ r$$ is the projection on the subspace $$\oplus_{L(g)\leq r}{\mathbb{C}}\delta_ g$$ of the Hilbert space $$\ell^ 2(\Gamma)$$. Then the author shows that $$T^{\infty}_ r(\Gamma)\subset H^{\infty}_ L(\Gamma)$$, and if $$H^{\infty}_ L(\Gamma)\subset C^*_ r(\Gamma)$$, then $$T^{\infty}_ r(\Gamma)=H^{\infty}_ L(\Gamma).$$
The author now gives a proof of an unpublished theorem of A. Connes to the effect that the technical algebra is stable under the holomorphic functional calculus. Here a subalgebra B of an algebra A is said to be stable under the holomorphic functional calculus if, for every $$n\in {\mathbb{N}}$$ and every $$a\in M_ n(B)$$, the element $$f(a)\in M_ n(B)$$ for every function f holomorphic on a neighborhood of the spectrum of a in $$M_ n(A)$$. Under the hypothesis that $$\Gamma$$ has property (RD), the author shows as a corollary that the inclusion of $$H^{\infty}_ L(\Gamma)\subset C^*_ r(\Gamma)$$ induces an isomorphism $$K_ i(H^{\infty}_ L(\Gamma))$$ onto $$K_ i(C^*_ r(\Gamma))$$ for $$i=0,1$$. Here the equivalence relation for $$K_ 1(H^{\infty}_ L(\Gamma))$$ is defined in terms of being connected by a piecewise linear path in $$GL_{\infty}.$$
The author also proves analogous theorems for the algebra $$H_ L^{1,\infty}(\Gamma)$$ of functions $$\phi$$ with $$\phi (1+L)^ s\in \ell^ 1(\Gamma)$$ for every $$s\geq 0$$. He also gives some examples. In particular, he considers the free group of rank n with its natural length function.
Reviewer: H.Halpern

### MSC:

 46L80 $$K$$-theory and operator algebras (including cyclic theory)
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### References:

  Bost, J.-B.: private communication. See also K-théorie des produits croisés et principe de Oka, C.R. Acad. Sci. Paris, 301, Série I, No. 5 (1985), 189-192.  Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation, Preprint IHES (1984). · Zbl 0647.46054  Connes, A.: Non-commutative differential geometry, Publ. Math. IHES 62 (1986), 41-144. · Zbl 0592.46056  Cuntz, J.: The K-groups for free products of C*-algebras, Proc. Symp. Pure Appl. Math. 38 (1982), Part 1, pp. 81-84. · Zbl 0502.46050  de la Harpe, P.: Reduced C*-algebras of discrete groups which are simple with a unique trace, in Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132, Springer, New York (1985), pp. 230-253.  Jolissaint, P.: Croissance d’un groupe de génération finie et fonctions lisses sur son dual, C. R. Acad. Sci. Paris, 300, Série I, 17 (1985), 601-604.  Jolissaint, P.: Rapidly decreasing functions in reduced C*-algebras of groups, to appear in Trans Amer. Math. Soc. · Zbl 0711.46054  Masuda, T.: Cyclic cohomology of the group algebra of free groups, J. Operator Theory 15 (1986), 345-357. · Zbl 0617.16016  Pedersen, G. K.: C*-Algebras and their Automorphism Groups, Academic Press, London (1979). · Zbl 0416.46043
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