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The author now gives a proof of an unpublished theorem of A. Connes to the effect that the technical algebra is stable under the holomorphic functional calculus.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


46L80 \(K\)-theory and operator algebras (including cyclic theory)
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