Invertibility and topological stable rank for semi-crossed product algebras (Q1173650)

Let \((X,\varphi)\) be a dynamical system, i.e. \(X\) is a compact Hausdorff space and \(\varphi: X\to X\) is a continuous surjection. Denote by \(K(\mathbb{Z}^ +,C(X))\) the free product of \(C(X)\) with a single operator \(U\) together with the relations \(fU=Uf\circ\varphi\) (\(f\in C(X)\)). Thus, the elements of the complex algebra \(K(\mathbb{Z}^ +,C(X))\) have the form \[ F=f_ 0+Uf_ 1+\dots+U^ n f_ n \qquad (n\in\mathbb{Z}^ +,\;f_ 1,\dots,f_ n\in C(X)). \] For \(x\in X\) define a representation \(\Pi_ x\) of \(K(\mathbb{Z}^ +,C(X))\) on \(\ell_ 2\) by the formulae \[ \begin{aligned} \Pi_ x(U)(\xi_ 0,\xi_ 1,\dots) & =(0,\xi_ 0,\xi_ 1,\dots)\qquad\text{and}\\ \Pi_ x(f)(\xi_ 0,\xi_ 1,\dots) &=(f(x)\xi_ 0,f\circ\varphi(x)\xi_ 1, f\circ\varphi^ 2(x)\xi_ 2\dots),\qquad f\in C(X).\end{aligned} \] The semi-crossed product \(\mathbb{Z}^ +\times_ \varphi C(X)\) is the completion of \(K(\mathbb{Z}^ +,C(X))\) with respect to the norm \(\| F\|=\sup_{x\in X}\| \Pi_ x(F)\|\). It is shown that to each \(F\in{\mathcal U}=\mathbb{Z}^ +\times_ \varphi C(X)\) corresponds a unique Fourier series \(\sum_{n=0}^ \infty U^ n f_ n\) and that \({\mathcal U}\) contains the disc algebra as a closed subalgebra. Invertibility and local invertibility in \({\mathcal U}\) is discussed. The invertible elements are not dense in \({\mathcal U}\), therefore the topological stable rank of \({\mathcal U}\) is greater than one. It equals two in case \(\varphi\) is a homeomorphism and \(C(X)\) has topological stable rank one. Along the classical line summability results for Fourier series are proved. Using these results the author shows that \(K\)-theory of \({\mathcal U}\) ``reduces'' to that of \(C(X)\), more precisely, \(K_ i({\mathcal U})\) is isomorphic to \(K_ i(C(X))\) for \(i=0,1\).