Stable ranks for the real function algebra \(C(X,\tau)\) (Q258991)

Let \(X\) be a compact Hausdorff space and \(C(X,\tau)=\{f\in C(X):f\circ\tau=\bar{f}\}\), where \(\tau\) is a topological involution on \(X\), cf. \textit{R. F. Arens} and \textit{I. Kaplansky} [``Topological representation of algebras'', Trans. Am. Math. Soc. 63, 457--481 (1948; Zbl 0032.00702)].NEWLINENEWLINEThe purpose of this paper is to calculate the Bass and topological stable ranks of \(C(X,\tau)\), thus extending the classical Vasershtein's results for the algebras \(C(X)\) and \(C(X,\mathbb{R})\). As a corollary of the main result (Theorem 5.9), the previous result of the authors [``Approximation by invertible elements and the generalized E-stable rank for \(A(\mathbf{D})_{\mathbb{R}}\) and \(C({\mathbf{D}})_{\mathrm {sym}}\)'', Math. Scand. 109, No. 1, 114--132 (2011; Zbl 1241.46031)] giving a characterization of the Bass and topological stable ranks for the algebra of continuous real-symmetric functions on real-symmetric compacta \(K\subseteq{\mathbb{C}}\) is recovered.