The covering dimension of a distinguished subset of the spectrum \(M(H^\infty)\) of \(H^\infty\) and the algebra of real-symmetric and continuous functions on \(M(H^\infty)\). (English) Zbl 1245.46042
Let \(H^\infty(\mathbb D)\) denote the Banach algebra of bounded analytic functions on the unit disk \(\mathbb D\) in \(\mathbb C\), and let \(\mathcal M(H^\infty)\) denote the maximal ideal space of \(H^\infty\). Also, let \(C(\mathcal M(H^\infty))_{\text{sym}}\) denote the algebra of all continuous complex valued functions \(f\) on \(\mathcal M(H^\infty)\) that satisfy \(f(z) = \overline{f(\overline z)}\) for all \(z\in\mathbb D\).
In the paper the author proves that the covering dimension, \(\dim E\), of the closure \(E\) of the interval \((-1,1)\) in \(\mathcal M(H^\infty)\) equals one, and that the covering dimension of the closure \(M^+\) of \(\mathbb D^+ =\{z\in \mathbb D: \text{Im\,} z >0\}\) in \(\mathcal M(H^\infty)\) is two. Using the result of F. D. Suárez [J. Funct. Anal. 123, No. 2, 233–263 (1994; Zbl 0808.46076)] that \(\dim\mathcal M(H^\infty) =2\), the author then proves that the Bass and topological stable ranks of \(C(\mathcal M(H^\infty))_{\text{sym}}\) equal two.
In the paper the author proves that the covering dimension, \(\dim E\), of the closure \(E\) of the interval \((-1,1)\) in \(\mathcal M(H^\infty)\) equals one, and that the covering dimension of the closure \(M^+\) of \(\mathbb D^+ =\{z\in \mathbb D: \text{Im\,} z >0\}\) in \(\mathcal M(H^\infty)\) is two. Using the result of F. D. Suárez [J. Funct. Anal. 123, No. 2, 233–263 (1994; Zbl 0808.46076)] that \(\dim\mathcal M(H^\infty) =2\), the author then proves that the Bass and topological stable ranks of \(C(\mathcal M(H^\infty))_{\text{sym}}\) equal two.
Reviewer: Manfred Stoll (Columbia)
MSC:
| 46J10 | Banach algebras of continuous functions, function algebras |
| 30H05 | Spaces of bounded analytic functions of one complex variable |
| 46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |
| 54F45 | Dimension theory in general topology |
Keywords:
covering dimension, spectrum for bounded analytic functions; Bass stable rank; topological stable rankCitations:
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