Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Abstract.

Let E be a separable infinite-dimensional Hilbert space, and let $H({\mathbb{D}}; {\mathcal{L}}(E))$ denote the algebra of all functions $f : {\mathbb{D}} \rightarrow {\mathcal{L}}(E)$ that are holomorphic. If ${\mathcal{A}}$ is a subalgebra of $H({\mathbb{D}}; {\mathcal{L}}(E))$ , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of ${\mathcal{A}}$ is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra $H^{\infty}({\mathbb{D}}; {\mathcal{L}}(E))$, the disk algebra $A({\mathbb{D}}; {\mathcal{L}}(E))$ and the Wiener algebra $W_{+}({\mathbb{D}}; {\mathcal{L}}(E))$ are all infinite.