Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

摘要

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