CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2007-22

August 2007

Stable Ranks of Banach Algebras of Operator-Valued H Functions

Amol Sasane

Let E be an infinite-dimensional Hilbert space, and let HL(E) denote the Banach algebra of all functions f : D → L(E) that are holomorphic and bounded, equipped with the supremum norm |f| := supz∈D |f(z)|L(E), f ∈ HL(E). We show that the Bass and topological stable ranks of HL(E) are infinite. If S is an open subset of T, then let ASL(E) denote the subalgebra of HL(E) of all functions that have a continuous extension to S. We also prove that ASL(E) has infinite Bass and topological stable ranks.

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