Noncommutative uniform algebras

Abstract

implies that (i) A is commutative, and further implies that (ii) A must be of a very speciÞc form, namely it must be isometrically isomorphic with a uniformly closed subalgebra of CC (X). We denote here by CC (X) the complex Banach algebra of all continuous functions on a compact set X. The obvious question about the validity of the same conclusion for real Banach algebras is immediately dismissed with an obvious counterexample: the non commutative algebra H of quaternions. However it turns out that the second part (ii) above is essentially also true in the real case. We show that the condition (1.1) implies, for a real Banach algebra A, that A is isometrically isomorphic with a subalgebra of CH (X) the algebra of continuous quaternion valued functions on a compact set X . That result is in fact a consequence of a theorem by Aupetit and Zemanek [1]. We will also present several related results and corollaries valid for real and complex Banach and topological algebras. To simplify the notation we assume that the algebras under consideration have units, however, like in the case of the Hirschfeldu Zelazko Theorem, the analogous results can be stated for none unital algebras as one can formally add a unit to such an algebra. We use standard notation, as can be found for example in [6]. For a real Banach algebra A with a unit e we denote by A−1 the set of invertible elements of A; for a ∈ A we deÞne the real and the complex spectrum and the corresponding spectral radius by well known formulas: σR (a) = {s ∈ R : a− se / ∈ A−1}, σC (a) = {s+ it ∈ C : (a− se) + te / ∈ A−1}, ,R (a) = sup {|λ| : λ ∈ σR (a)} , ,C (a) = sup {|λ| : λ ∈ σC (a)} , where we put ,R (a) = 0 if the set σR (a) is empty. It is well known [6] that σC (a) = {λ ∈ C : a − λe / ∈ A−1 C }, where AC is the complexiÞcation of A, and that ,C (a) = lim n pkank.