The Dirichlet problem and spectral theory of operator algebras

Abstract: The main result of the paper is an extension of the Dirichlet problem from (closures of) bounded open domains U to arbitrary compact subsets X of the complex plane, i.e. the closure of the corresponding space of functions which are harmonic in a neighbourhood of X and equipped with the supremums norm on X is shown to be isometric with the space of continuous functions C (/delta X) on its Shilov boundary (a given compact subset of X). This is used to define an extension of holomorphic function calculus with respect to certain (weakly normal) elements x of a unital operator algebra A to a completely isometric harmonic function calculus into the enveloping operator system of A. It is also shown that in case of a super C*-algebra A (operator algebra with involution) any weakly normal superpositive element x has a square root in A.