Subnormal Operators and Representations of Algebras of Bounded Analytic Functions and other Uniform Algebras Thomas L. Mller , Robert F. Olin2 and o James E. Thomson CHAPTER I INTRODUCTION Let G be a bounded domain in the plane C and let H°°(G) denote the Banach algebra of bounded analytic functions on G. Let x denote the function whose value at X is X for every XeC. This paper is concerned with the theory of the continuous algebra homomorphisms from H°°(G) into B(M) that send 1 to 1 and x to S where S is a subnormal operator acting on a separable Hilbert space X. The Banach algebra 8(M) consists of the bounded operators on )i. Received by the editors February 28, 1986. Some of the results in Sections 2 and 3 appear in the first author's Ph.D thesis written under the supervision of Robert Olin. 2 The last two authors were partially supported by a grant from the National Science Foundation during the preparation of this paper. 1

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