Finite dimensional semisimple Q-algebras

A Q-algebra can be represented as an operator algebra on an infinite dimensional Hilbert space. However we do not know whether a finite n-dimensional Q-algebra can be represented on a Hilbert space of dimension n except n = I, 2. It is known that a 2-dimensional Q-algebra is just a 2-dimensional commutative operator algebra on a 2-dimensional Hilbert space. In this paper we study a finite n-dimensional semisimple Q-algebra on a finite n-dimensional Hilbert space. In particular we describe a 3-dimensional Q-algebra of the disc algebra on a 3-dimensional Hilbert space. Our studies are related to the Pick interpolation problem for a uniform algebra. (c) 2006 Elsevier Inc. All rights reserved.