On a problem of Barnes and Duncan

Abstract

Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach »-algebra /'(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Fund. Anal. 18 (1975), 96-113.) dealing with the case where P has two elements. Mathematics subject classification (1985 Revision): 43A65Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:We equip R with its natural involution * in which each element of P is Hermitian. When P contains exactly two elements Barnes and Duncan [2] have shown that the Banach *-algebra ^i(R) has a separating family of finite dimensional *-representations. We show that this result is in fact true for an arbitrary P. It follows that t l (R) is •-semisimple. Let S be a monoid i.e. a semigroup with an identity element 1. By a representation n of S we shall mean a bounded map n from S into the set of all bounded linear operators on a (real or complex) Hilbert space H such that TI(1) is the identity operator and n(xy) = n{x)n(y) for all x,yeS. Each representation of S has a unique extension to a representation of the Banach algebra t l (S). Tensor products and direct sums may be formed in a similar way to those of group representations. Definition.A representation n of a monoid S on a Hilbert space H will be called formally real is there is an orthonormal basis {e,|ie/} of H for which (n(x)e s \e r } is real for all x6S and all r,sel.