Vol. 39, No. 1, 1971

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On representing F∗-algebras

Robert Morgan Brooks

Vol. 39 (1971), No. 1, 51–69

The purpose of this paper is to obtain a concrete representation for F-algebras with identity: a Frechet algebra with involution for which there exists a determining sequence of B-seminorms. The main result is Theorem 3.4 which is described here. Let A be an F-algebra with identity. Let {(πλ,Hλ);λ Λ} be a complete family of irreducible Hilbert space representations of A. Let H = Σλ Hλ, define E Λ to be equicontinuous provided supλEπλ(a)< (a A), and let X = {x H : Supp(x)is equicontinuous}. The linear space X is given the final topology τf determined by the family {HEJ = [x H : Supp(x) E] : E equicontinuous} of subspaces of X. Let Xf be (X,τf) and let (Xf) be all operators on X which have an adjoint relative to the inner product inherited from H such that both the operator and its adjoint are τf-continuous. This algebra will be endowed with the topology 𝒯b of bounded convergence. Let +(X) be all operators which have adjoints. It has a natural topology F+ described in §2. Define π;A →ℒa(X) by π(a){xλ} = {πλ(a)xλ} for a A and x = {xλ}∈ X. Then π(A) +(X) = (Xf), and (1) π : A ((Xf),𝒯b) is a topological -isomorphism (into) and (2) π;A (+(X),𝒯+) is a topological -isomorphism (into).

Mathematical Subject Classification 2000
Primary: 46K10
Received: 10 September 1970
Published: 1 October 1971
Robert Morgan Brooks