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
supλ∈E∥πλ(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 ℒ∗(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 ℒ+(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) ⊆ℒ+(X) = ℒ∗(Xf), and (1) π : A → (ℒ∗(Xf),𝒯b−) is a topological
∗-isomorphism (into) and (2) π;A → (ℒ+(X),𝒯+−) is a topological ∗-isomorphism
(into).
