The primary purpose of this
paper is to investigate the structures of functionals and homomorphisms of
unbounded operator algebras called symmetric #-algebras, EC#-algebras and
EW#-algebras. First, we give the definitions and the fundamental properties of
such algebras. In particular, we define several locally convex topologies on
such algebras; a weak topology, a strong topology, a σ-weak topology and a
σ-strong topology. In the next section, we study the elementary operations on
EW#-algebras. We can define induced and reduced EW#-algebras, the
product of EW#-algebras and homomorphisms called an induction and an
amplification. In the final two sections, we obtain the main results (Theorem 4.8
and 5.5) which are described here. It is shown that a linear functional f
on a closed EW#-algebra A on D is weakly continuous (resp. σ-weakly
continuous) if and only if f(A) =∑i=1n(Aξi∣ηi),A ∈ A;ξi,ηi∈ D(i = 1,2,⋯,n)
(resp. f(A) =∑n=1∞(Aξn∣ηn);ξn,ηn∈ D(n = 1,2,⋯) and Σn=1∞∥Tξn∥2< ∞,
∑n=1∞∥Tηn∥2< ∞ for all T ∈ A). Also, it is shown that a σ-weakly continuous
homomorphism of a closed EW#-algebra A onto a closed EW#-algebra B is
decomposed in the following three types; a spatial isomorphism, an induction and an
amplification.
