Vol. 65, No. 1, 1976

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On a class of unbounded operator algebras

Atsushi Inoue

Vol. 65 (1976), No. 1, 77–95

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(iηi),A A;ξii D(i = 1,2,,n) (resp. f(A) = n=1(nηn);ξnn D(n = 1,2,) and Σn=1n2 < , n=1n2 < 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.

Mathematical Subject Classification 2000
Primary: 46L99
Secondary: 46K15
Received: 17 December 1975
Published: 1 July 1976
Atsushi Inoue