Unbounded generalizations of left Hilbert algebras

Abstract

An unbounded left Hilbert algebra which is a generalization of the notation of a left Hilbert algebra to the unbounded case is defined and studied. The first primary purpose of this paper is to investigate the basic properties of an unbounded left Hilbert algebra. The second purpose is to study unbounded operator algebras (called left E # -algebras) of an unbounded left Hilbert algebra. The third purpose is to investigate an unbounded left Hilbert algebra M ξ o formed by a closed EW # -algebra M with a generating and separating vector ξ 0. The final purpose is to study an unbounded left Hilbert algebra K U formed by an unbounded Hilbert algebra U over U 0 and a positive self-adjoint invertible operator K′ affiliated with the right von Neumann algebra V 0( U 0) of the Hilbert algebra U 0.