Ideal categories of the normed algebra of finite rank bounded operators on a Hilbert space

Abstract

In this paper, we introduce the normal category [Formula: see text] [[Formula: see text]] of principal left [right] ideals of the normed algebra [Formula: see text] of all finite rank bounded operators on a Hilbert space [Formula: see text] and show that [Formula: see text] and [Formula: see text] are isomorphic using Hilbert space duality. We also prove that the semigroup of all normal cones in [Formula: see text] is isomorphic to the semigroup of all finite rank operators on [Formula: see text]. Further, we construct bounded normal cones in [Formula: see text] such that the set of all bounded normal cones in [Formula: see text] forms a normed algebra isomorphic to the normed algebra [Formula: see text].