Structure of quasi operator systems

Abstract

In this paper we investigate algebraic structure of quasi operator spaces and quasi operator systems. We call a subspace of a unital complex ∗-algebra A [resp. self-adjoint subspace of A containing 1A] a quasi operator space [resp. quasi operator system]. Our keys in this investigation are the bounded subalgebra A0 of A, a C∗-seminorm on A0, and a new notion of algebraic bound.Our main goal is to show that many of fundamental results concerning operator systems are of algebraic nature. We obtain an algebraic characterization of operator systems which improves a classic result due to Choi–Effros. Moreover we show that A0 is the largest T∗-subalgebra of A. Then we extend the classical relationships among positivity, boundedness, complete positivity and complete boundedness to quasi operator systems. Schwarz inequality for 2-positive maps and Smith’s theorem are also extended to quasi operator spaces.Several examples are provided to show the relations and distinctions of various classes of spaces under consideration, with their classic analogs and also to show up to what extent our conclusions are true.