Introduction

Abstract

A locally convex quasi *-algebra is a pair consisting of a locally convex space ${\mathfrak A}[\tau ]$ containing densely a *-algebra ${{\mathfrak A}}_{\scriptscriptstyle 0}$, whose multiplication and involution extend to ${\mathfrak A}$, in the sense that for any $a \in {\mathfrak A}$ and $x \in {{\mathfrak A}}_{\scriptscriptstyle 0}$, the elements ax, xa belong to ${\mathfrak A}$, in such a way that (ax)^∗ = x ^∗a ^∗, (xa)^∗ = a ^∗x ^∗ and moreover the left and right multiplications of the elements of ${\mathfrak A}$ by a fixed $x \in {{\mathfrak A}}_{\scriptscriptstyle 0}$, as well as the involution on ${\mathfrak A}[\tau ]$ are continuous. A typical example is obtained by taking as ${\mathfrak A}$ the completion of a locally convex *-algebra ${{\mathfrak A}}_{\scriptscriptstyle 0}[\tau ]$ with continuous involution and separately continuous multiplication.