#### Representable functionals for a class of locally convex quasi *-algebras.

###### Ponente(s): María Stella Adamo -

###### Locally convex quasi *-algebras were introduced by G. Lassner in the late 1980s to deal with some quantum models not fitting in the Haag–Kastler framework. A way to study such models, concerning for example quantum statistical mechanics, is to consider locally convex quasi *-algebras, for which Banach quasi *-algebras constitute a particular class. For example, Banach quasi *-algebras can be obtained by taking the completion of a ∗−algebra A0 with respect to a norm for which the multiplication in A0 is separately continuous, but not jointly continuous.

In the (locally convex) quasi *-algebra setting, a relevant role is played by representable functionals. Roughly speaking, a linear functional will be called representable if it allows a GNS-like construction. These kinds of functionals can be seen as the counterpart of positive functionals for C*-algebras but they are defined in a more general framework in which the multiplication is only partially defined. It is known that for C*-algebras, it is always possible to find non-trivial positive functionals, and these are always continuous. In the case of Banach quasi *-algebras, we have to restrict to the class of those that are fully representable to find non-trivial representable functionals. However, these functionals may not be continuous.

In this talk, we discuss the problem of continuity for these functionals in the setting of Banach quasi *-algebras and some related results in a broader context of the locally convex case. We begin our discussion by looking at the properties of representable (and continuous) functionals, especially in the simplest case of Hilbert quasi *-algebras. This discussion leads naturally to look at the problem of continuity for these functionals. Hence, we examine the approaches to study this problem, and if time permits, we will discuss some applications.

The talk is based on joint work with C. Trapani and with M. Fragoulopoulou.