Representable functionals and derivations on Banach quasi *-algebras

Locally convex quasi *-algebras, in particular Banach quasi *-algebras, have been deeply investigated by many mathematicians in the last decades in order to describe quantum physical phenomena (see \cite{ankar, ankar1, Ant1, Bag2, Bag6, Frag3,ino, ino1,kschm,Trap3,FragCt}). Banach quasi *-algebras constitute the framework of this thesis. They form a special family of locally convex quasi *-algebras, whose topology is generated by a single norm, instead of a separating family of seminorms (see, for instance, \cite{Bag1,Bag4,Bag5,btt_meas}). The first part of the work concerns the study of representable functionals and their properties. The analysis is carried through the key notions of \textit{fully representability} and \textit{*-semisimplicity}, appeared in the literature in \cite{Ant1,Bag1,Bag5,Frag2}. In the case of Banach quasi *-algebras, these notions are equivalent up to a certain \textit{positivity condition}. This is shown in \cite{AT}, by proving first that every sesquilinear form associated to a representable functional is everywhere defined and continuous. In particular, Hilbert quasi *-algebras are always fully representable. The aforementioned result about sesquilinear forms allows one to select {\em well behaved} Banach quasi *-algebras where it makes sense to reconsider in a new framework classical problems that are relevant in applications (see \cite{Bade,Brat1,HP,Kish,Sakai,trap,weigt,WZ1,WZ2}). One of them is certainly that of derivations and of the related automorphisms groups (for instance see \cite{AT2,Alb,Ant4,Bag8,Brat2}). Definitions of course must be adapted to the new situation and for this reason we introduce weak *-derivations and weak automorphisms in \cite{AT2}. We study conditions for a weak *-derivation to be the generator of such a group. An infinitesimal generator of a continuous one-parameter group of uniformly bounded weak *-automorphisms is shown to be closed and to have certain properties on its spectrum, whereas, to acquire such a group starting with a certain closed * derivation, extra regularity conditions on its domain are required. These results are then applied to a concrete example of weak *-derivations, like inner qu*-derivation occurring in physics. Another way to study representations of a Banach quasi *-algebra is to construct new objects starting from a finite number of them, like \textit{tensor products} (see \cite{ada,fiw,fiw1,hei,hel,lau,lp,sa}). In \cite{AF} we construct the tensor product of two Banach quasi *-algebras in order to obtain again a Banach quasi *-algebra tensor product. We are interested in studying their capacity to preserve properties of their factors concerning representations, like the aforementioned full representability and *-semisimplicity. It has been shown that a fully representable (resp. *-semisimple) tensor product Banach quasi *-algebra passes its properties of representability to its factors. About the viceversa, it is true if only the pre-completion is considered, i.e. if the factors are fully representable (resp. *-semisimple), then the tensor product pre-completion normed quasi *-algebra is fully representable (resp. *-semisimple). Several examples are investigated from the point of view of Banach quasi *-algebras.