Lecture Notes in Mathematics

2 primary works

Partial Inner Product Spaces

by J-P. Antoine and Camillo Trapani

Published 4 February 2010

Partial Inner Product (PIP) Spaces are ubiquitous, e.g. Rigged Hilbert spaces, chains of Hilbert or Banach spaces (such as the Lebesgue spaces L^p over the real line), etc. In fact, most functional spaces used in (quantum) physics and in signal processing are of this type. The book contains a systematic analysis of PIP spaces and operators defined on them. Numerous examples are described in detail and a large bibliography is provided. Finally, the last chapters cover the many applications of PIP spaces in physics and in signal/image processing, respectively.

As such, the book will be useful both for researchers in mathematics and practitioners of these disciplines.

Locally Convex Quasi *-Algebras and their Representations

by Maria Fragoulopoulou and Camillo Trapani

Published 8 April 2020

This book offers a review of the theory of locally convex quasi *-algebras, authored by two of its contributors over the last 25 years. Quasi *-algebras are partial algebraic structures that are motivated by certain applications in Mathematical Physics. They arise in a natural way by completing a *-algebra under a locally convex *-algebra topology, with respect to which the multiplication is separately continuous. Among other things, the book presents an unbounded representation theory of quasi *-algebras, together with an analysis of normed quasi *-algebras, their spectral theory and a study of the structure of locally convex quasi *-algebras. Special attention is given to the case where the locally convex quasi *-algebra is obtained by completing a C*-algebra under a locally convex *-algebra topology, coarser than the C*-topology.
Introducing the subject to graduate students and researchers wishing to build on their knowledge of the usual theory of Banach and/or locally convex algebras, this approach is supported by basic results and a wide variety of examples.