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 usualtheory of Banach and/or locally convex algebras, this approach is supported by basic results and a wide variety of examples.