
Author(s)  Power, S.C. 

Title  Limit algebras: an introduction to subalgebrasof C*algebras 
Publisher  Longman 
Year of publication  1992 
Reviewed by  Miroslav Englis 
The process of taking direct limits is a basic tool for obtaining new objects from the old ones. This book is mostly concerned with applications of this process to finite dimensional (that is, matrix) C*algebras, and the main theme is how various properties of the limit algebra can be deduced from those of the limited ones and their connecting homomorphisms. The topics discussed include: AF and Glimm (UHF) algebras, matrix units and canonical maximal abelian subalgebras, the isomorphism problem for AF C*algebras and regular canonical subalgebras (topological binary relations and the essential support), chordal algebras, triangularity and semisimplicity, some homology (K0, Elliott's theorem), crossed products, and more.
Each chapter is supplemented by a set of exercises and some bibliographical notes. The book contains an index, a list of notation, and an extensive bibliography (14 pp.). The exposition is good, though it could probably be even better if it were sometimes a little more relaxed and discursive  a large portion of it is just in the definitiontheoremproof style. There is also hardly any need to use doublespaced typing for a manuscript prepared by LaTeX, but this may have been dictated by the publisher's demand for the uniform appearance of the RNM series.
There are other books on this subject, or on closely related ones: Davidson's Nest Algebras (Pitman 1988), Goodearl's Notes on real and complex C*algebras (Shiva 1982), or Stratila and Voiculescu's Representations of AF algebras and of the group U(°) (Springer 1975), for instance. The present volume, however, contains some new results, as well as many older ones which have not been hitherto available in the book form; and this gives the book value.
The prerequisites have been deliberately kept to a minimum (basic linear algebra and functional analysis), so as to make the book accessible to graduate students and nonspecialists; since, however, the topic is rather specialised, the main audience will probably remain limited to those who actively work in the C*algebra theory. I believe that the book deserves to be recommended for the departmental libraries of the former, and will certainly make a valued contribution to either departmental or personal libraries of the latter.