×

Characterizations of \(C^*\)-algebras. The Gelfand-Naimark theorems. (English) Zbl 0597.46056

This is an off-mainstream book on \(C^*\)-algebras. It is concerned with the characterization of \(C^*\)-algebras by different sets of axioms, thus, in particular, with the Gelfand-Naimark theorems and various generalizations. Much of the work in the first part of the book is caused by the fact that the authors define a \(C^*\)-algebra by the condition \(\| X^*X\| =\| X^*\| \| X\|\) rather than by the usual condition \(\| X^*X\| =\| X\|^ 2\) (which is more natural and easier to work with). The latter condition immediately implies the former, but to establish the converse requires a lot of work (contained in chapter 3). Chapters 3, 6 (hermitian and symmetric *- algebras), 7 (a further weakening of the \(C^*\)-axioms, 9 (locally \(C^*\)-equivalent algebras) contain results that have not yet appeared in book form before. The material in chapters 5 (*-representations and Hilbert space), 8 (geometrical characterizations of \(C^*\)-algebras), 10 (applications of the characterization theorems) can also be found elsewhere.
The choice of material and of references seems sometimes somewhat erratic. For instance, it is not clear why the authors do not include, or at least mention, the characterization of \(C^*\)-algebras by the existence of square roots due to Katznelson in the commutative case and valid, thanks to a result of the reviewer presented in chapter 9, also in the non-commutative case. At other occasions one feels a lack of comments on connections between results in the book and furthergoing or related research on the subject. In summary the book presents an attractive part of \(C^*\)-algebra theory and makes accessible to the non-specialist a body of results that are usually only scratched in books on operator algebras and would otherwise be difficult to locate in the literature. Its usefulness is enhanced by a large number of exercices. It should be a useful complement to the existing treatises on \(C^*\)-algebra theory.
Reviewer: J.Cuntz

MSC:

46L05 General theory of \(C^*\)-algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis