This monograph is devoted to new variants of periodic cyclic homology, namely to analytic cyclic homology and local cyclic homology. To a great extent, it is based on the author’s thesis. The author first presents shortcomings of periodic cyclic homology and shows that it is useful to introduce analytic and local cyclic homology. He also investigates relations of these new cyclic homologies to Alain Connes’ entire cyclic homology. A central part of the monograph studies homological properties of these new cyclic homologies, for example invariance of analytic cyclic homology under homotopies of bounded variation and under analytically nilpotent extensions, the excision theorem, and invariance under the passage to isoradial subalgebras. It is also worth mentioning the relation between local and analytic cyclic homology and K-theory (Chern-Connes character and the universal coefficient theorem). Prerequisites for reading the monograph are functional analysis and homological algebra. For this reason, the author included chapter 1 (Bornological vector spaces and inductive systems) and chapter 3 (The spectral radius of bounded subsets and its applications). As the author himself states, these two chapters can be useful even for mathematicians not at all interested in cyclic homology. Necessary notions from algebra are also included in the appendix “Algebraic preliminaries”. The monograph is designed primarily for specialists, including postgraduate students. But in any case we must say that it is very well written. The index and the notation and symbols section are also very helpful.

Reviewer:

jiva