A survey of some aspects of non-commutative geometry. (English) Zbl 0840.19004

Cuntz, J.

Jahresber. Dtsch. Math.-Ver. 95, No. 2, 60-84 (1993).

This survey paper gives a concise introduction to \(K\)- and \(KK\)-theory for \(C^*\)-algebras in a version that uses universal algebras and is due to the author himself [see \(K\)-Theory 1, 31-51 (1987; Zbl 0636.55001)]. The paper discusses the natural connection between the author’s version of \(KK\) and cyclic cohomology [cf. A. Connes and the author, Commun. Math. Phys. 114, No. 3, 515-526 (1988; Zbl 0664.46067)]. The simplest example of a universal algebra, \(q \mathbb{C}\), is also worked out in detail.

Reviewer: C.Farsi (MR 94c:46137)

MSC:

19K35	Kasparov theory (\(KK\)-theory)
46L87	Noncommutative differential geometry
19D55	\(K\)-theory and homology; cyclic homology and cohomology
46L80	\(K\)-theory and operator algebras (including cyclic theory)

Keywords:

\(KK\)-theory; \(C^*\)-algebras; cyclic cohomology; universal algebra