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

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)