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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.

##### 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)