An Introduction to KK-Theory

KK-theory is one of the most important achievements of the field of Noncommutative Geometry. KK-theory was invented by Kasparov (Izv Akad Nauk SSSR Ser Mat 39(4):796–838, 1975; Izv. Akad Nauk SSSR Ser Ma. 44(3):571–636, 719, 1980; Invent Math 91:147–201, 1988), motivated by ideas of Atiyah (J Math Oxford Ser 19:113–140, 1968), and was further developed by Connes and Skandalis (Publ Res Inst Math Sci 20(6):1139–1183, 1984), who introduced an axiomatic approach to the intersection product (the composition operation in the category KK) and produced applications to families and foliation index theorems. The article (Baaj and Julg (C R Acad Sci Paris Sér I Math 296(21):875–878, 1983)) gives an important description of KK-theory in terms of unbounded operators.