KK-Theory of C*-Categories and the Analytic Assembly Map

We define KK-theory spectra associated to C ? -categories and look at certain instances of the Kasparov product at this level. This machinery is used to give a description of the analytic assembly map as a natural map of spectra. Contents 1 Preliminaries 1 1.1 C ? -categories and their K-theory . . . . . . . . . . . . . . . . . . 1 1.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Hilbert modules over C ? -categories 7 2.1 Hilbert Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Some Computations . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 KK-theory 17 3.1 Definition of KK-theory . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Assembly 24 4.1 Finitely Generated Projective Hilb...