By Cordero L.A. (ed.)

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This is sufficient for the Baum—Connes conjecture, but not for our generalized Guillemin-Sternberg conjecture. 12 Cf. [Landsman (2003)] for an exposition of the link between the analytic assembly map and C*-algebraic deformation quantization, following Connes's discussion of this map in E-theory [Connes (1994)]. 13 A complication arises when M does not admit a G invariant Spinc structure. For techniques to overcome this cf. [Hilsum and Skandalis (1987); Paradan (2001b)]. 42 N. P. Landsman K0{C*{G)) under the map K0(C*{G)) -> Z induced in K-theory by the *-homomorphism / >-> / G dx f(x) from C*(G) to C.

This means (cf. [Guillemin, Ginzburg and Karshon (2002)], App. 5 When M is compact, the operators fo± determined by the Spin0 structure (P, =) have finite-dimensional kernels, whose dimensions define the quantization of (M,w) as Q(M,w) = index(#)+) = dimker(#>+) - dimker($>~). 4) In fact, the corresponding Hilbert space operators p^ are Fredholm, and by elliptic regularity index(p ) coincides with the Fredholm index dimker(^>+) - dimker(^)~) of p+. This notion of quantization just associates an integer to (M,u>).

J)+tE+(S),E-(S)]. 9) indeed maps this element to G-index(^)+). Apart from the dual pairpt <— M —» g l , the momentum map associated to the action G (J M equally well leads to a dual pair Q*_ <— M~ —> pt. This is to be quantized by an element of KK{C*(G),C) = K°(C*(G)), the socalled Kasparov representation ring of G (cf. [Higson and Roe (2000)]). This time, we interpret the Hilbert spaces L 2 ^ * ) as C*(G)-C Hilbert bimodules, where the pertinent representations TT* of C* (G) are given by a very slight adaptation of the procedure sketched in the preceding paragraph: to obtain left actions instead of right actions, we now put 7r ± (/) = JG dx f(x)U±(x).

### Differential geometry by Cordero L.A. (ed.)

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