By Cortinas G., et al. (eds.)

Given that its inception 50 years in the past, K-theory has been a device for realizing a wide-ranging kin of mathematical buildings and their invariants: topological areas, jewelry, algebraic kinds and operator algebras are the dominant examples. The invariants diversity from attribute sessions in cohomology, determinants of matrices, Chow teams of types, in addition to strains and indices of elliptic operators. therefore K-theory is remarkable for its connections with different branches of arithmetic. Noncommutative geometry develops instruments which permit one to think about noncommutative algebras within the similar footing as commutative ones: as algebras of services on (noncommutative) areas. The algebras in query come from difficulties in a variety of components of arithmetic and mathematical physics; regular examples contain algebras of pseudodifferential operators, workforce algebras, and different algebras bobbing up from quantum box thought. to check noncommutative geometric difficulties one considers invariants of the proper noncommutative algebras. those invariants contain algebraic and topological K-theory, and likewise cyclic homology, chanced on independently through Alain Connes and Boris Tsygan, which are looked either as a noncommutative model of de Rham cohomology and as an additive model of K-theory. There are fundamental and secondary Chern characters which move from K-theory to cyclic homology. those characters are suitable either to noncommutative and commutative difficulties and feature functions starting from index theorems to the detection of singularities of commutative algebraic forms. The contributions to this quantity symbolize this variety of connections among K-theory, noncommmutative geometry, and different branches of arithmetic.

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A/ be its kernel. The two canonical embeddings A ! A t A are sections for the folding homomorphism. A/. A/ ! B. Theorem 46. `2 N/. If F W S ! C is C -stable and split-exact, then F is homotopy invariant. This is a deep result of Nigel Higson [24]; a simple proof can be found in [11]. Besides basic properties of quasi-homomorphisms, it uses that inner endomorphisms act identically on C -stable functors (Example 40). Actually, the literature only contains Theorem 46 for functors on C alg. But the proof in [11] works for functors on categories S as above.

Proof. The Choi–Effros Lifting Theorem asserts that any extension of A has a completely positive contractive section. A; B/ if A is nuclear and G acts properly on A (see also [55]). It should be possible to weaken properness to amenability here, but I am not aware of a reference for this. Cred G/ of reduced group C -algebras [4]. We shall compare the approach of Davis and Lück [14] using homotopy theory for G-spaces and its counterpart in bivariant K-theory formulated in [38]. To avoid technical difficulties, we assume that the group G is discrete.

Alternatively, we can use G-equivariantly cp-split extensions in G-C sep. Q/ ! I ! E ! Q in KK . Such triangles are called extension triangles. A triangle in KKG is exact if and only if it is isomorphic to the extension triangle of a G-equivariantly cp-split extension [38]. Theorem 56. With the suspension automorphism and exact triangles defined above, KKG is a triangulated category. S/ if S Â G-C sep is closed under suspensions, G-equivariantly cp-split extensions, and Morita–Rieffel equivalence as in Theorem 50.