On graded $K$-theory, elliptic operators and the functional calculus

Abstract

Let A be a graded C * -algebra. We characterize Kasparov's K-theory groupK 0 (A) in terms of graded * -homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.

Positive scalar curvature and product formulas for secondary index invariants

“…The long exact sequence in K-theory associated to this short exact sequence yields the desired (equivariant) Higson-Roe sequence after identifying K-homology with the K-theory of C * L (X) Γ via the local index map. The main technical novelty of this paper is that we combine Yu's localization algebras with the description of K-theory for graded C * -algebras due to Trout [22]. That is, we view the K 0 -group of a Z 2 -graded C * -algebra A as the set of homotopy classes of graded * -homomorphisms S → A ⊗K, where S is equal to C 0 (R) but graded into even and odd functions.…”

Coarse indices of twisted operators

“…A more detailed exposition of E-theory, which is based on a slightly different definition, is found in [GHT00]. Furthermore, we treat K-theory of graded C * -algebras simply as the special case K(B) = E(C, B), which is essentially the spectral picture of K-theory of [Tro00].…”

“…In this special case, it is not even necessary to use asymptotic morphisms, because E(C, B) ∼ = S, B ⊗ K 0 by [HG04, Proposition 1.3]. This is the spectral picture of K-theory of [Tro00].…”

A non-commutative model for higher twistedK-theory

“…Thus, K n → ΩK n+1 is a weak equivalence for n 1, which proves the isomorphism in (a). By [33,Theorem 4.7] and Bott periodicity, we have natural isomorphisms…”

“…Following [14], we denote the C *algebra C 0 (R) graded by odd and even functions by S. This is a coassociative, counital, coalgebra with comultiplication Δ : S → S ⊗ S and counit : S → C (see [17,21]). It has the universal property that, for any graded σ-unital C * -algebra B, there is a correspondence between essential graded * -homomorphisms ϕ : S → B and unbounded, self-adjoint, regular, odd operators T : B → B (see [33,Proposition 3.1]). Moreover, let C 1 be the complex Clifford algebra spanned by the even element 1 and the odd element c with c 2 = 1 and let e ∈ K be a fixed rank 1-projection.…”