Seminar

The natural isomorphism of Rørdam groups R(X; A, B) with ideal-system equivariant KK-groups KK(X; A, B)

Classification of operator algebras: complexity, rigidity, and dynamics

01 February 14:00 - 15:00

Eberhard Kirchberg - Humboldt-Universität zu Berlin

Let A and B stable separable C*-algebras with A exact and B strongly purely infinite, X := Prim(B) and Ψ: I(B) → I(A) an “action of X on A” that is non-degenerate, lower semi-continuous and monotone upper semi- continuous. Consider the corresponding non-degenerate nuclear *-monomorphism h0 from A to B with h0 ⊕ h0 unitarily homotopic to h0 that defines the action Ψ coming from the Embedding Theorem. Let SB := C0((−∞, ∞), B) the suspension of B. The infinite repeat H := δ∞ ◦ h0 := h0 ⊕ h0 ⊕ · · · of h0 in M(B) ⊆ M(SB) and H0 := πSB ◦ H can be used to describe KK(C; A, B) =: KK(X; A, B) as the kernel of the natural map K1(H0(A)I ∩ E) → K1(E) ∼= K1(B) where Q(SB) := M(SB)/SB. The natural monomorphism from Cb([0, ∞), B)/C0([0, ∞), B) onto an ideal of Q(SB) defines the canonical epimorphism from R(C; A, B) onto KK(C; A, B). It turns out that this epimorphism is injective if and only if R(C; A, B) is homotopy invariant with respect to B. That it is surjective and that the homotopy invariance of R(C; A, B) with respect to B implies injectivity can be seen from this particular picture. We can see from an general abstract model G(h0; A, E) of R(C; A, B) and G(H0; A, E) of KK(C; A, B) and the natural group morphism k 1→ k ⊕ H0 from G(h0; A, E) into G(H0; A, E) that the natural map is surjective. I outline a direct proof of the injectivity of this epimorphism. It uses that Kasparov’s proof of the homotopy invariance of KK(A, B) generalizes to the KK(C; A, B), and that this gives a certain decomposition result for unitaries in M(CB) that commute modulo CB with H0(A). That the homotopy invariance of R(C; A, B) with respect to B implies injectivity can be seen directly from above given particular picture.
Organizers
Marius Dadarlat
Purdue University
Søren Eilers
University of Copenhagen
Asger Törnquist
University of Copenhagen

Program
Contact

Søren Eilers

eilers@math.ku.dk

Other
information

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