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