$E$-Theory is a Special Case of $KK$-Theory

Abstract

Let A and B be C * -algebras, A separable, and B σ-unital and stable. It is shown that there are natural isomorphismsis the generalized Calkin algebra and K denotes the C * -algebra of compact operators of an infinite dimensional separable Hilbert space.

Pictures of $KK$ -theory for real $C^{*}$ -algebras and almost commuting matrices

“…Finally, we mention that with a similar application of Theorem 3.10, we can easily obtain the following real analog of Theorem 5.8 of [31], showing that Etheory is a special case of KK-theory. Since M(B ⊗ K R ) is KK-trivial, it follows that E(A, M(B ⊗ K R )) = 0.…”

“…Let ε ′ = γ • ε where ε is the isomorphism of Theorem 4.6. We know that ε ′ is an isomorphism in the complex case by Theorem 5.8 of [31]. So it suffices by Theorem 3.10 to show that ε ′ fits in a commutative square in the same way that ε does in the proof of Theorem 4.6.…”

“…Since then, many alternate but equivalent or closely related pictures of KK-theory have been introduced and developed by various authors ( [10], [11], [17], [18], [31], [37]). The ability to move among the various pictures has contributed immensely to the utility of KK-theory as a tool for solving problems.…”

The Calkin algebra has outer automorphisms

“…The homomorphism ψ will be a "folding" of an asymptotic morphism from B to M (E)/E obtained from ϕ, as in Section 3 of [7]. See especially Lemma 3.5 and the discussion after Remark 3.6 in [7].…”

“…Proposition 1.4 is a relative version of the construction of Section 3 of [7], which, starting from an asymptotic morphism from a separable C*-algebra to an outer multiplier algebra, produces a true homomorphism. In this context, "relative" means that if the restriction of the asymptotic morphism to a subalgebra comes from a homomorphism, then the homomorphism we construct can be chosen to agree with that homomorphism on the subalgebra.…”