A KK-like picture for E-theory of C*-algebras

2017/03/31 by Manuilov, Vladimir · 2 citations
#46L80 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1703.10781

Abstract

Let A, B be separable C*-algebras, B stable. Elements of the E-theory group E(A,B) are represented by asymptotic homomorphisms from the second suspension of A to B. Our aim is to represent these elements by (families of) maps from A itself to B. We have to pay for that by allowing these maps to be even further from *-homomorphisms. We prove that E(A,B) can be represented by pairs (φ⁺,φ^-) of maps from A to B that are not necessarily asymptotic homomorphisms, but have the same deficiency from being ones. Not surprisingly, such pairs of maps can be viewed as pairs of asymptotic homomorphisms from some C*-algebra C that surjects onto A, and the two maps in a pair should agree on the kernel of this surjection. We give examples of full surjections C→ A, i.e. those, for which all classes in E(A,B) can be obtained from pairs of asymptotic homomorphisms from C.

Cited by

• Scalability and asymptotic adjunction
• On generalized morphisms associated to endofunctors of C*-algebras

Related

• E-theory is a special case of KK-theory
• Asymptotically split extensions and E-theory
• Asymptotic homomorphisms into the Calkin algebra
• A topology on E-theory
• Extensions of C*-algebras and translation invariant asymptotic homomorphisms