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

## TL;DR

This paper introduces a new way to represent elements of E-theory for C*-algebras using pairs of maps from the algebra itself, broadening the understanding of asymptotic homomorphisms.

## Contribution

It provides a novel representation of E-theory elements via pairs of maps from the algebra, extending the classical asymptotic homomorphism approach.

## Key findings

- E(A,B) can be represented by pairs of maps with the same deficiency from being homomorphisms
- Pairs of maps can be viewed as asymptotic homomorphisms from a surjecting C*-algebra C
- Examples of full surjections C→A show all classes in E(A,B) can be obtained from such pairs

## 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 $(\varphi^+,\varphi^-)$ 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 $C\to A$, i.e. those, for which all classes in $E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$.