\(E\)-theory with \(*\)-homomorphisms (Q1923960)

The aim of this paper is to show how \(E\)-theory introduced by \textit{A. Connes} and \textit{N. Higson} [C. R. Acad. Sci. Paris, Sér. I 311, No. 2, 101-106 (1990; Zbl 0717.46062)], see also \textit{M. Dadarlat} [\(K\)-theory 8, No. 5, 465-482 (1994; Zbl 0821.19003)] can be constructed by using only \(*\)-homomorphisms. The homomorphisms which are used are those given by equivalence classes of asymptotic homomorphisms. So, the steps of previous beginning works are followed by translating everything into the language of homomorphisms; the product and the excision property are given easier descriptions in this context. Moreover, ocasionally, some general results which might bring a better understanding of certain \(E\)-theoretical constructions (the Puppe exact sequence, for example) are indicated.