Strongly Asymptotic Morphisms on Separable Metrisable Algebras

Asymptotic morphisms on C* algebras and their compositions were introduced by Connes and Higson. This paper considers definitions of asymptotic morphisms on separable metrisable algebras, and a compatibility condition is given which allows the composition of such morphisms. A class of algebras is defined, with the property that every bounded set has compact closure, where the compatibility conditions are automatically satisfied. Three examples are given in detail, the first involving a non-normable algebra due to Elliott, Natsume and Nest. The second is integration with respect to a certain quasi-commutative spectral measure on the algebra of paths on a C* algebra, and the third the equivalence between the suspensions of the mapping cone and the ideal for a short exact sequence of C* algebras.