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 morphi sms oil separable metrisable algebras, and a compatibility condition is giv en which allows the composition of such morphisms. A class of algebras is d efined, with the property that every bounded set has compact closure, where the compatibility conditions are automatically satisfied. Three examples a re given in detail, the first involving a non-normable algebra due to Ellio tt, Natsume and Nest. The second is integration with respect to a certain q uasi-commutative spectral measure on the algebra of paths on a C* algebra, and the third the equivalence between the suspensions of the mapping cone a nd the ideal for a short exact sequence of C* algebras. (C) 2000 Academic P ress.