Families of homomorphisms in non-commutative gelfand theory: Comparisons and examples

In non-commutative Gelfand theory, families of Banach algebra homomorphisms, and particularly families of matrix representations, play an important role. Depending on the properties imposed on them, they are called sufficient, weakly sufficient, partially weakly sufficient, radical-separating or separating. I n this paper these families are compared with one another. Conditions are given under which the defining properties amount to the same. Where applicable, examples are presented to show that they are genuinely different.