Abstract
A version of Ascoli's Theorem (equating compact and equicontinuous sets) is presented in the context of convergence spaces. This theorem and another, (involving equicontinuity) are applied to characterize compact subsets of quasi-multipliers of a $C^*$-algebra B, and to characterize the compact subsets of the state space of B. The classical Ascoli Theorem states that, for pointwise pre-compact families F of continuous functions from a locally compact space Y to a complete Hausdorff uniform space Z, equicontinuity of F is equivalent to relative compactness in the compact-open topology([4] 7.17). Though this is one of the most important theorems of modern analysis, there are some applications of the ideas inherent in this theorem which arc not readily accessible by direct appeal to the theorem. When one passes to so-called "non-commutative analysis", analysis of non-commutative $C^*$-algebras, the analogue of Y may not be relatively compact, while the conclusion of Ascoli's Theorem still holds. Consequently it seems plausible to establish a more general Ascoli Theorem which will directly apply to these examples.