Finite-Dimensional Approximations and Semigroup Coactions for Operator Algebras

Abstract The residual finite-dimensionality of a ${\textrm{C}}^{\ast }$-algebra is known to be encoded in topological property its space representations, stating that finite-dimensional representations should dense therein. We extend this paradigm general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge greater generality, we exhibit novel tools for constructing approximations. One such tool notion residually coaction semigroup on an algebra, which allows us construct approximations algebras functions and semigroups. Our investigation intimately related the question when algebra inherited by maximal }$-cover, establish many cases interest.