  We present some general theorems about operator algebras that are algebras of
functions on sets, including theories of local algebras, residually finite
dimensional operator algebras and algebras that can be represented as the
scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use
these to further develop a quantized function theory for various domains that
extends and unifies Agler's theory of commuting contractions and the
Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous
factorization theorems, prove that the algebras that we obtain are dual
operator algebras and show that for many domains, supremums over all commuting
tuples of operators satisfying certain inequalities are obtained over all
commuting tuples of matrices.
