  The matrix normed structure of the unitization of a (non-selfadjoint)
operator algebra is determined by that of the original operator algebra. This
yields a classification up to completely isometric isomorphism of
two-dimensional unital operator algebras. This allows to define invariant
distances on the spectrum of commutative operator algebras analogous to the
Caratheodory distance for complex manifolds. Moreover, unitizations of
two-dimensional operator algebras with zero multiplication provide a rich class
of counterexamples. Especially, several badly behaved quotients of function
algebras are exhibited. Recently, Arveson has developed a model theory for
d-contractions. Quotients of the operator algebra of the d-shift are much more
well-behaved than quotients of function algebras. Completely isometric
representations of these quotients are obtained explicitly. This provides a
generalization of Nevanlinna-Pick theory. An important property of quotients of
the d-shift algebra is that their quotients of finite dimension r have
completely isometric representations by rxr-matrices. Finally, the class of
commutative operator algebras with this property is investigated.
