It has been known since the 1950's that every contractive representation of the disk algebra is completely contractive. Surprisingly, for just about as long it has been an open problem to decide whether or not every contractive representation of the algebra of bounded analytic functions on the disk is completely contractive. An analogous problem of Davidson asks whether or not every contractive representation of a nest algebra is completely contractive. These algebras are examples of logmodular algebras.

We will present new proofs of some of the known results on these problems together with an operator space proof of a key result in Banach space theory. This talk is based on joint work with Mrinal Raghupathi.