  We study the question of whether or not contractive representations of
logmodular algebras are completely contractive. We prove that a 2-contractive
representation of a logmodular algebra extends to a positive map on the
enveloping C*-algebra, which we show generalizes a result of Foias and Suciu on
uniform logmodular algebras. Our proof uses non-commutative operator space
generalizations of classical results on 2-summing maps and semispectral
measures. We establish some matrix factorization results for uniform logmodular
algebras
