  We study the Hopf structure of a class of dual operator algebras
corresponding to certain semigroups. This class of algebras arises in dilation
theory, and includes the noncommutative analytic Toeplitz algebra and the
multiplier algebra of the Drury-Arveson space, which correspond to the free
semigroup and the free commutative semigroup respectively. The preduals of the
algebras in this class naturally form Hopf (convolution) algebras. The original
algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the
sense of Effros and Ruan. We study these algebras from this perspective, and
obtain a number of results about their structure.
