We show that limit algebras
having interpolating spectrum are characterized by the property that all locally
contractive representations have ∗-dilations. This extends a result for digraph
algebras by Davidson. It is an open question if such a limit algebra is the limit of a
direct system of digraph algebras with interpolating digraphs, although a positive
answer would allow one to obtain one direction of our result directly from Davidson’s.
Instead, we give a ‘local’ construction of digraph algebras with interpolating digraphs
and use this to extend representations.
Tree algebras (in the sense of Davidson, Paulsen, and Power) have been
characterized by a commutant lifting property among digraph algebras with
interpolating digraphs. We show that the analogous result holds for limit algebras,
i.e., limit algebras with the analogous spectral condition are characterized by the
same commutant lifting property among the limit algebras with interpolating
spectrum.
