Abstract

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.

Authors