DILATIONS OF LIMIT ALGEBRAS AND INTERPOLATING SPECTRUM

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 Po-wer) have been characterized by a commutant lifting prop-erty 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 char-acterized by the same commutant lifting property among the limit algebras with interpolating spectrum. Dilation theory for operator algebras has its origin in the Sz.-Nagy dilation theorem, that every contractive operator on Hilbert space has a unitary dilation [SN]. That is, given a contractive operator X on a Hilbert space H, there a Hilbert space K containing H and a unitary operator U on K so that Xn = PHUnjH for n 1. Ando’s theorem [An] extends this result to two operators, i.e., any pair of commuting contractions has a unitary dilation. This fails for three or more operators [Pa]. Closely related to Ando’s theorem is the Sz.-Nagy-Foias commutant lifting theorem [SNF1], that given a contraction T with unitary dilation U and a contraction X commuting with T, there is a contractive dilation of X commuting with U. For a unied treatment of these topics, see the monograph [SNF2]. From one point of view, these results are about dilating representations of the disk algebra. By von Neumann’s inequality, each contraction T induces a contractive representation of disk algebra, given by f 7! f(T). Conversely, a contractive representation of disk algebra gives a contractive operator, the image of the function f(z) = z. Thus, Sz.-Nagy’s theorem is equivalent t