Dilations of limit algebras and interpolating spectrum

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. 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 X n = PHU n jH 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-Foia s 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 f7! 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 to