# Journal of Operator Theory

Volume 34, Issue 2, Fall 1995 pp. 347-380.

Boundary sets for a contraction**Authors**: Bernard Chevreau (1), George R. Exner (2) and Carl M. Pearcy (3)

**Author institution:**(1) U.F.R. de Mathematiques et d'Informatique, 351 Cours de la Liberation, 33405 Talence Cedex, FRANCE

(2) Department of Mathematics, Bucknell University, Lewisburg, PA 17837, U.S.A.

(3) Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, U.S.A.

**Summary:**For any absolutely continuous contraction operator T on Hilbert space we produce a Borel set X_T contained in the unit circle $\mathbb T$; X_T localizes a sequence condition which, obtaining on all of $\mathbb T$, is equivalent to the membership of T in $\mathbb A_{\aleph_0}$ (the most restrictive of the classes of contractions arising from the Scott Brown theory). By consideration of X_T along with other subsets of $\mathbb T$ arising naturally from the minimal isometric dilation and minimal coisometric extension of T, we improve known results on the structure of dual operator algebras. Further results include a new characterization of membership in the class $\mathbb A_{1,\aleph_0}$ and that $T \in \mathbb A_{1,\aleph_0}$ implies $T^n \in \mathbb A_{n,\aleph_0}$

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