# Journal of Operator Theory

Volume 53, Issue 1, Winter 2005 pp. 159-167.

Boundary Representations for Families of Representations of Operator Algebras and Spaces**Authors**: Michael A. Dritschel (1) and Scott A. McCullough (2)

**Author institution:**(1) epartment of Mathematics, School of Mathematics and Statistics, Merz Court, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK

(2) Department of Mathematics, University of Florida, Box 118105, Gaines\-ville, FL 32611-8105, USA

**Summary:**In analogy with the peak points of the Shilov boundary of a uniform algebra, Arveson defined the notion of boundary representations among the completely contractive representations of a unital operator algebra. However, he was unable to show that such representations always exist. Dropping his original condition that such representations should be irreducible, we show that a family of representations (in Agler's sense) of either an operator algebra or an operator space has boundary representations. This leads to a direct proof of Hamana's result that all unital operator algebras have enough such boundary representations to generate the C*-envelope.

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