BOUNDARY REPRESENTATIONS FOR FAMILIES OF REPRESENTATIONS OF OPERATOR ALGEBRAS AND SPACES

In analogy with the peak points of the Shilov boundary of a uni- form algebra, Arveson defined the notion of boundary representations among the completely contractive representations of a unital operator algebra. How- ever, he was unable to show that such representations always exist. Drop- ping his original condition that such representations should be irreducible, we show that a family of representations (in Agler's sense) of either an opera- tor 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.