### Topics in Operator Theory, Lecture 7: boundary representations

In this lecture we will present a proof that boundary representations exist in abundance, following Davidson and Kennedy’s breakthrough paper. Davidson and Kennedy’s paper was in the spirit of Arveson’s paper from 1969, and followed Arveson’s solution in the separable case from 2007. (BTW, I wrote about Davidson and Kennedy’s solution in a an old blog post).

#### 1. The unique extension property and maximal representations

Recall the definition of a boundary representation.

Our setting will be of an operator system $S$ contained in a C*-algebra $B = C^*(S)$. Recall that earlier we discussed the situation of a unital operator algebra $A \subseteq B = C^*(A)$, and later we extended our attention to unital operator spaces. In this post we will consider only operator systems, but there will be no loss of generality (because every unital completely contractive map $A \mapsto B(H)$ extends to a unique unital completely positive map $S: A + A^* \to B(H)$, and vice versa).

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