A proof of the boundary theorem

Davidson, Kenneth R.

doi:10.1090/S0002-9939-1981-0603599-3

A proof of the boundary theorem
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by Kenneth R. Davidson PDF

Proc. Amer. Math. Soc. 82 (1981), 48-50 Request permission

Abstract:

This note contains a simple proof of the following theorem of Arveson: If $\mathcal {A}$ is an irreducible subspace of $\mathcal {B}(H)$, then the identity map ${\phi _0}(A) = A$ on $\mathcal {A}$ has a unique completely positive extension to $\mathcal {B}(H)$ if and only if the quotient map $q$ by the compact operators is not completely isometric on $\mathcal {S} = [\mathcal {A} + \mathcal {A}^*]$.

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