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# A Proof of the Boundary Theorem

Kenneth R. Davidson
Proceedings of the American Mathematical Society
Vol. 82, No. 1 (May, 1981), pp. 48-50
DOI: 10.2307/2044314
Stable URL: http://www.jstor.org/stable/2044314
Page Count: 3
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## Abstract

This note contains a simple proof of the following theorem of Arveson: If $\mathscr{A}$ is an irreducible subspace of $\mathscr{B}(H)$, then the identity map $\phi_0(A) = A$ on $\mathscr{A}$ has a unique completely positive extension to $\mathscr{B}(H)$ if and only if the quotient map $q$ by the compact operators is not completely isometric on $\mathscr{S} = \lbrack \mathscr{A} + \mathscr{A}^\ast \rbrack$.

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