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Universal compressions of representations of $$H^{\infty}(G)$$. (English) Zbl 0627.46062
Let A be an algebra, and let $$\Phi$$ and $$\Psi$$ be two representations of A on Hilbert spaces H nd K, respectively. We say that $$\Psi$$ is a compression of $$\Phi$$ up to similarity if there exists a subspace M of H, and an invertible operator X:M$$\to K$$ such that $$\Psi (a)=XP_ M\Phi (a)X^{-1}$$ for all a in A. It is shown for $$A=H^{\infty}(G)$$, where G is a bounded open subset of $${\mathbb{C}}^ n$$, that there is a class of representations $$\Phi$$ with the property that “almost any” representation $$\Psi$$ of A is a compression of $$\Phi$$ up to similarity. Such a representation $$\Phi$$ is, for instance, the representation of $$H^{\infty}(G)$$ as multiplication operators on the Bergman space associated with G. The techniques used are related with dual algebras and complete boundedness.

##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 47A20 Dilations, extensions, compressions of linear operators
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##### References:
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