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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.

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47A20 Dilations, extensions, compressions of linear operators
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