zbMATH — the first resource for mathematics

Homogeneous operators and systems of imprimitivity. (English) Zbl 0867.47023
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 67-76 (1995).
Let \(\Omega\subset \mathbb{C}^d\) be a bounded symmetric domain and let \({\mathcal A}(\Omega)\) be the linear space of all functions holomorphic in some neighbourhood of \(\overline{\Omega}\), endowed with the sup norm. Let \(G\) be the connected component of identity in the group of biholomorphic automorphisms of \(\Omega\). Let also \({\mathcal H}\) be a separable Hilbert space and let \({\mathcal L}({\mathcal H})\) be the algebra of all bounded linear operators on \({\mathcal H}\). A bounded homomorphism \(\rho:{\mathcal A}(\Omega)\to{\mathcal L}({\mathcal H})\) is said to be homogeneous if for each \(\varphi\in G\), \(\rho(f)\) is unitarily equivalent to \(\rho(\varphi\cdot f)\) for all \(f\in{\mathcal A}(\Omega)\) via a fixed unitary operator \(U_\varphi\), where \((\varphi\cdot f)(\omega)= f(\varphi^{-1}(\omega))\).
The aim of this work is to initiate a study of homogeneous homomorphisms. The authors exhibit a large class of examples and then show that these homomorphisms are tractable. The motivation of this study is the equality \(U_\varphi \rho(f)U^*_\varphi= \rho(\varphi\cdot f)\), valid for irreducible homogeneous homomorphisms, called the imprimitivity relation, which formally resembles the Mackey systems of imprimitivity.
For the entire collection see [Zbl 0819.00022].

47B38 Linear operators on function spaces (general)
22E46 Semisimple Lie groups and their representations
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)