# 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].

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