The classical bounded symmetric domains

Abstract

The definition of bounded symmetric domains is geometric in nature and does not seem, at first glance, to be connected to the category of operators on Hilbert spaces. Nevertheless, we have already seen in the previous chapter that the spin factor has certain properties found in operator spaces, such as spectral decomposition, representation by Pauli matrices, and Peirce decomposition. Surprisingly, most BSDs are unit balls of operator spaces. Such domains are calledclassical domains.They provide a familiar setting in which to introduce some of the more abstract concepts connected with BSDs.