On the double commutant of Cowen–Douglas operators

Let T be a Cowen–Douglas operator. In this paper, we study the von Neumann algebra V⁎(T) consisting of operators commuting with both T and T⁎ from a geometric viewpoint. We identify operators in V⁎(T) with connection-preserving bundle maps on E(T), the holomorphic Hermitian vector bundle associated to T. By studying such bundle maps, the structure of V⁎(T) as well as information on reducing subspaces of T can be determined.