On the double commutant of Cowen-Douglas operators (Q630793)

It is shown as the main result of this paper that for any Cowen-Douglas operator \(T\in B_n(\Omega)\) [\textit{M. J. Cowen} and \textit{R.G. Douglas}, Acta Math. 141, 187--261 (1978; Zbl 0427.47016)], the von Neumann algebra \(V^*(T)\) consisting of operators commuting with \(T\) and \(T^*\) is isomorphic the commutant of matrix algebra \(M({\mathcal N}, \otimes {\mathcal M})\in M_n(C)\) which is generated by the curvature \({\mathcal K}\) and its covariant derivatives \({\mathcal K}_{z^i, \overline{z}^j}\), where \({\mathcal K}\) is the curvature of the bundle \((E_T, D)\) determined by \(T\). By this theorem, the authors give a geometrical characterization of \(V^*(T)\).