Title: Operator theory on noncommutative varieties

Authors: Gelu Popescu

Issue: Volume 55 (2006), Issue 2, 389-442

Abstract: We develop a dilation theory on noncommutative varieties determined by row contractions $T := [T_1, \ldots, T_n]$ subject to constraints such as $$p(T_1, \ldots, T_n) = 0, \quad p \in \mathcal{P},$$ where $\mathcal{P}$ is a set of noncommutative polynomials. The model $n$-tuple is the universal row contraction $[B_1, \ldots, B_n]$ satisfying the same constraints as $T$, which turns out to be, in a certain sense, the \emph{maximal constrained piece} of the $n$-tuple $[S_1, \ldots, S_n]$ of left creation operators on the full Fock space on $n$ generators. The theory is based on a class of \emph{constrained Poisson kernels} associated with $T$ and representations of the $C^*$-algebra generated by $B_1, \ldots, B_n$ and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation.\par A characteristic function $\Theta_T$ is associated with any (constrained) row contraction $T$ and it is proved that $$I - \Theta_T \Theta_T^* = K_T K_T^*,$$ where $K_T$ is the (constrained) Poisson kernel of $T$. Consequently, for \emph{pure constrained} row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp.~commuting) row contraction $T$ can be expressed only in terms of the (resp.~constrained) characteristic function of $T$.\par We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.