 
Summary: PROJECTIVE MODULES ON FOCK SPACES
Alvaro Arias
Abstract. A Hilbert module over the free algebra generated by n noncommutative variables
is a Hilbert space H with n bounded linear operators. In this paper we use Hilbert module
language to study the semiinvariant subspaces of a family of weighted Fock spaces and their
quotients that includes the Full Fock space, the symmetric Fock space, the Dirichlet algebra,
and the reproducing kernel Hilbert spaces with a NevanlinnaPick kernel. We prove a com
mutant lifting theorem, obtain explicit resolutions and characterize the strongly orthogonally
projective subquotients of each algebra. We use the symbols associated with the commutant
lifting theorem to prove that two minimal projective resolutions are unitarily equivalent.
1. Introduction and Preliminaries
In [DPa], Douglas and Paulsen reformulated a part of single variable operator theory,
including aspects of the NagyFoias dilation theory, into the Hilbert module language
and proposed it as a guide to study multivariate function algebras. This approach was
strengthened by Muhly and Solel in [MS1], who studied more general operator algebras.
In this paper we use the Hilbert module language to study the semiinvariant subspaces of
a family of weighted Fock spaces and their quotients. This family includes the Full Fock
space, the symmetric Fock space, the Dirichlet algebra, and the reproducing kernel Hilbert
spaces with a NevanlinnaPick kernel.
We first prove a commutant lifting theorem, based on the recent paper of Clancy and
