arXiv subjects
Publications and source records attributed to Ronald G. Douglas.
In this note we settle some technical questions concerning finite rank quasi-free Hilbert modules and develop some useful machinery. In particular, we provide a method for determining when two such modules are unitarily equivalent. Along the way we obtain representations for module maps and study how to determine the underlying holomorphic structure on such modules.
Consider a Hilbert space obtained as the completion of the polynomials C[z} in m-variables for which the mnonomials are orthogonal. If the commuting weighted shifts defined by the coordinate functions are essentially normal, then the same is true for their restrictions to invariant subspaces spanned by monomials. This generalizes the result of Arveson [4] in which the Hilbert space is the m-shift Hardy space H_m^2. He establishes his result for the case of finite multiplicity and shows the self-commutators lie in the Schatten p-class for p > m. We establish our result at the same level of generality. We also discuss the K-homology invariant defined in these cases.
For a quasi-free module over a function algebra $A(Ω)$, we define an analogue of the Berezin transform and relate this to the quotient of the C*-algebra it generates modulo the commutator ideal.
Let $\cl{M}$ be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(Ω)$, where $Ω\subseteq \bb{C}^m$ is a bounded domain. Let $\cl{M}_0\subseteq \cl{M}$ be the submodule of functions vanishing to order $k$ on a hypersurface $\cl{Z} \subseteq Ω$. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules $\cl{Q}=\cl{M} \ominus \cl{M}_0$. The invariants are given explicitly in the particular case of $k = 2$.
We determine the ideal structure of the Toeplitz C*-algebra on the bidisk.
The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $Ω$ in complex $m$ space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $β$ satisfying $\|ϕf \| \leq β\|ϕ\|_X \|f\|$ for some compact subset $X$ of $Ω$ and $ϕ$ in $\mathcal{A}$, $f$ in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.