Index theory has had profound impact on many branches of mathematics. In this note we discuss the context for a new kind of index theorem. We begin, however, with some operator-theoretic results. In [11] Berger and Shaw established that finitely cyclic hyponormal operators have trace-class self-commutators. In [9], [31] Berger and Voiculescu extended this result to operators whose self-commutat...

Given an ideal a in A[x1, . . . , xn] where A is a Noetherian integral domain, we propose an approach to compute the Krull dimension of A[x1, . . . , xn]/a, when the residue class ring is a free A-module. When A is a field, the Krull dimension of A[x1, . . . , xn]/a has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetheri...

We consider projectivity and injectivity of Hilbert C*-modules in the categories of Hilbert C*-(bi-)modules over a fixed C*-algebra of coefficients (and another fixed C*-algebra represented as bounded module operators) and bounded (bi-)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characteri...

Let k be a field, and let R = k[x1, x2, x3]. Given a Hilbert function H for a cyclic module over R, we give an algorithm to produce a stable ideal I such that R/I has Hilbert function H and uniquely minimal graded Betti numbers among all R/J with the same Hilbert function, where J is another stable ideal in R. We also show that such an algorithm is impossible in more variables and disprove a re...

Normality of bounded and unbounded adjointable operators is discussed. If T is an adjointable operator on a Hilbert C*-module which has polar decomposition, then T is normal if and only if there exists a unitary operator U which commutes with T and T ∗ such that T = U T ∗. Kaplansky’s theorem for normality of the product of bounded operators is also reformulated in the framework of Hilbert C*-m...

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the AlgL-module U is initiated, where L is a completely distributive subspace lattice on a Hilbert space H. Furthermore, as an application of single elements, we study module isomorphisms between norm closed AlgN -modules, where N is a nest, and obtain the following result: Sup...

In this paper we study the unitary equivalence between Hilbert modules over a locally C∗-algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally C∗-algebra and show that a Hilbert module over a Fréchet locally C∗-algebra is countably generated if and only if the locally C∗-algebra of all ”compact” operators has an approximate unit. 2000 Mathemat...

We associate a sheaf model to a class of Hilbert modules satisfying a natural finiteness condition. It is obtained as the dual to a linear system of Hermitian vector spaces (in the sense of Grothendieck). A refined notion of curvature is derived from this construction leading to a new unitary invariant for the Hilbert module. A division problem with bounds, originating in Douady’s privilege, is...

With every E0–semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0–semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and a...

We determine the ideal structure of the Toeplitz C∗-algebra on the bidisk 0 Introduction A large part of doing research in mathematics is asking the right question. Posing a timely question can trigger thought and provoke insights, even when the matter has no good resolution. That is what happened in the case at hand. After devoting much time to the study of Hilbert modules and quotient Hilbert...