Quotients of standard Hilbert modules

By W. Arveson. 2005

72 citations recorded by the source corpus.

We initiate a study of Hilbert modules over the polynomial algebra that are obtained by completing with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity version of one of these. Standard Hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra, and their quotients by submodules give rise to universal solutions of nonlinear relations. Essentially all of the basic Hilbert modules that have received attention over the years are standard, including the Hilbert module of the -shift, the Hardy and Bergman modules of the unit ball, modules associated with more general domains in , and those associated with projective algebraic varieties. We address the general problem of determining when a quotient of an essentially normal standard Hilbert module is essentially normal. This problem has been resistant. Our main result is that it can be ``linearized'' in that the nonlinear relations defining the submodule can be reduced, appropriately, to linear relations through an iteration procedure, and we give a concrete description of linearized quotients.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/a40459eba74414362a8a639344ca144c925112b4

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A harmonic analysis approach to essential normality of principal submodules (2011, background): “In [3,4] Arveson raised the interesting question of whether homogeneous polynomial ideals lead to C∗-algebras of essentially normal operators.”
A new kind of index theorem (2005, background): “(Also, see [6], [27], for some subsequent work on this topic.”
Essentially normal Hilbert modules and K-homology (2008, methodology): “Therefore, the submodule [zn 1 + zn 2 + zn 3] of 3-shift Hilbert module on which focused by Arveson [5,7], is p-essentially normal for p > 3.”
Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients (2013, methodology): “This conjecture attracted a lot of attention [23, 24, 52, 53, 56, 57, 58, 62, 66, 74, 75, 76, 80, 81, 108], where the conjecture was proved in particular classes of submodules, but it is still far from being solved.”
The isomorphism problem for some universal operator algebras (2010, background): “is commutative [6]. This conjecture is still open; the most up-to-date results can be found in [22]. There are several signiﬁcant consequences of this conjecture treated in the literature (see, e.g., [7]). We will see below that another consequence is a connection between the ∗-algebraic structure o”