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Summary: QUOTIENTS OF STANDARD HILBERT MODULES
WILLIAM ARVESON
Abstract. We initiate a study of Hilbert modules over the polynomial
algebra A = C[z1, . . . , zd] that are obtained by completing A with re-
spect 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 d-shift, the Hardy
and Bergman modules of the unit ball, modules associated with more
general domains in Cd
, and those associated with projective algebraic
varieties.
We address the general problem of determining when a quotient H/M
of an essentially normal standard Hilbert module H 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 M
can be reduced, appropriately, to linear relations through an iteration
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