On a Question of Arveson about Ranks of Hilbert modules

Details On a Question of Arveson about Ranks of Hilbert modules On a Question of Arveson about Ranks of Hilbert modules

2001-04-25

arxiv.org/abs/math/0104246v1

Mathematics

Operator Algebras

13 pages

Scientific paper

It's well known that the functional Hilbert space over the unit ball in $B_{d} \in C^d$, with kernel function $K(z,w)=\frac{1}{1-z_{1}w_{1}-... -z_{d}w_{d}}$, admits a natural $A(B_{d})$-module structure. We show the rank of a nonzero submodule is infinity if and only if the submodule is of infinite codimension. Together with Arveson's dilation theory, our result shows that Hilbert modules stand in stark contrast with Hilbert basis theorem for algebraic modules. This result answers a question of Arveson.

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