Curvature invariant for Hilbert modules over free semigroup algebras
by
Gelu Popescu
University of Texas at San Antonio

We introduce a notion of relative curvature (resp. Euler characteristic) for finite rank contractive Hilbert modules over C F_n⁺, the complex free semigroup algebra generated by the free semigroup F_n⁺ on n generators.

Asymptotic formulas and basic properties for both the curvature and the Euler characteristic are established. In particular, it is shown that the ``standard'' relative curvature invariant of a Hilbert module \" is a nonnegative number less than or equal to the ``rank'' of \", and it depends only on the properties of the completely positive map \phi_T(X): = \sum_i=1ⁿ T_iXT_i^*, where [T₁, ... , T_n] is the row contraction of (not necessarily commuting) operators uniquely determined by the C F_n⁺-module structure of \". Moreover, we proved that for every t >= 0 there is a Hilbert module \" such that \textcurv (\")=t.

The module structure defined by the left creation operators on the full Fock space F²(H_n) on n generators occupies the position of the rank-one free module in the algebraic theory. We obtain a complete description of the closed submodules (resp. quotients) of the free Hilbert module F²(H_n) and calculate their curvature invariant. It is shown that the curvature is a complete invariant for the finite rank submodules of the free Hilbert module F²(H_n)\otimes\K, where \K is a finite dimensional Hilbert space.

Date received: April 14, 2000