Curvature Invariant for Hilbert Modules over Free Semigroup Algebras

Abstract

We introduce a notion of relative curvature (resp. Euler characteristic) for finite rank contractive Hilbert modules over $\text{C}$$\text{F}$⁺_n, the complex free semigroup algebra generated by the free semigroup $\text{F}$⁺_n on n generators. Asymptotic formulas and basic properties for both the curvature and the Euler characteristic are established. In particular, it is shown that the standard relative curvature invariant (resp. Euler characteristic) of a Hilbert module $\text{H}$ is a nonnegative number less than or equal to the rank of $\text{H}$, and it depends only on the properties of the completely positive map φ_T(X)≔∑ⁿ_i=1 T_iXT*_i, where [T₁, …, T_n] is the row contraction of (not necessarily commuting) operators uniquely determined by the $\text{C}$$\text{F}$⁺_n-module structure of $\text{H}$. Moreover, we prove that for every t⩾0 there is a Hilbert module $\text{H}$ such that curv($\text{H}$)=χ($\text{H}$)=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)⊗$\text{K}$, where $\text{K}$ is a finite dimensional Hilbert space. A noncommutative version of the Gauss–Bonnet–Chern theorem from Riemannian geometry is obtained for graded Hilbert modules over $\text{C}$$\text{F}$⁺_n. In particular, it is proved that the curvature and Euler characteristic coincide for certain classes of pure Hilbert modules. Our investigation is based on noncommutative Poisson transforms, noncommutative dilation theory, and harmonic analysis on Fock spaces.