Euler characteristic on noncommutative polyballs

TL;DR

This paper introduces an Euler characteristic for algebraic modules in noncommutative polyballs, establishing its properties, invariance, and geometric connections, including a noncommutative Gauss-Bonnet-Chern theorem.

Contribution

It defines a new invariant called the Euler characteristic for modules in noncommutative polyballs and proves its fundamental properties and applications.

Findings

Euler characteristic is a complete unitary invariant for certain subspaces.

Its range covers all non-negative real numbers.

An analogue of the Gauss-Bonnet-Chern theorem is established in this setting.

Abstract

In this paper we introduce and study the Euler characteristic associated with algebraic modules generated by arbitrary elements of certain noncommutative polyballs. We provide several asymptotic formulas and prove some of its basic properties. We show that the Euler characteristic is a complete unitary invariant for the finite rank Beurling type invariant subspaces of the tensor product of full Fock spaces $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$, and prove that its range coincides with the interval $[0,\infty)$. We obtain an analogue of Arveson's version of the Gauss-Bonnet-Chern theorem from Riemannian geometry, which connects the curvature to the Euler characteristic. In particular, we prove that if M is an invariant subspace of $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$, $n_i\geq 2$, which is graded (generated by multi-homogeneous polynomials), then the curvature and the Euler characteristic of the orthocomplement of M coincide.