Singularity of the generator subalgebra in $q$-Gaussian algebras

TL;DR

This paper proves that in $q$-Gaussian algebras, the generator subalgebra is singular for all $q$ between -1 and 1, extending understanding of their algebraic structure.

Contribution

It establishes the singularity of the generator subalgebra in $q$-Gaussian algebras for all relevant $q$ values, a new result in operator algebra theory.

Findings

Generator subalgebra is singular in $q$-Gaussian algebras

Result holds for all $q$ in (-1, 1)

Applicable to separable real Hilbert spaces of dimension ≥ 2

Abstract

Given $-1 < q < 1$ and a separable real Hilbert space with dimension no less than 2, we prove that the generator subalgebra in the $q$-Gaussian algebra is singular.