Heat-smoothing for holomorphic subalgebras of free group von Neumann algebras

TL;DR

This paper establishes sharp heat-smoothing inequalities for holomorphic subalgebras in free group von Neumann algebras, extending known results from Gaussian spaces to non-commutative settings.

Contribution

It proves sharp heat-smoothing inequalities for holomorphic subalgebras in free group von Neumann algebras, generalizing Gaussian space results to non-commutative operator algebras.

Findings

Sharp inequalities for holomorphic subalgebras in free group von Neumann algebras

Results extend to q-Gaussian algebras and quantum tori

Weaker heat-smoothing results with optimal order in free group von Neumann algebras

Abstract

The heat semigroup on discrete hypercubes is well-known to be contractive over $L_p$-spaces for $1<p<\infty$. A question of Mendel and Naor \cite{MN14} concerns a stronger contraction property in the tail spaces, which is known as the heat-smoothing conjecture. Eskenazis and Ivanisvili \cite{EI20} considered a Gaussian analog of this conjecture and resolved some special cases. In particular, they proved that heat-smoothing type conjecture holds for holomorphic functions in the Gaussian spaces with sharp constants. In this paper, we prove analogous sharp inequalities for holomorphic subalgebras of free group von Neumann algebras. Similar results also hold for $q$-Gaussian algebras and quantum tori. In the case of free group von Neumann algebras, the weaker formulation of heat-smoothing is proved with optimal order.