Heat properties for groups

TL;DR

This paper explores heat properties related to groups, analyzing convergence, semigroups, and the impact of properties like Kazhdan's (T) and Haagerup on heat equations in group C*-algebras.

Contribution

It introduces new heat properties for countably infinite groups and investigates their relation to group properties like Kazhdan's (T) and Haagerup.

Findings

Kazhdan's property (T) obstructs the weakest heat property.

Groups with the Haagerup property satisfy the strongest heat property.

The strongest heat property ensures unique solutions to the heat problem.

Abstract

We revisit Fourier's approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan's property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum.