Localization and Non-propagation in a Groupoid Framework

TL;DR

This paper studies the properties of normal elements in the C* algebra of certain groupoids, providing estimates on their functional calculus and implications for the local behavior of their evolution groups.

Contribution

It introduces a framework for representing normal elements of groupoid C* algebras as operators on dense orbits and establishes norm estimates related to their functional calculus.

Findings

Norm estimates for products of functional calculus elements and multiplication operators.

Uniform bounds on the local behavior of the evolution group of these operators.

Representation of algebra elements as normal operators on dense orbits.

Abstract

Normal elements (or multipliers) of the C* algebra of a certain class of locally compact groupoids admit a natural faithful representation as normal operators on the $L^2$-space of a dense orbit of the groupoid. We prove norm estimates on the product between elements of the functional calculus of these operators and multiplication operators, subject to suitable restrictions expressed in terms of the orbit structure of the groupoid. As a consequence, one gets uniform estimates on the local behavior of the evolution group of the operators.