[Paper Review] Spectral Theory in a Twisted Groupoid Setting: Spectral Decompositions, Localization and Fredholmness

Marius Măntoiu, Victor Nistor|arXiv (Cornell University)|Dec 11, 2018

Advanced Operator Algebra Research41 references4 citations

TL;DR

This paper develops a spectral theory framework for twisted groupoid C*-algebras associated with amenable, Hausdorff, second countable locally compact groupoids endowed with a continuous 2-cocycle. It establishes spectral decompositions, essential spectrum and Fredholm properties via boundary orbit analysis, and proves localization and non-propagation theorems for operators in the algebra, with applications to magnetic pseudo-differential operators on nilpotent groups.

ABSTRACT

We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.

Motivation & Objective

• To establish a comprehensive spectral theory for bounded operators in the regular representations of twisted groupoid C*-algebras.
• To analyze the essential spectrum, essential numerical range, and Fredholm properties of operators associated with natural quotients of twisted groupoid C*-algebras.
• To derive spectral decompositions for regular representations associated with units in free, locally closed orbits, using spectral data from boundary orbits.
• To prove localization and non-propagation properties for operators in the essential spectrum, particularly when the groupoid restriction at the boundary is totally intransitive.
• To apply the framework to magnetic pseudo-differential operators on nilpotent Lie groups, including the Heisenberg group, via twisted Wiener-Hopf operators.

Proposed method

• Utilizes twisted groupoid C*-algebras defined on amenable, Hausdorff, second countable, locally compact groupoids with a continuous 2-cocycle.
• Applies the regular representation and vector representation constructions, particularly in the case of standard groupoids with open dense orbits and trivial isotropy.
• Employs groupoid extensions and results from Brown and an Huef to realize twisted C*-algebras as ideals and direct summands in untwisted C*-algebras of extended groupoids.
• Introduces a compactification $X = M igsqcup X_∞$ of the groupoid's unit space, where $X_∞$ is the boundary at infinity, and equips it with a topology $τ(X)$ to analyze asymptotic behavior.
• Defines the algebra $C_{\sf p}(M)$ of bounded continuous functions on $M$ that are asymptotically constant along fibers $M_n^{\rm out}$, and uses it to characterize asymptotic spectral behavior.
• Applies the abstract localization result (Theorem 5.1) to show that spectral projections outside the spectrum of boundary operators $H_n$ exhibit rapid decay in the fiber directions away from compact sets.

Experimental results

Research questions

• RQ1 How can spectral decompositions of regular representations be expressed in terms of spectral data from boundary orbits of free, locally closed groupoid orbits?
• RQ2 What is the precise characterization of the essential spectrum and Fredholm property for operators in the twisted groupoid C*-algebra under boundary intransitivity?
• RQ3 In what sense do spectral projections associated with the essential spectrum exhibit non-propagation or localization in the fiber directions?
• RQ4 How do the spectral properties of magnetic pseudo-differential operators on nilpotent groups emerge from the twisted groupoid framework?
• RQ5 Under what conditions does the essential spectrum of an operator in the twisted algebra avoid certain spectral sets, leading to localization?

Key findings

• For any normal element $F \in L^{\infty,1}_{\rm cont}(\Xi) \subset C^*(\Xi,\omega)$ and a point $n \in X_\infty$, if $\kappa \in C_0(\mathbb{R})$ has support disjoint from the spectrum of the boundary operator $H_n$, then the spectral projection $\kappa(H_0)$ decays rapidly in the fiber directions away from compact sets.
• The operator norm $\| \mathbf{1}_{\{m \notin K \mid \sf p(m) \in E\}} \kappa(H_0) \|_{\mathbb{B}(\mathcal{H}_0)} \leq \epsilon$ for some neighborhood $E$ of $n$ and compact $K \subset M$, uniformly in $\epsilon > 0$.
• For self-adjoint $F$, the evolution $e^{itH_0}\kappa(H_0)u$ exhibits uniform decay in norm outside compact sets in the fiber, uniformly in $t \in \mathbb{R}$ and $u \in \mathcal{H}_0$.
• The compactification $X = M \sqcup X_\infty$ is Hausdorff and compact, with $M$ dense in $X$, and the topology $\mathcal{T}(X)$ ensures continuity of functions that are asymptotically constant along fibers.
• The algebra $C_{\mathcal{T}(X)}(M)$ of restrictions to $M$ of continuous functions on $X$ strictly contains $C_0(M) + C_{\sf p}(M)$, providing a natural setting for asymptotic spectral analysis.
• When the boundary groupoid $\Sigma = \bigsqcup_{n \in X_\infty} \Sigma_n$ is totally intransitive, the spectral localization results hold with explicit control over decay rates in the fiber directions.