Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1404.5494

Spectral Triples on Carnot Manifolds

Stefan Hasselmann
View original source ↗Read source document ↗Export metadata ↓

Abstract

We analyze whether one can construct a spectral triple for a Carnot manifold $M$, which detects its Carnot-Carathéodory metric and its graded dimension. Therefore we construct self-adjoint horizontal Dirac operators $D^H$ and show that each horizontal Dirac operator detects the metric via Connes' formula, but we also find that in no case these operators are hypoelliptic, which means they fail to have a compact resolvent. First we consider an example on compact Carnot nilmanifolds in detail, where we present a construction for a horizontal Dirac operator arising via pullback from the Dirac operator on the torus. Following an approach by Christian Bär to decompose the horizontal Clifford bundle, we detect that this operator has an infinite dimensional kernel. But in spite of this, in the case of Heisenberg nilmanifolds we will be able to discover the graded dimension from the asymptotic behavior of the eigenvalues of this horizontal Dirac operator. Afterwards we turn to the general case, showing that any horizontal Dirac operator fails to be hypoelliptic. Doing this, we develop a criterion from which hypoellipticity of certain graded differential operators can be excluded by considering the situation on a Heisenberg manifold, for which a complete characterization of hypoellipticity in known by the Rockland condition. Finally, we show how spectral triples can be constructed from horizontal Laplacians via the Heisenberg pseudodifferential calculus developed by Richard Beals and Peter Greiner. We suggest a few of these constructions, and discuss under which assumptions it may be possible to get an equivalent metric to the Carnot-Carathéodory metric from these operators. In addition, we mention a formula by which the Carnot-Carathéodory metric can be detected from arbitrary horizontal Laplacians.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefan Hasselmann. 2014-04-22. Spectral Triples on Carnot Manifolds. https://arxiv.org/abs/1404.5494

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Maximal Ergodic Theorems for Operators with Finite Peripheral Spectrum

Let $\mathcal M$ be a semifinite von Neumann algebra and $T : \mathcal{M} \to \mathcal{M}$ be a positive $L_\infty-L_1$ contraction in the sense of Junge-Xu, of which the numerical range, when viewed as an operator on $L_2(\mathcal M),$ is contained in a closed polygon with vertices on the unit circle. In this article, we prove that there exists a positive constant $C_p(T)$ such that \begin{equation}\label{abstract1stin} \Big\|\sup_{n \ge 0}\!^{+} T^n x \Big\|_p \le C_p(T)\, \|x\|_p \end{equation} for all \( x \in L_p(\mathcal{M}) \), $1<p<\infty$ extending some noncommutative maximal ergodic inequalities proved by Junge-Xu \cite{junge-Xu} and later generalized by Bekjan \cite{Bekjan2008}. In the commutative setting, similar inequalities as in \eqref{abstract1stin} hold for arbitrary $L_\infty-L_1$ contractions with the same condition in the numerical range, yielding a vast generalization of a classical maximal ergodic theorem of Stein \cite{Stein-ergodic-theorem} proved in 1960s. Moreover, we establish a noncommutative weak-type maximal inequality for convolution powers which was proved by Calderón and Bellow \cite{Bellow-Calderon} in the classical setting, complementing our strong type noncommutative maximal ergodic inequalities. Our method relies on several new polynomial identities, suitable square function estimates tailored to fit our setting and generalization of Stein's method of embedding maximal function into analytic family of operators. However, we show that even in the classical setting, the variational inequality extending \eqref{abstract1stin} holds for arbitrary operators described above, precisely when the spectrum meets the unit circle only at $1.$

math.OA↗
arXiv2026

The noncommutative topological factor theorem for rank-one product lattices

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that, for lattices in connected semisimple real Lie groups with finite center and no compact factors, the scalar-expectation case of the corresponding classification is equivalent to ordinary ITAP.

math.OA↗
arXiv2026

Representation stability for compact and discrete quantum groups

We study approximate representations of locally compact quantum groups and prove stability results in the sense of Ulam in this context. Our main result is that compact and amenable discrete quantum groups are representation stable. We also show that an analogous stability result holds for unitary compressions of general amenable locally compact quantum groups without the assumption of compactness or discreteness.

math.OA↗