Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.26049

Noncommutative protori and inductive spectral triples

Remus Floricel·Patrick Melanson
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study inductive limits of higher-dimensional noncommutative tori, which we call noncommutative protori. We compute the Elliott invariants for broad classes of unital and nonunital systems, including toric maps, Morita-corner embeddings, and dimension-changing and proper embeddings. For the resulting simple limits we determine explicitly the ordered $K$-groups, trace cone, scale, and projection scale, yielding concrete classification criteria. We also construct compatible spectral triples and locally compact spectral triples on these limits via Fourier- and Morita-compatible Dirac structures.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Remus Floricel, Patrick Melanson. 2026-05-25. Noncommutative protori and inductive spectral triples. https://arxiv.org/abs/2605.26049

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↗