arXiv subjects

Remus Floricel

Publications and source records attributed to Remus Floricel.

10 recordsLinked to original sources

arXiv2026

Noncommutative protori and inductive spectral triples

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.

math.OA↗

arXiv2026

Bures--Kuratowski metrics and simplicial complexes for completely bounded maps

Let $A$ be a unital $C^*$-algebra and $H$ a Hilbert space. The cone $\CP(A,B(H))$ of completely positive maps carries the Bures metric $\beta$, closely related to the cb-norm. We introduce a family of Bures--Kuratowski (BK) metrics on $\CB(A,B(H))$ that extend $\beta$ exactly on $\CP(A,B(H))$. The construction combines a Kuratowski embedding of the Bures cone, based at an anchor $\theta\in\CP(A,B(H))$, with a regular-representation Hausdorff coordinate arising from universal regular models. Each BK metric admits an $\ell^p$-wedge decomposition, splitting $\CB(A,B(H))$ into the Bures cone and a non-CP component attached at $\theta$. We then study Vietoris--Rips and \v{C}ech complexes of BK metric spaces. The wedge formula yields explicit criteria for mixed simplices, a join-type description of the mixed Rips complex, and ball-intersection criteria for mixed \v{C}ech simplices. For finite point clouds, this makes the mixed simplicial geometry computable from the two component metrics and reveals new homological features arising from the interaction between the CP and non-CP sectors.

math.OA↗

arXiv2026

Factorizing random sets and type III Arveson systems

We develop a representative-level framework for the Liebscher-Tsirelson random-set construction of Arveson systems from stationary factorizing measure types. We introduce the notion of a measurable factorizing family of probability measures on hyperspaces of closed subsets of time intervals and prove that every such family canonically generates an Arveson system. Within this framework we obtain a purely measure-theoretic characterization of spatiality: positive normalized units correspond exactly to dominated families of measures that factorize strictly. We then present a general mechanism for constructing type III Arveson systems via infinite products of measurable factorizing families. Starting from a type II$_0$ seed satisfying a quantitative Hellinger-smallness condition, we form a marked infinite product indexed by $[0,1]\times\mathbb N$ and show, using Kakutani's criterion, that the resulting product system admits no units. This yields a robust construction principle for type III random-set systems. As an application we analyze zero sets of Brownian motion. After anchor-adapted localization and Palm-type uniformization, the Brownian seed satisfies the required overlap estimates, and the associated infinite-product construction produces explicit examples of type III random-set systems, as anticipated in the work of Tsirelson and Liebscher.

math.PR↗

arXiv2024

Product systems arising from L\'evy processe

This paper investigates the structure of product systems of Hilbert spaces derived from Banach space-valued L\'evy processes. We establish conditions under which these product systems are completely spatial and show that Gaussian L\'evy processes with non-degenerate covariance always give rise to product systems of type I. Furthermore, we construct a continuum of non-isomorphic product systems of type $\rm{II}\sb\infty$ from pure jump L\'evy processes.

math.PR↗

arXiv2024

Convergence of Tsirelson convolution systems of probability spaces

We associate two specific projective systems of probability spaces with any Tsirelson convolution system. If the projective limits of these systems exist, then we call the convolution system convergent and $K$-convergent, respectively. It is shown that convergent convolution systems give rise to continuous products of probability spaces, while $K$-convergent convolution systems lead to flow systems. We investigate the relationship between convergence and $K$-convergence, as well as their connections to two-parameter product systems of Hilbert spaces.

math.PR↗

arXiv2022

$C^*$-subproduct and product systems

We introduce and study two-parameter subproduct and product systems of $C^*$-algebras as the operator-algebraic analogues of, and in relation to, Tsirelson's two-parameter product systems of Hilbert spaces. Using several inductive limit techniques, we show that (i) any $C^*$-subproduct system can be dilated to a $C^*$-product system; and (ii) any $C^*$-subproduct system that admis a unit, i.e., a co-multiplicative family of projections, can be assembled into a $C^*$-algebra, which comes equipped with a one-parameter family of comultiplication-like homomorphisms. We also introduce and discuss co-units of $C^*$-subproduct systems, consisting of co-multiplicative families of states, and show that they correspond to idempotent states of the associated $C^*$-algebras. We then use the GNS construction to obtain Tsirelson subproduct systems of Hilbert spaces from co-units, and describe the relationship between the dilation of a $C^*$-suproduct system and the dilation of the Tsirelson subproduct system of Hilbert spaces associated with a co-unit. All these results are illustrated concretely at the level of $C^*$-subproduct systems of commutative $C^*$-algebras.

math.OA↗

arXiv2018

Approximately Clean Quantum Probability Measures

A quantum probability measure--or quantum measurement--is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al (2005) for finite-dimensional Hilbert space, and was studied subsequently by Kahn (2007) and Pellonpää (2011). The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of `approximately clean quantum probability measures' and characterise this property for measures whose range determines a finite-dimensional operator system.

quant-ph↗

arXiv2017

On inductive limit spectral triples

Given an inductive system of spectral triples $\{(A_j,\H_j,D_j)\}_j$, we find conditions under which the triple $(\limind A_j,\limind H_j,\limind D_j)$ is a spectral triple. We also analyze and describe some classical examples of spectral triples in terms of these conditions.

math.QA↗

arXiv2017

The Ricci Curvature in Noncommutative Geometry

Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function of the full Laplacian of the de Rham complex, localized by smooth endomorphisms of the cotangent bundle and their trace. We use this formulation to introduce the Ricci functional in a noncommutative setting and in particular for curved noncommutative tori. This Ricci functional uniquely determines a density element, called the Ricci density, which plays the role of the Ricci operator. The main result of this paper provides an explicit computation of the Ricci density when the conformally flat geometry of the noncommutative two torus is encoded by the modular de Rham spectral triple.

math.QA↗

arXiv2003

A note on cocycle-conjugate endomorphisms of von Neumann algebras

We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm{I}\sb{\infty}.$

math.OA↗