arXiv subjects

Jon Bannon

Publications and source records attributed to Jon Bannon.

8 recordsLinked to original sources

arXiv2026

Weak Factorization and Product Systems Over Groupoids

We give a product-system description of generalized higher-rank graphs with the weak factorization property. For such a graph with degree-zero groupoid G, we prove that weak factorization is equivalent to multiplication inducing coherent bijections between balanced products of its homogeneous G-G-bisets. Consequently, generalized higher-rank graphs with fixed degree-zero groupoid are equivalent to normalized product systems of groupoid bisets over N^k. For countable left-cancellative graphs, these bisets linearize canonically to product systems of C*(G)-correspondences. Finite alignment implies compact alignment, and the resulting Nica-Toeplitz algebra agrees canonically with Spielberg's full category algebra. Under row-finiteness modulo G and the no-sources condition, the corresponding Cuntz-Pimsner quotient is the boundary groupoid algebra; with injective left actions, the same conclusion holds for the Cuntz-Nica-Pimsner algebra. We also characterize the R-condition by the existence of a strict multiplicative splitting and relate such splittings to higher-rank graph/groupoid Zappa-Szep products. A cancellative rank-two example shows that strict splittings need not exist.

math.OA↗

arXiv2026

Non-isomorphism of rings of integer-coefficient holomorphic functions on disks of varying radius

For $\rho \in (0,1]$, let $R(\rho) = \mathbb{Z}[[z]] \cap O(B(0,\rho))$ denote the ring of power series with integer Taylor coefficients converging on the open disk $B(0,\rho)$. We prove that these rings are pairwise non-isomorphic as abstract rings. Three ingredients drive the proof: the ideal $(z)$ is the unique principal ideal with quotient $\mathbb{Z}$, so any isomorphism sends $z$ to a generator $g$ of $(z)$; every isomorphism is substitution by $g$, because it respects the $(z)$-adic filtration; and a Hadamard gap series with a natural boundary at $|w| = \rho_1$ forces the image $g(B(0,\rho_2))$ into $B(0,\rho_1)$, after which the Schwarz lemma and integrality of coefficients force $g = \pm z$ and $\rho_1 = \rho_2$.

math.RA↗

arXiv2026

Integer Coefficient Power Series with Prescribed Zero Sets

We prove that a discrete effective divisor on the open unit disk $\mathbb{D}$ is the zero divisor of a holomorphic function on $\mathbb{D}$ with integer Taylor coefficients if and only if it is invariant under complex conjugation. The construction uses a one-parameter deformation of the Weierstrass elementary factors in which each modified factor of order $n$ leaves all Taylor coefficients of degree $\leq n$ unchanged while shifting the coefficient of degree $n+1$ by a controlled affine amount. These modified factors act as elementary jet-correction operators: the triangular structure of the coefficient map permits an inductive rounding scheme compatible with canonical-product convergence. As a consequence, every holomorphic function on $\mathbb{D}$ differs from one with Gaussian-integer Taylor coefficients by multiplication by a nowhere-vanishing holomorphic factor.

math.CV↗

arXiv2023

Quasinormalizers in crossed products of von Neumann algebras

We study the relationship between the dynamics of the action $\alpha$ of a discrete group $G$ on a von Neumann algebra $M$, and structural properties of the associated crossed product inclusion $L(G) \subseteq M \rtimes_\alpha G$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $L(G) \subseteq M \rtimes_\alpha G$ such as the Haagerup Approximation Property, and is essential to capturing "almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorems for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.

math.OA↗

arXiv2019

Full factors and co-amenable inclusions

We show that if $M$ is a full factor and $N \subset M$ is a co-amenable subfactor with expectation, then $N$ is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if $M$ is a full factor and $σ\colon G \curvearrowright M$ is an outer action of a compact group $G$, then $σ$ is automatically minimal and $M^G$ is a full factor which has w-spectral gap in $M$. Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion $N\subset M$ of von Neumann algebras are equivalent, thus closing the cycle of implications given in Anantharaman-Delaroche's paper in 1995.

math.OA↗

arXiv2019

On Noncommutative Joinings

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful identification of joinings with equivariant quantum channels between $W^{\ast}$-dynamical systems that provides noncommutative generalizations of many fundamental results of classical joining theory. We obtain fully general analogues of the main classical disjointness characterizations of ergodicity, primeness and mixing phenomena.

math.OA↗

arXiv2019

Noncommutative Joinings II

This paper is a continuation of the authors' previous work on noncommutative joinings, and contains a study of relative independence of W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-dynamical $G$-system $\mathfrak{M}$ with compact subsystem $\mathfrak{N}$ is disjoint relative to $\mathfrak{N}$ from its maximal compact subsystem $\mathfrak{M}_{K}$ if and only if $\mathfrak{N}\cong\mathfrak{M}_{K}$. This generalizes recent work of Duvenhage, which established the result for $G$ abelian.

math.OA↗

arXiv2019

The Modular Symmetry of Markov Maps

A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.

math.OA↗