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We study the fluctuation entropy for two-dimensional matter systems with an internal symmetry coupled to Jackiw--Teitelboim(JT) gravity joined to a Minkowski region. The fluctuation entropy is the Shannon entropy associated to probabilities…
We study embeddings of tracial $\mathrm{W}^*$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques. Jung implicitly and Hayes explicitly defined $1$-bounded entropy…
A distance between von Neumann algebras is introduced, depending on a further norm inducing the $w^*$-topology on bounded sets. Such notion is related both with the Gromov-Hausdorff distance for quantum metric spaces of Rieffel and with the…
We study the ergodic theory of a one-parameter family of interval maps T_alpha arising from generalized continued fraction algorithms. First of all, we prove the dependence of the metric entropy of T_alpha to be Hoelder-continuous in the…
In this paper we introduce the concept of the upper free orbit-dimension of a finite von Neumann algebra, and we derive some of its basic properties. Using this concept, we are able to improve most of the applications of free entropy to…
For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{*n} * L\mathbb{F}_r$ for $n \in \mathbb{N}$ and $r \in (1, \infty]$ and use them to obtain an interpolation $\mathcal{F}_{s,r}(A)$ for all real numbers $s>0$…
As an analogue of the topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M, \tau)$ and apply it to generalize the main results of [AHO23], showing that for a…
We discuss quantum entanglement between fast and slow degrees of freedom, in a two dimensional (2D) large $N_c$ gauge theory with Dirac quarks, quantized on the light front. Using the 't Hooft wave functions, we construct the reduced…
We compute the entanglement entropy in a composite system separated by a finitely ramified boundary with the structure of a self-similar lattice graph. We derive the entropy as a function of the decimation factor which determines the…
We consider the entanglement entropy for a free $U(1)$ theory in $3 + 1$ dimensions in the extended Hilbert space definition. By taking the continuum limit carefully we obtain a replica trick path integral which calculates this entanglement…
For $x\in (0,1)$, let $\langle d_1(x),d_2(x),d_3(x),\cdots \rangle$ be the Engel series expansion of $x$. Denote by $\lambda(x)$ the exponent of convergence of the sequence $\{d_n(x)\}$, namely \begin{equation*} \lambda(x)= \inf\left\{s…
We calculate the almost sure Hausdorff dimension of uniformly random self-similar fractals. These random fractals are generated from a finite family of similarities, where the linear parts of the mappings are independent uniformly…
This article continues the investigation of the tracial geometry of classifiable $\mathrm{C}^*$-algebras that have real rank zero and stable rank one. Using the language of optimal transport, we describe several situations in which the…
We consider semi-group BMO-spaces associated with arbitrary von Neumann algebras and prove interpolation theorems. This extends results by Junge-Mei for the tracial case. We give examples of multipliers on free Araki-Woods algebras and in…
In three dimensions, the pure Maxwell theory with compact U(1) gauge group is dual to a free compact scalar, and flows from the Maxwell theory with non-compact gauge group in the ultraviolet to a non-compact free massless scalar theory in…
We show that, for many choices of finite tuples of generators $X = (x_1, \dots , x_d)$ of a tracial von Neumann algebra $(M, \tau)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property…
A reduction formula for compressions of von Neumann algebras arising as free products is proved. This shows that the fundamental group is all of the positive reals for some such algebras. Additionally, by taking a sort of free product with…
We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. We notice, for a specific choice of $f \in \mathcal{H}$, that the associated element in…
Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X,mu), preserving the measure mu. If T is an automorphism of the equivalence relation defined by G then T can be extended to an automorphism alpha_T of the…
We consider the entanglement entropy for a spacetime region and its spacelike complement in the framework of algebraic quantum field theory. For a M\"obius covariant local net satisfying a certain nuclearity property, we consider the von…