arXiv subjects

Brent Nelson

Publications and source records attributed to Brent Nelson.

At least 19 recordsLinked to original sources

arXiv2026

Irreducibility and weak spectral gap in free product von Neumann algebras

Consider a free product $(M,\varphi)= (M_1,\varphi_1)* (M_2,\varphi_2)$ of non-trivial von Neumann algebras. Using amalgamated free product techniques, we establish irreducibility and weak spectral gap results for free product subalgebras and the centralizer subalgebra of the free product state. In particular, it is shown that, under a mild dimension constraint, the inclusion of the diffuse summand of these subalgebras into the corresponding corner of $M^\omega$ is irreducible for any free ultrafilter $\omega\in \beta\mathbb{N}\setminus \mathbb{N}$. We also obtain analogous irreducibility results for graph products of von Neumann algebras.

math.OA↗

arXiv2025

Strong convergence to operator-valued semicirculars

We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices $\eta = (\eta_{i,j})_{i,j \in I}$. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of $\eta_{i,j}$, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a $B$-valued semicircular family and generators of the base algebra $B$. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.

math.OA↗

arXiv2025

On the genericity of irreducible subfactors

We show that finitely generated irreducible $\mathrm{II}_1$ subfactors are generic in the following sense. Given a separable $\mathrm{II}_1$ factor $M$ and an integer $n\geq 2$, equip the set of $n$-tuples of self-adjoint operators in $M$ with norm at most $1$ with the metric $d(x,y) = \max_{1\leq i \leq n} \|x_i - y_i\|_2$. Then the set of $n$-tuples that generate an irreducible subfactor of $M$ forms a dense $G_\delta$ set in this metric space. On the way to proving this result, we show that closable derivations vanish on the anticoarse space associated to their kernels, which leads to new applications of conjugate systems in free probability.

math.OA↗

arXiv2025

Almost unimodular groups

We show that a locally compact group has open unimodular part if and only if the Plancherel weight on its group von Neumann algebra is almost periodic. We call such groups almost unimodular. The almost periodicity of the Plancherel weight allows one to define a Murray-von Neumann dimension for certain Hilbert space modules over the group von Neumann algebra, and we show that for finite covolume subgroups this dimension scales according to the covolume. Using this we obtain a generalization of the Atiyah-Schmid formula in the setting of second countable almost unimodular groups with finite covolume subgroups. Additionally, for the class of almost unimodular groups we present many examples, establish a number of permanence properties, and show that the formal degrees of irreducible and factorial square integrable representations are well behaved.

math.OA↗

arXiv2025

Simplicity of Cuntz-Pimsner algebras of quantum graphs

Let $\mathcal{G}$ be a quantum graph without quantum sources and $E_\mathcal{G}$ be the quantum edge correspondence for $\mathcal{G}.$ Our main results include sufficient conditions for simplicity of the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ in terms of $\mathcal{G}$ and for defining a surjection from the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ onto a particular relative Cuntz-Pimsner algebra for $E_\mathcal{G}$. As an application of these two results, we give the first example of a quantum graph with distinct quantum Cuntz-Krieger and local quantum Cuntz-Krieger algebras. We also characterize simplicity of $\mathcal{O}_{E_\mathcal{G}}$ for some fundamental examples of quantum graphs, including rank-one quantum graphs on a single full matrix algebra, complete quantum graphs, and trivial quantum graphs. Along the way, we provide an equivalent condition for minimality of $E_\mathcal{G}$ and sufficient conditions for aperiodicity of $E_\mathcal{G}$ in terms of the underlying quantum graph $\mathcal{G}$.

math.OA↗

arXiv2024

General solidity phenomena and anticoarse spaces for type $\mathrm{III}_1$ factors

By developing a theory of anticoarse spaces in the purely infinite setting and using 1-bounded entropy techniques along with recent strong convergence results in random matrix theory, we show that free Araki--Woods factors offer the first examples of type $\mathrm{III}$ factors satisfying vastly general degrees of indecomposability phenomena. Notably this includes strong solidity with respect to any weakening of the normalizer currently in the literature and the Peterson--Thom property.

math.OA↗

arXiv2024

Remarks on C*-discrete inclusions

We obtain a Galois correspondence between the lattice of intermediate C*-discrete subalgebras intermediate to a given irreducible C*-discrete inclusion, and characterize these as targets of compatible expectations under a traciality condition. Furthermore, we define freeness for actions of tensor categories on C*-algebras, and show simplicity is preserved under taking reduced crossed products. Finally, we show under certain constraints that A-valued semicircular systems give rise to C*-discrete inclusions, and thus are crossed products by an action of a tensor category. Along the way, we show the set of single algebraic generators of a dualizable bimodule forms an open subset.

math.OA↗

arXiv2024

Uniqueness of extremal almost periodic states on the injective type III$_{1}$ factor

Let $R_\infty$ denote the Araki--Woods factor -- the unique separable injective type III$_{1}$ factor. For extremal almost periodic states $φ, ψ\in (R_\infty)_*$, we show that if $Δ_φ$ and $Δ_ψ$ have the same point spectrum then $ψ= φ\circ α$ for some $α\in $ Aut$(R_\infty)$. Consequently, the extremal almost periodic states on $R_\infty$ are parameterized by countable dense subgroups of $\mathbb{R}_+$, up to precomposition by automorphisms. As an application, we show that KMS states for generalized gauge actions on Cuntz algebras agree (up to an automorphism) with tensor products of Powers states on their von Neumann completions.

math.OA↗

arXiv2024

Murray-von Neumann dimension for strictly semifinite weights

Given a von Neumann algebra $M$ equipped with a faithful normal strictly semifinite weight $\varphi$, we develop a notion of Murray-von Neumann dimension over $(M,\varphi)$ that is defined for modules over the basic construction associated to the inclusion $M^\varphi \subset M$. For $\varphi=\tau$ a faithful normal tracial state, this recovers the usual Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either a type $\mathrm{III}_\lambda$ factor with $0<\lambda <1$ or a full type $\mathrm{III}_1$ factor with $\text{Sd}(M)\neq \mathbb{R}$, then amongst extremal almost periodic weights the dimension function depends on $\varphi$ only up to scaling. As an application, we show that if an inclusion of diffuse factors with separable preduals $N\subset M$ is with expectation $\mathcal{E}$ and admits a compatible extremal almost periodic state $\varphi$, then this dimension quantity bounds the index $\text{Ind}{\mathcal{E}}$, and in fact equals it when the modular operators $\Delta_\varphi$ and $\Delta_{\varphi|_N}$ have the same point spectrum. In the pursuit of this result, we also show such inclusions always admit Pimsner-Popa orthogonal bases.

math.OA↗

arXiv2024

On the structure of graph product von Neumann algebras

We undertake a comprehensive study of structural properties of graph products of von Neumann algebras equipped with faithful, normal states, as well as properties of the graph products relative to subalgebras coming from induced subgraphs. Among the technical contributions in this paper include a complete bimodule calculation for subalgebras arising from subgraphs. As an application, we obtain a complete classification of when two subalgebras coming from induced subgraphs can be amenable relative to each other. We also give complete characterizations of when the graph product can be full, diffuse, or a factor. Our results are obtained in a broad generality, and we emphasize that they are new even in the tracial setting. They also allow us to deduce new results about when graph products of groups can be amenable relative to each other.

math.OA↗

arXiv2024

Random permutation matrix models for graph products

Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain random permutation matrices, we construct random matrix models for graph independence with amalgamation over the diagonal matrices. This yields a new probabilist,ic proof that graph products of sofic groups are sofic.

math.OA↗

arXiv2024

Strong 1-boundedness, $L^2$-Betti numbers, algebraic soficity, and graph products

We show that graph products of non trivial finite dimensional von Neumann algebras are strongly 1-bounded when the underlying *-algebra has vanishing first L2-Betti number. The proof uses a combination of the following two key ideas to obtain lower bounds on the Fuglede-Kadison determinant of matrix polynomials in a generating set: a notion called ''algebraic soficity'' for *-algebras allowing for the existence of Galois bounded microstates with asymptotically constant diagonals; a probabilistic construction of the authors of permutation models for graph independence over the diagonal.

math.OA↗

arXiv2023

Ergodic Quantum Processes on Finite von Neumann Algebras

Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual and let $(\Omega, \mathbb{P})$ be a probability space. A bounded positive random linear operator on $L^1(M,\tau)$ is a map $\gamma : \Omega \times L^1(M,\tau) \to L^1(M,\tau)$ so that $\tau(\gamma_\omega(x)a)$ is measurable for all $x\in L^1(M,\tau)$ and $a\in M$, and $x\mapsto \gamma_\omega(x)$ is bounded, positive, and linear almost surely. Given an ergodic $T\in Aut(\Omega, \mathbb{P})$, we study quantum processes of the form $\gamma_{T^n \omega}\circ \gamma_{T^{n-1}\omega} \circ \cdots \circ \gamma_{T^m\omega}$ for $m,n\in \mathbb{Z}$. Using the Hennion metric introduced in [MS22], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when $\gamma_\omega$ is the predual of a normal positive linear map on $M$. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [MS22].

math.OA↗

arXiv2023

Discrete Inclusions of C*-algebras

We introduce the category of C*-discrete inclusions of C*-algebras $A\subset B$ with a faithful conditional expectation $E:B\twoheadrightarrow A$. This class includes many examples such as finite Watatani index inclusions, and also abundant infinite index inclusions like crossed products by outer actions of discrete (quantum) groups and unitary tensor categories. We prove irreducible $(A'\cap B= \mathbb{C}1)$ C*-discrete inclusions are precisely crossed products by outer actions of unitary tensor categories and certain C*-algebra objects.

math.OA↗

arXiv2022

Orientifold Calabi-Yau Threefolds with Divisor Involutions and String Landscape

We establish an orientifold Calabi-Yau threefold database for $h^{1,1}(X) \leq 6$ by considering non-trivial $\mathbb{Z}_{2}$ divisor exchange involutions, using a toric Calabi-Yau database (http://www.rossealtman.com/toriccy/). We first determine the topology for each individual divisor (Hodge diamond), then identify and classify the proper involutions which are globally consistent across all disjoint phases of the Kähler cone for each unique geometry. Each of the proper involutions will result in an orientifold Calabi-Yau manifold. Then we clarify all possible fixed loci under the proper involution, thereby determining the locations of different types of $O$-planes. It is shown that under the proper involutions, one typically ends up with a system of $O3/O7$-planes, and most of these will further admit naive Type IIB string vacua.The geometries with freely acting involutions are also determined. We further determine the splitting of the Hodge numbers into odd/even parity in the orbifold limit. The final result is a class of orientifold Calabi-Yau threefolds with non-trivial odd class cohomology $h^{1,1}_{-}(X / σ^*) \neq 0$.

hep-th↗

arXiv2022

Quantum edge correspondences and quantum Cuntz-Krieger algebras

Given a quantum graph $\mathcal{G}=(B,ψ,A)$, we define a C*-correspondence $E_\mathcal{G}$ over the noncommutative vertex C*-algebra $B$, called the quantum edge correspondence. For a classical graph $\mathcal{G}$, $E_\mathcal{G}$ is the usual graph correspondence spanned by the edges of $\mathcal{G}$. When the quantum adjacency matrix $A\colon B\to B$ is completely positive, we show that $E_\mathcal{G}$ is faithful if and only if $\ker(A)$ does not contain a central summand of $B$. In this case, we show that the Cuntz-Pimsner algebra $\mathcal{O}_{E_\mathcal{G}}$ is isomorphic to a quotient of the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ defined by Brannan, Eifler, Voigt, and Weber. Moreover, the kernel of the quotient map is shown to be generated by "localized" versions of the quantum Cuntz-Krieger relations, and $\mathcal{O}_{E_\mathcal{G}}$ is shown to be the universal object associated to these local relations. We study in detail some concrete examples and make connections with the theory of Exel crossed products.

math.OA↗

arXiv2022

On Free Stein Dimension

We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a given set of generators, we show that (approximate) algebraic relations produce (non-approximate) bounds on the free Stein dimension. Particular treatment is given to the case of separable abelian von Neumann algebras, where we show that free Stein dimension is a von Neumann algebra invariant. In addition, we show that under mild assumptions $L^2$-rigidity implies free Stein dimension one. Finally, we use limits superior/inferior to extend the free Stein dimension to a von Neumann algebra invariant -- which is substantially more difficult to compute in general -- and compute it in several cases of interest.

math.OA↗

arXiv2021

Gravitational Waves from Dark Yang-Mills Sectors

Dark Yang-Mills sectors, which are ubiquitous in the string landscape, may be reheated above their critical temperature and subsequently go through a confining first-order phase transition that produces stochastic gravitational waves in the early universe. Taking into account constraints from lattice and from Yang-Mills (center and Weyl) symmetries, we use a phenomenological model to construct an effective potential of the semi quark-gluon plasma phase, from which we compute the gravitational wave signal produced during confinement for numerous gauge groups. The signal is maximized when the dark sector dominates the energy density of the universe at the time of the phase transition. In that case, we find that it is within reach of the next-to-next generation of experiments (BBO, DECIGO) for a range of dark confinement scales near the weak scale.

hep-ph↗