Neumann Algebra Research Articles

Orthogonal unitary bases and a subfactor conjecture

We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of M n ( C ) M_n(\mathbb {C}) admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta [Glasg. Math. J. 64 (2022), pp. 586–602], showing that any finite-index regular inclusion N ⊆ M N\subseteq M of I I 1 II_1 -factors admits an orthonormal unitary Pimsner-Popa basis.

Hermitian operators and isometries on symmetric operator spaces

Let \mathcal{M} be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space \mathcal{H} equipped with a semifinite faithful normal trace \tau . Let E(\mathcal{M},\tau) be a symmetric operator space affiliated with \mathcal{M} , whose norm is order continuous and is not proportional to the Hilbertian norm {\parallel{\cdot}\parallel}_2 on L_2(\mathcal{M},\tau) . We obtain a general description of all bounded hermitian operators on E(\mathcal{M},\tau) . This is the first time that the description of hermitian operators on a symmetric operator space (even for a noncommutative L_p -space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.

Spectral Gap Characterizations of Property (T) for II1 Factors

Abstract For II$_1$ factors, we show that property (T) is equivalent to having weak spectral gap in any inclusion into a larger tracial von Neumann algebra. We also show that not having non-zero almost central vectors in weakly mixing bimodules characterizes property (T) for II$_1$ factors, which allows us to obtain a stronger characterization of property (T) where only weak spectral gap in any irreducible inclusion is required.

Isometric actions on Lp-spaces: dependence on the value of p

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

Geometry of infinite dimensional unitary groups: Convexity and fixed points

In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full unitary group, the unitary Schatten perturbations of the identity and unitary groups of finite von Neumann algebras. The Finsler structures are defined by translation of different norms on the tangent space at the identity. We first prove a convexity result for the metric derived from the operator norm on the full unitary group. We also prove strong convexity results for the squared metrics in Hilbert-Schmidt unitary groups and unitary groups of finite von Neumann algebras. In both cases the tangent spaces are endowed with an inner product defined with a trace. These results are applied to fixed point properties and to quantitative metric bounds in certain rigidity problems. Radius bounds for all convexity and fixed point results are shown to be optimal.

Heat-smoothing for holomorphic subalgebras of free group von Neumann algebras

The heat semigroup on discrete hypercubes is well-known to be contractive over L p L_p -spaces for 1 > p > ∞ 1>p>\infty . A question of Mendel and Naor [Publ. Math. Inst. Hautes Études Sci. 119 (2014), pp. 1–95] concerns a stronger contraction property in the tail spaces, which is known as the heat-smoothing conjecture. Eskenazis and Ivanisvili [Israel J. Math. (2022), pp. 1–17] considered a Gaussian analog of this conjecture and resolved some special cases. In particular, they proved that heat-smoothing type conjecture holds for holomorphic functions in the Gaussian spaces with sharp constants. In this paper, we prove analogous sharp inequalities for holomorphic subalgebras of free group von Neumann algebras. Similar results also hold for q q -Gaussian algebras and quantum tori. In the case of free group von Neumann algebras, the weaker formulation of heat-smoothing is proved with optimal order.

Semi-Fredholm theory in $$C^{*}$$-algebras

Kečkić and Lazović introduced an axiomatic approach to Fredholm theory by considering Fredholm type elements in a unital $$C^{*}$$ -algebra as a generalization of $$C^{*}$$ -Fredholm operators on the standard Hilbert $$C^{*}$$ -module introduced by Mishchenko and Fomenko and of Fredholm operators on a properly infinite von Neumann algebra introduced by Breuer. In this paper, we establish semi-Fredholm theory in unital $$C^{*}$$ -algebras as a continuation of the approach by Kečkić and Lazović. We introduce the notion of a semi-Fredholm type element and a semi-Weyl type element in a unital $$C^{*}$$ -algebra. We prove that the difference between the set of semi-Fredholm elements and the set of semi-Weyl elements is open in the norm topology, that the set of semi-Weyl elements is invariant under perturbations by finite type elements, and several other results generalizing their classical counterparts. Also, we illustrate applications of our results to the special case of properly infinite von Neumann algebras and we obtain a generalization of the punctured neighbourhood theorem in this setting.

The free field: Realization via unbounded operators and Atiyah property

Let X1,…,Xn be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the noncommutative distribution of X1,…,Xn. More precisely, X1,…,Xn generate the free skew field if and only if there exist no non-zero finite rank operators T1,…,Tn such that ∑i[Ti,Xi]=0. Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of X1,…,Xn. Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field.We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of Charlesworth and Shlyakhtenko.

Some applications of group-theoretic Rips constructions to the classification of von Neumann algebras

In this paper we study various von Neumann algebraic rigidity aspects for the property (T) groups that arise via the Rips construction developed by Belegradek and Osin in geometric group theory \cite{BO06}. Specifically, developing a new interplay between Popa's deformation/rigidity theory \cite{Po07} and geometric group theory methods we show that several algebraic features of these groups are completely recognizable from the von Neumann algebraic structure. In particular, we obtain new infinite families of pairwise non-isomorphic property (T) group factors thereby providing positive evidence towards Connes' Rigidity Conjecture. In addition, we use the Rips construction to build examples of property (T) II$_1$ factors which posses maximal von Neumann subalgebras without property (T) which answers a question raised in an earlier version of \cite{JS19} by Y. Jiang and A. Skalski.

Wasserstein distance between noncommutative dynamical systems

We introduce and study a class of quadratic Wasserstein distances on spaces consisting of generalized dynamical systems on a von Neumann algebra. We emphasize how symmetry of such a Wasserstein distance arises, but also study the asymmetric case. This setup is illustrated in the context of reduced dynamics, and a number of simple examples are also presented.

On the Commutativity of States in von Neumann Algebras

The notion of commutativity of two normal states on a von Neumann algebra was defined some time ago by means of the Pedersen–Takesaki theorem. In this note, we aim at generalising this notion to an arbitrary number of states, and obtaining some results on so defined joint commutativity. Also relations between commutativity and broadcastability of states are investigated.

Ring derivations of Murray–von Neumann algebras

Let M be a type II1 von Neumann algebra, S(M) be the Murray–von Neumann algebra associated with M and let A be a ⁎-subalgebra in S(M) with M⊆A. We prove that any ring derivation D from A into S(M) is necessarily inner. Further, we prove that if A is an EW⁎-algebra such that its bounded part Ab is a W⁎-algebra without finite type I direct summands, then any ring derivation D from A into LS(Ab) — the algebra of all locally measurable operators affiliated with Ab, is an inner derivation. We also give an example showing that the condition M⊆A is essential. At the end of this paper, we provide several criteria for an abelian extended W⁎-algebra such that all ring derivations on it are linear.

A note on the column–row property

AbstractIn this article, we study the following question asked by Michael Hartz in a recent paper (Every complete Pick space satisfies the column-row property, to appear in Acta Mathematica): which operator spaces satisfy the column–row property? We provide a complete classification of the column–row property (CRP) for noncommutative $L_{p}$ -spaces over semifinite von Neumann algebras. We study other relevant properties of operator spaces that are related to the CRP and discuss their existence and nonexistence for various natural examples of operator spaces.

Relative Haagerup property for arbitrary von Neumann algebras

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

Dilations of Markovian semigroups of measurable Schur multipliers

AbstractUsing probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of self-adjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm {B}({\mathrm {L}}^2(X))$ of bounded operators on the Hilbert space ${\mathrm {L}}^2(X)$ , where X is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$ -automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh’s ${\mathrm {H}}^\infty $ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to ${\mathrm {BMO}}$ -spaces. We also give an answer to a question of Steen, Todorov, and Turowska on completely positive continuous Schur multipliers.

Large N algebras and generalized entropy

We construct a Type II∞ von Neumann algebra that describes the large N physics of single-trace operators in AdS/CFT in the microcanonical ensemble, where there is no need to include perturbative 1/N corrections. Using only the extrapolate dictionary, we show that the entropy of semiclassical states on this algebra is holographically dual to the generalized entropy of the black hole bifurcation surface. From a boundary perspective, this constitutes a derivation of a special case of the QES prescription without any use of Euclidean gravity or replicas; from a purely bulk perspective, it is a derivation of the quantum-corrected Bekenstein-Hawking formula as the entropy of an explicit algebra in the G → 0 limit of Lorentzian effective field theory quantum gravity. In a limit where a black hole is first allowed to equilibrate and then is later potentially re-excited, we show that the generalized second law is a direct consequence of the monotonicity of the entropy of algebras under trace-preserving inclusions. Finally, by considering excitations that are separated by more than a scrambling time we construct a “free product” von Neumann algebra that describes the semiclassical physics of long wormholes supported by shocks. We compute Rényi entropies for this algebra and show that they are equal to a sum over saddles associated to quantum extremal surfaces in the wormhole. Surprisingly, however, the saddles associated to “bulge” quantum extremal surfaces contribute with a negative sign.

Unitarily invariant norms on finite von Neumann algebras

We give a characterization of all the unitarily invariant norms on a finite von Neumann algebra acting on a separable Hilbert space. The characterization is analogous to von Neumann’s characterization for the $$n\times n$$ complex matrices and the characterization in Fang et al. (J Funct Anal 255(1):142–183, 2008) for $$II_{1}$$ factors.

NONLINEAR BI-SKEW LIE-TYPE DERIVATIONS ON *-ALGEBRAS

Let 𝒜 be a unital ∗-algebra. Under some mild conditions on 𝒜, we show that ϕ is a nonlinear bi-skew Lie-type derivation on 𝒜 if and only if ϕ is an additive ∗-derivation. As applications, nonlinear bi-skew Lie-type derivations on prime ∗-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Monotonic multi-state quantum f-divergences

We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

Ke Li’s Lemma for Quantum Hypothesis Testing in General Von Neumann Algebras

A lemma stated by Li (Ann Statist 42:171-189, 2014) has been used in, for example, Nilanjana et al. (J Math Phys 57:062207, 2016), Kaur et al. (Phys Rev A 96:062318, 2017), Rouzé, Datta (IEEE Trans Inform Theory 64:593–612, 2018), Tomamichel, Tan (Comm Math Phys 338:103-137, 2015) and Wilde et al. (IEEE Trans Inform Theory 63:1792-1817, 2017) for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here, we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields a new proof of quantum Stein’s lemma with slightly weaker assumption, as well as immediate generalizations of its second-order asymptotics, for example the main results in Nilanjana et al. (J Math Phys 57:062207, 2016) and Li (Ann Statist 42:171-189, 2014).