Neumann Algebra Research Articles

Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras

For a right-angled Coxeter system $(W,S)$ and $q>0$, let $\mathcal{M}_q$ be the associated Hecke von Neumann algebra, which is generated by self-adjoint operators $T_s, s \in S$ satisfying the Hecke relation $(\sqrt{q}\: T_s - q) (\sqrt{q} \: T_s + 1) = 0$ as well as suitable commutation relations. Under the assumption that $(W, S)$ is irreducible and $\vert S \vert \geq 3$ it was proved by Garncarek that $\mathcal{M}_q$ is a factor (of type II$_1$) for a range $q \in [\rho, \rho^{-1}]$ and otherwise $\mathcal{M}_q$ is the direct sum of a II$_1$-factor and $\mathbb{C}$. In this paper we prove (under the same natural conditions as Garncarek) that $\mathcal{M}_q$ is non-injective, that it has the weak-$\ast$ completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case $\mathcal{M}_q$ is a strongly solid algebra and consequently $\mathcal{M}_q$ cannot have a Cartan subalgebra. In the general case $\mathcal{M}_q$ need not be strongly solid. However, we give examples of non-hyperbolic right-angled Coxeter groups such that $\mathcal{M}_q$ does not possess a Cartan subalgebra.

Recovery map stability for the data processing inequality

Let be a finite dimensional von Neumann algebra and a von Neumann subalgebra of it. For states and on , let and be the corresponding states induced on . The data processing inequality implies that where is the relative entropy. Petz proved that there is equality if and only if , where is the Petz recovery map. We prove a quantitative version of Petz’s theorem. In it simplest form, our bound iswhere is the relative modular operator. Since , this yields a bound that is independent of . We also prove an analogous result with a more complicated constant in which the roles of and are interchanged on the right.Quantum information theoretic inequalities are usually much harder to prove, or differ from, their classical counterparts because classical proofs often rely on conditioning argument that do not carry over to the quantum setting. In particular, quantum conditional expectations rarely preserve expectations—something that always happens in the classical setting. We also prove a simple theorem characterizing states and subalgebras for which conditional expectations do preserve expectation with respect to , illuminating the quantum obstacle to the existence of nicely behaved conditional expectations and the origins the Petz recovery map.

Lattice isomorphisms between projection lattices of von Neumann algebras

Abstract Generalizing von Neumann’s result on type II $_1$ von Neumann algebras, I characterise lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, I give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.

Ergodic properties of Markov semigroups in von Neumann algebras

We investigate ergodic properties of Markov semigroups in von Neumann algebras with the help of the notion of constrictor, which expresses the idea of closeness of the orbits of the semigroup to some set, as well as the notion of generalised averages, which generalises to arbitrary abelian semigroups the classical notions of Ces`aro, Borel, or Abel means. In particular, mean ergodicity, asymptotic stability, and structure properties of the fixed-point space are analysed in some detail.

The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps

We study the unique solution m of the Dyson equation -m(z)^{-1} = z1\!\!1 - a + S[m(z)] on a von Neumann algebra \mathcal{A} with the constraint \text{Im}\, m\ge 0 . Here, z lies in the complex upper half-plane, a is a self-adjoint element of \mathcal{A} and S is a positivity-preserving linear operator on \mathcal{A} . We show that m is the Stieltjes transform of a compactly supported \mathcal{A} -valued measure on \mathbb{R} . Under suitable assumptions, we establish that this measure has a uniformly 1/3 -Hölder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of m near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy–Widom universality for the edge eigenvalue statistics for correlated random matrices [the first author et al., Ann. Probab. 48, No. 2, 963–1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [ G. Cipolloni et al., Pure Appl. Anal. 1, No. 4, 615–707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203–1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases.

Von Neumann代数上的可导映射与导子

von Neumann代数上的可导映射与导子

An extension of the Beurling-Chen-Hadwin-Shen theorem for noncommutative Hardy spaces associated with finite von Neumann algebras

In 2015, Yanni Chen, Don Hadwin and Junhao Shen proved a noncommutative version of Beurling’s theorem for a continuous unitarily invariant norm α on a tracial von Neumann algebra (M, τ) such that α is one dominating with respect to τ . The role of H∞ is played by a maximal subdiagonal algebra A. In the talk, we first will show that if α is a continuous normalized unitarily invariant norm on (M, τ), then there exists a faithful normal tracial state ρ on M and a constant c > 0 such that α is a c times one norm-dominating norm on (M,ρ). Moreover, ρ(x) = τ(xg), where x ∈M , g is positive in L1(Z, τ), where Z is the center of M . Here c and ρ are not unique. However, if there is a c and ρ so that the Fuglede-Kadison determinant of g is positive, then Beurling-Chen-Hadwin-Shen theorem holds for L(α)(M, τ). The key ingredients in the proof of our result include a factorization theorem and a density theorem for for L(α)(M,ρ). Talk time: 2016-07-19 03:00 PM— 2016-07-19 03:20 PM Talk location: Cupples I Room 207

A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems

In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.

Actions, Quotients and Lattices of Locally Compact Quantum Groups

We prove a number of property (T) permanence results for locally compact quantum groups under exact sequences and the presence of invariant states, analogous to their classical versions. Along the way we characterize the existence of invariant weights on quantum homogeneous spaces of quotient type, and relate invariant states for LCQG actions on von Neumann algebras to invariant vectors in canonical unitary implementations, providing an application to amenability. Finally, we introduce a notion of lattice in a locally compact quantum group, noting examples provided by Drinfeld doubles of compact quantum groups. We show that property (T) lifts from a lattice to the ambient LCQG, just as it does classically, thus obtaining new examples of non-classical, non-compact, non-discrete LCQGs with property (T).

Сходимость по мере и τ-компактность τ-измеримых операторов, ассоциированных с полуконечной алгеброй фон Неймана

Сходимость по мере и τ-компактность τ-измеримых операторов, ассоциированных с полуконечной алгеброй фон Неймана

INNER AMENABLE GROUPOIDS AND CENTRAL SEQUENCES

We introduce inner amenability for discrete probability-measure-preserving (p.m.p.) groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations that either are stable or have a nontrivial central sequence in their full group.

The order topology on duals of $C^\ast $-algebras and von Neumann algebras

For a von Neumann algebra $\mathcal M$, we study the order topology associated to the hermitian part $\mathcal M_*^s$, and to intervals of the predual $\mathcal M_*$. It is shown that the order topology on $\mathcal M_*^s$ coincides with the topology indu

Complex Interpolation of Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.

A few observations on Weaver's quantum relations

Recently, a notion of quantum relation over a von Neumann algebra M has been introduced by Weaver. That definition generalizes the concept of a relation over a set. We prove that quantum relations over M are in bijective correspondence with weakly closed left ideals in M⊗ehM, where ⊗eh represents the extended Haagerup tensor product. The key step of the proof is showing a double annihilator relation between operator bimodules and the bimodular maps annihilating them. As an application, we study invariant quantum relations over a group von Neumann algebra.

Cartan Triples

AbstractWe introduce the class of Cartan triples as a generalization of the notion of a Cartan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi’s theorem to this setting. This context contains that of Fulman’s generalization of Cartan MASAs and we discuss his generalization in an appendix.

Absolutely compatible pairs in a von Neumann algebra

Let a, b be elements in a unital C∗-algebra with 0 ≤ a, b ≤ I. The element a is absolutely compatible with b if |a − b| + |I − a − b| = I. In this note, some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra are found. These characterizations are applied to measure how far are two absolute compatible positive elements in the closed unit ball from being mutually orthogonal or commuting. In the case of 2 by 2 matrices, the results admit a geometric interpretation. Namely, non-commutative matrices of the form a = ( t α ) and b = ( x β ) with x, t ∈ (0, 1)\{ 1 }, |α|2 < t(1 − t) α¯ 1 − t β 1 − x 2 and |β|2 < x(1 − x), are absolutely compatible if, and only if, the corresponding point b = (x, lRe(β), S'm(β)) in R3 lies in the ellipsoid Ea = {x ∈ R3 : d2(x, a) + d2(x, a,) = 1}, where d2 denotes the Euclidean distance in R3, and the elements a and a, are (t, lRe(α), S'm(α)) and (1 − t, −lRe(α), −S'm(α)), respectively. The description of absolutely compatible pairs of positive 2 by 2 matrices is applied to determine absolutely compatible pairs of positive elements in the closed unit ball of Mn.

Absolutely compatible pairs in a von Neumann algebra

Let a, b be elements in a unital C∗-algebra with 0 ≤ a, b ≤ I. The element a is absolutely compatible with b if |a − b| + |I − a − b| = I. In this note, some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra are found. These characterizations are applied to measure how far are two absolute compatible positive elements in the closed unit ball from being mutually orthogonal or commuting. In the case of 2 by 2 matrices, the results admit a geometric interpretation. Namely, non-commutative matrices of the form a = ( t α ) and b = ( x β ) with x, t ∈ (0, 1)\{ 1 }, |α|2 < t(1 − t) α¯ 1 − t β 1 − x 2 and |β|2 < x(1 − x), are absolutely compatible if, and only if, the corresponding point b = (x, lRe(β), S'm(β)) in R3 lies in the ellipsoid Ea = {x ∈ R3 : d2(x, a) + d2(x, a,) = 1}, where d2 denotes the Euclidean distance in R3, and the elements a and a, are (t, lRe(α), S'm(α)) and (1 − t, −lRe(α), −S'm(α)), respectively. The description of absolutely compatible pairs of positive 2 by 2 matrices is applied to determine absolutely compatible pairs of positive elements in the closed unit ball of Mn.

MEASURABLE -SEMIGROUPS ARE CONTINUOUS

AbstractLet $G$ be a second countable locally compact Hausdorff topological group and $P$ be a closed subsemigroup of $G$ containing the identity element $e\in G$. Assume that the interior of $P$ is dense in $P$. Let $\unicode[STIX]{x1D6FC}=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be a semigroup of unital normal $\ast$-endomorphisms of a von Neumann algebra $M$ with separable predual satisfying a natural measurability hypothesis. We show that $\unicode[STIX]{x1D6FC}$ is an $E_{0}$-semigroup over $P$ on $M$.

Duals of quantum semigroups with involution

We define a category $\mathcal{QSI}$ of quantum semigroups with involution which carries a corepresentation-based duality map $M\mapsto \widehat M$. Objects in $\mathcal{QSI}$ are von Neumann algebras with comultiplication and coinvolution, we do not suppose the existence of a Haar weight or of a distinguished spatial realisation. In the case of a locally compact quantum group $\mathbb G$, the duality $\;\widehat{\ }\;$ in $\mathcal{QSI}$ recovers the universal duality of Kustermans: $\widehat{L^\infty(\mathbb G)} = C_0^u(\hat {\mathbb G})^{**}= \widehat{ C_0^u(\mathbb G)^{**}}$, and $\widehat{L^\infty(\hat{\mathbb G})} = C_0^u(\mathbb G)^{**} = \widehat{ C_0^u(\hat{\mathbb G})^{**}}$. Other various examples are given.

Witnessing incompatibility of quantum channels

We introduce the notion of incompatibility witness for quantum channels, defined as an affine functional that is non-negative on all pairs of compatible channels and strictly negative on some incompatible pair. This notion extends the recent definition of incompatibility witnesses for quantum measurements. We utilize the general framework of channels acting on arbitrary finite-dimensional von Neumann algebras, thus allowing us to investigate incompatibility witnesses on measurement-measurement, measurement-channel, and channel-channel pairs. We prove that any incompatibility witness can be implemented as a state discrimination task in which some intermediate classical information is obtained before completing the task. This implies that any incompatible pair of channels gives an advantage over compatible pairs in some such state discrimination task.