Von Neumann Algebra Of Operators Research Articles

Sir Vaughan Jones. 31 December 1952—6 September 2020

Vaughan Jones made fundamental contributions to mathematics and mathematical physics, bringing together disparate areas of operator algebras, knots, links and low-dimensional topology in mathematics, and statistical mechanics, quantum field theory and quantum information in physics, while opening up the new field of quantum topology. The key that unlocked all this was his seminal work on subfactor theory in von Neumann algebras of operators, which led to a new invariant of links: the Jones polynomial. For this he was awarded the Fields Medal in 1990.

On Period, Cycles and Fixed Points of a Quantum Channel

We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.

On τ-essentially Invertibility of τ-measurable Operators

Let $$ \mathcal{M} $$ be a von Neumann algebra of operators on a Hilbert space and τ be a faithful normal semifinite trace on $$ \mathcal{M} $$ . Let I be the unit of the algebra $$ \mathcal{M} $$ . A τ-measurable operator A is said to be τ-essentially right (or left) invertible if there exists a τ-measurable operator B such that the operator I − AB (or I − BA) is τ-compact. A necessary and sufficient condition for an operator A to be τ-essentially left invertible is that A∗A (or, equivalently, $$ \sqrt{A^{\ast }A} $$ ) is τ-essentially invertible. We present a sufficient condition that a τ-measurable operator A not be τ-essentially left invertible. For τ-measurable operators A and P = P2 the following conditions are equivalent: 1. A is τ-essential right inverse for P; 2. A is τ-essential left inverse for P; 3. I − A,I − P are τ-compact; 4. PA is τ-essential left inverse for P. For τ-measurable operators A = A3, B = B3 the following conditions are equivalent: 1. B is τ-essential right inverse for A; 2. B is τ-essential left inverse for A. Pairs of faithful normal semifinite traces on $$ \mathcal{M} $$ are considered.

When weak and local measure convergence implies norm convergence

Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M . Let t τ be the measure topology on the ⁎-algebra S ( M , τ ) of all τ -measurable operators. We prove that for B ∈ S ( M , τ ) + the sets I B = { A ∈ S ( M , τ ) h : − B ≤ A ≤ B } and K B = { A ∈ S ( M , τ ) : A ⁎ A ≤ B } are convex and t τ -closed in S ( M , τ ) . In this case, we have I B = { B T B : T ∈ M h and ‖ T ‖ ≤ 1 } and, for invertible B , we describe the set of extreme points of the set I B . Let M be an atomic von Neumann algebra. We prove that an operator B ∈ S ( M , τ ) + is τ -compact if and only if the set I B is t τ -compact. The t τ -compactness of I B for all τ -compact operators B characterizes these algebras.

Paranormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from Mand k = 1, 2; if an operator T from P1 admits the bounded inverse T−1 then T−1 lies in P1. If a bounded operator T lies in P1 then T is normaloid, Tn belongs to P1 and a rearrangement μt(Tn) ≥ μt(T )n for all t > 0 and natural n. If a τ-measurable operator T is hyponormal and Tn is τ-compact operator for some natural number n then T is both normal and τ-compact. If an operator T lies in P1 then T 2 belongs to P1. If M= B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators onH. If a τ-measurable operator A is q-hyponormal (1 ≥ q > 0) and |A*| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q-cohyponormal) operator is normal. Consider a τ-measurable nilpotent operator Z ≠ 0 and numbers a, b ∈ R. Then an operator Z*Z − ZZ* + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.

Tingley's problem through the facial structure of operator algebras

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C⁎-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.

The Hadamard determinant inequality — Extensions to operators on a Hilbert space

A generalization of classical determinant inequalities like Hadamard's inequality and Fischer's inequality is studied. For a version of the inequalities originally proved by Arveson for positive operators in von Neumann algebras with a tracial state, we give a different proof. We also improve and generalize to the setting of finite von Neumann algebras, some ‘Fischer-type’ inequalities by Matic for determinants of perturbed positive-definite matrices. In the process, a conceptual framework is established for viewing these inequalities as manifestations of Jensen's inequality in conjunction with the theory of operator monotone and operator convex functions on [0,∞). We place emphasis on documenting necessary and sufficient conditions for equality to hold.

On τ-Compactness of Products of τ-Measurable Operators

Let \(\mathcal {M}\) be a von Neumann algebra of operators on a Hilbert space \(\mathcal {H}\), τ be a faithful normal semifinite trace on \(\mathcal {M}\). We obtain some new inequalities for rearrangements of τ-measurable operators products. We also establish some sufficient τ-compactness conditions for products of selfadjoint τ-measurable operators. Next we obtain a τ-compactness criterion for product of a nonnegative τ-measurable operator with an arbitrary τ-measurable operator. We construct an example that shows importance of nonnegativity for one of the factors. The similar results are obtained also for elementary operators from \(\mathcal {M}\). We apply our results to symmetric spaces on \((\mathcal {M}, \tau )\). The results are new even for the *-algebra \(\mathcal {B}(\mathcal {H})\) of all linear bounded operators on \(\mathcal {H}\) endowed with the canonical trace τ = tr.

Two classes of τ-measurable operators affiliated with a von Neumann algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from M and k = 1, 2; if an operator T from P1 admits the bounded inverse T−1, then T−1 lies in P1. We establish some new inequalities for rearrangements of operators from P1. If a τ-measurable operator T is hyponormal and Tn is τ-compact for some natural number n, then T is both normal and τ-compact. If M = B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators on H.

Concerning the theory of τ-measurable operators affiliated to a semifinite von Neumann algebra

Let M be a von Neumann algebra of operators in a Hilbert space H, let τ be an exact normal semifinite trace on M, and let L 1(M, τ) be the Banach space of τ-integrable operators. The following results are obtained. If X = X*, Y = Y* are τ-measurable operators and XY ∈ L 1(M, τ), then YX ∈ L 1(M, τ) and τ(XY) = τ(YX) ∈ R. In particular, if X, Y ∈ B(H)sa and XY ∈ G 1, then YX ∈ G 1 and tr(XY) = tr(YX) ∈ R. If X ∈ L 1(M, τ), then $$\tau \left( {X*} \right) = \tau \left( X \right)$$ . Let A be a τ-measurable operator. If the operator A is τ-compact and V ∈ M is a contraction, then it follows from V* AV = A that V A = AV. We have A = A 2 if and only if A = |A*||A|. This representation is also new for bounded idempotents in H. If A = A 2 ∈ L 1(M, τ), then $$\tau \left( A \right) = \tau \left( {\sqrt {\left| A \right|} \left| {A*} \right|\sqrt {\left| A \right|} } \right) \in {\mathbb{R}^ + }$$ . If A = A 2 and A (or A*) is semihyponormal, then A is normal, thus A is a projection. If A = A 3 and A is hyponormal or cohyponormal, then A is normal, and thus A = A* ∈ M is the difference of two mutually orthogonal projections (A + A 2)/2 and (A 2 − A)/2. If A,A 2 ∈ L 1(M, τ) and A = A 3, then τ(A) ∈ R.

The Schur–Horn problem for normal operators

We consider the Schur–Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well known to be extremely difficult, and in fact, it remains open for matrices of size greater than 3 . We show that the infinite-dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I∞, II, and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.

On the Yang-Mills mass gap problem

Theorem 4.1 of the author’s paper “Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm”, RJMP 21 (2), 169–188 (2014) claims the relative ellipticity of cutoff Yang-Mills quantum energy-mass operators in von Neumann algebras with regular traces. This implies that the spectra of cutoff self-adjoint Yang-Mills energy-mass operators in a nonperturbative quantum Yang-Mills theory (with an arbitrary compact simple gauge group) are nonnegative sequences of the eigenvalues converging to +∞. The spectra are self-similar in the inverse proportion to the running coupling constant. In particular, they have self-similar positive spectral mass gaps. Presumably, this is a solution of the Yang-Mills Millennium problem.

Normal Limits of Nilpotent Operators in von Neumann Algebras

We will investigate the intersection of the normal operators with the norm closure of the nilpotent operators in von Neumann algebras. A complete description of said intersection will be given for type I and type III von Neumann algebras with separable predual whereas difficulties and restrictions in the type II setting will be discussed. In addition, the closure of the span of the nilpotent operators and the distance from a projection to the nilpotent operators will be examined in the context of von Neumann algebras.

Local convergence in measure on semifinite von Neumann algebras, II

Suppose that ℳ is a von Neumann algebra of operators on a Hilbert space \(\mathcal{H}\) and τ is a faithful normal semifinite trace on ℳ. The set Open image in new window of all τ-measurable operators with the topology t τ of convergence in measure is a topological *-algebra. The topologies of τ-local and weakly τ-local convergence in measure are obtained by localizing t τ and are denoted by t τ1 and t wτ1, respectively. The set Open image in new window with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in Open image in new window with respect to the topologies t τ1 and t wτ1 are proved. S.M. Nikol’skii’s theorem (1943) is extended from the algebra \(\mathcal{B}(\mathcal{H})\) to semifinite von Neumann algebras. The following theorem is proved: For a von Neumann algebra ℳ with a faithful normal semifinite trace τ, the following conditions are equivalent: (i) the algebra ℳ is finite; (ii) t wτ1 = t τ1; (iii) the multiplication is jointly t τ1-continuous from Open image in new window to Open image in new window ; (iv) the multiplication is jointly t τ1-continuous from Open image in new window to Open image in new window ; (v) the involution is t τ1-continuous from Open image in new window to Open image in new window .

Unitary orbits of normal operators in von Neumann algebras

The starting points for this paper are simple descriptions of the norm and strong* closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of sigma-finite algebras, any two normal operators have the same norm-closed unitary orbit if and only if they have the same strong*-closed unitary orbit if and only if they have the same strong-closed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strong-closed unitary orbit of any operator, not necessarily normal, meets the center in the (non-void) left essential central spectrum of Halpern. One corollary is a type III Weyl-von Neumann-Berg theorem involving containment of essential central spectra.

Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces

Let H be a separable complex Hilbert space, A a von Neumann algebra in ℒ( H),a faithful, normal state on A . We prove that if a sequence ( X n : n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σ ∞ n=1 n −2 Var( X n ) log 2( n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L 2 = L 2 ( A , φ ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complex separable Hilbert space H 0 , and let {T t , t ≥ 0} be a uniformly continuous quantum dynamical semigroup of completely positive unital maps on A. The infinitesimal generator L of {T t } is a bounded linear operator on the Banach space A. For any Hilbert space K, denote by B(K) the von Neumann algebra of all bounded operators on K. Christensen and Evans [3] have shown that L has the form [formula] where π is a representation of A in B(K) for some Hilbert space K, R: H 0 → K is a bounded operator satisfying the 'minimality' condition that the set {(RX-π(X)R)u, u∈H 0 , X∈A} is total in K, and K 0 is a fixed element of A. The unitality of {T t } implies that L(1) = 0, and consequently K 0 =iH-½RR, where H is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, π, R, H) constitutes the set of Christensen-Evans (CE) parameters which determine the CE generator L of the semigroup {T t }. It is quite possible that another set (K', π, R', H') of CE parameters may determine the same generator L. In such a case, we say that these two sets of CE parameters are equivalent. In Section 2 we study this equivalence relation in some detail.

Simulated annealing and quantum detailed balance

The analogue of simulated annealing is considered for time-inhomogeneous evolutions of a von Neumann algebra of operators, whose instantaneous generator at each timet satisfies the quantum detailed balance condition with respect to a faithful normal state which depends on time through a suitable cooling schedule. Convergence to the (nonfaithful) limiting state is proved under the usual kinds of assumptions. The approach is interesting in view of possible applications to stochastic Ising models and to Boltzmann machines.

Convex combinations of unitary operators in von Neumann algebras

The unitary rank of an element T in a von Neumann algebra U is defined as the smallest number u ( T ) for which there is a convex combination of unitaries from U of length u ( T ) and equalling T . We determine the unitary rank for every element T in the closed unit ball of U in terms of the index i ( T ) of T and the distance α ( T ) from T to the group of invertible elements in U . If i ( T ) = 0, then u ( T ) ⩽ 2; if i ( T ) ≠ 0, then u ( T ) = n when n − 1 < 2(1 − α ( T )) −1 ⩽ n , and u ( T ) = ∞ when α ( T ) = 1. We also determine precisely which asymmetric convex decompositions of T can be realized. We show that only an approximate version of these results holds in a general C ∗ -algebra.