The Schur–Horn problem for normal operators

Let A be a semifinite and properly infinite von Neumann algebra of operators of a complex Hilbert space H. Call C e A compact (relative to A) if it is the limit in the norm of elements with finite support. The investigation of analytic properties of compact elements of A, generalizing welt-known classical theorems on compact operators, was started in [2] and is continued in the present paper. Let N i (resp. Ri) be the null (resp. range) projection of(1 C ) ~ where i is a natural number. Contrary to the classical case, the sequences N1 =R2 > R3 ~ "" need not be stationary, even if A is of type I. The question of the finiteness (resp. cofiniteness) of the projection Noo = sup Ni (resp. R~ = infRi) was left open in [2]. In the present paper it is shown that No~ is in fact always finite (and R~ is cofinite). It was shown already in [2] that if N® is finite, then inf(N®, Ro0) = 0 and sup(No~,R~)= 1. This is a decomposition of H in the sense of lattice theory or perhaps continuous geometry. In the present paper this result is refined by introducing the concepts of essentially closed subspaces and essentially topological direct sums. It is shown that the algebraic sum No~(H)+R®(H) is essentially topological direct and essentially closed. The operator 1 C decomposes into the sum of two closed operators affiliated with A, one summand being essentially nilpotent on Noo(H), the other one essentially regular on Roo(H). Throughout this paper H denotes a complex Hilbert space and A a properly infinite semifinite von Neumann algebra of linear operators of H. § 1. Preliminaries

A quasi-expectation of a C ∗ {C^ \ast } -algebra A on a C ∗ {C^\ast } -subalgebra B is a bounded linear projection Q : A → B Q:A \to B such that Q ( x a y ) = x Q ( a ) y ∀ x , y ∈ B , a ∈ A Q(xay) = xQ(a)y \forall x,y \in B,a \in A . It is shown that if M is a von Neumann algebra of operators on Hilbert space H for which there exists a quasi-expectation of B ( H ) B(H) on M, then there exists a projection of norm one of B ( H ) B(H) on M, i.e. M is injective. Further, if M is an amenable von Neumann subalgebra of a von Neumann algebra N, then there exists a quasi-expectation of N on M ′ ∩ N M’ \cap N . These two facts yield as an immediate corollary the recent result of A. Connes that all amenable von Neumann algebras are injective.

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.

The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II 1 _1 . In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra M M , under which every local or 2-local derivation on the algebra of measurable operators affiliated with M M is automatically becomes a derivation. We also give examples of commutative algebras of measurable operators admitting local and 2-local derivations which are not derivations.

Operator algebras come in many flavours. For the purpose of this article, however, the term is only used for one of two kinds of self-adjoint algebras of operators on Hilbert space, viz., C*- algebras (which are norm-closed) or von Neumann algebras (which are closed in the topology of pointwise strong convergence or equivalently, in the weak-* topology it inherits as a result of being a Banach dual space). To be fair, there are a number of people in India (eg., Gadadhar Misra, Tirthankar Bhattacharyya, Jaydeb Sarkar, Santanu Dey, etc.,) who work on non-selfadjoint algebras, mostly from the point of view of connections with complex function theory; but in the interest of restricting the size of this paper, I confine myself here to selfadjoint algebras. I apologise for ways in which my own personal taste and limitations colour this depiction of operator algebras. Another instance of this arbitrary personal taste is a decision to concentrate on the work of younger people. Thus, the work of the more senior people who have worked in operator algebras is only seen via their collaborations with younger people: e.g., KRP via Srinivasan and Rajarama Bhat, Kalyan Sinha via Debashish, Partha, Arup, Raja, etc,. and me via Vijay, Srinivasan and Panchugopal. Not long ago, interest in operator algebras in India was restricted to the three centres of the Indian Statistical Institute. Now, I am happy to note that it has spread to IMSc, some IITs, IISERs, NISER, JNU,.... My role in this article has been merely that of compiling inputs from many active Indian operator algebraists that came to my mind. I wrote soliciting a response from a certain number of them, then put together the responses received. (I apologise to those people who were omitted in this process). My colleague, Partha, with the help of his collaborator Arup, agreed to take care of the C*-related inputs, while I take care of the von Neumann-related ones with the help of my collaborator Vijay. What follows are some areas of ongoing research done in von Neumann algebras in India and some names of people doing such work: (a) subfactors and planar algebras, (Vijay Kodiyalam of IMSc, Chennai); (b) quantum dynamical systems and complete positivity (Rajarama Bhat of ISI, Bengaluru); (c) E0 semigroups (R. Srinivasan of CMI, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar), and (d) Masas in II1 von Neumann algebras and free Araki-Woods factors (Kunal Mukherjee of IIT, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar).

The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

Let r be a normal semifinite trace on a von Neumann algebra, and let f be a continuous convex function on the interval [0, oo) with f (0) = 0. For a positive element a of the algebra and a positive contraction a on the algebra, the following inequality is obtained: r(f (c((a))) Zf((Ari,r,i)), where f is a convex function on the real line, A is a selfadjoint operator, and {rqi} is an orthonormal set (see [11], for example). Slightly more general, if {Pi} is a pairwise orthogonal family of projections and we set a(A) = >nPiAPi for a selfadjoint operator A, then tr a (f (A)) > tr f (a (A)) (see [2, 6, and 9]). Going further in this direction Brown and Kosaki [3] have recently proved that r(f (v*av)) < r(v* f (a)v), where r is a faithful normal semifinite trace on a von Neumann algebra, v is a contraction, and a is a positive element of the algebra. (In fact, a may be unbounded and affiliated with the algebra.) In the present paper we obtain a similar inequality for an arbitrary positive contraction a on the algebra and show that (*) r(f (a (a))) < r(a(f (a))). Here f is supposed to be a continuous convex function with f(O) = 0. (Note that (*) holds without r if f is operator convex; cf. Theorem IV.1 in [1].) The inequality (*) supports the experience that inequalities involving operators are more simply treatable provided both sides are inside of a trace (see [3 and 8] for further evidences). The idea of our proof is to approximate the operators a and a (a) by diagonal ones by means of their spectral resolution and to apply the classical Jensen's inequality in Received by the editors November 19, 1984 and, in revised form, June 24, 1985 and December 17, 1985. 1980 Mathematics Subject Cai Secondary 26A51, 47B15.

The Banach-Stone theorem says, in the language we shall use, that if two commutative C*-algebras are isomorphic, then they are as C*-algebras. In this formulation there are two natural question: whether the theorem holds for noncommutative C*-algebras, and whether it is possible to replace isometrically isomorphic with isomorphic. A complete answer to the first question was given by Kadison in [10], and M. Cambern settled the second in the affirmative in [5]. In Section 3 we show that Cambern's result may be partially extended to noncommutative von Neumann algebras having the property P ([9, 18, 19]). When we have the result of Section 3 that a nearly isometric completely positive map of a von Neumann algebra having property P is close to a * isomorphism between the algebras, we have nearly proved that von Neumann algebras whose unit balls are close in the Hausdorff metric are unitarily equivalent by a unitary close to the identity. The only thing we have to prove then is that if q) is an isomorphism of a v o n Neumann algebra A having property P onto a v o n Neumann algebra B acting on the same Hilbert space, and 4~ satisfies, for some positive k less than one and any x in A,

On homomorphisms of an orthogonally decomposable Hilbert space

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

A non-commutative version of probability theory is outlined, based on the concept of aΣ*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory ofΣ*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. TheΣ*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering theΣ*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schroedinger operators on manifolds, quantum percolation and quasi crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z*z. We show that a large class of L∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F3). These results generalize some of what is known about the quasinilpotent DT–operator.

This paper characterizes the self-adjoint part of C*-algebras and von Neumann algebras among normed Jordan algebras. It also explains how the associative product is determined by a general notion of orientation which is related to dynamics and reflects the dual role of physical variables as observables and as generators of one-parameter groups of motions of the state space (Schrodinger picture). This concept of orientation bridges the approach in Connes' characterization of the natural cone of a von Neumann algebra and our own characterization of the state space of a C*-algebra.