V. S. Sunder

An invitation to von Neumann algebras

0 Introduction.- 0.1 Basic operator theory.- 0.2 The predual L(H)*.- 0.3 Three locally convex topologies on L(H).- 0.4 The double commutant theorem.- 1 The Murray - von Neumann Classification of Factors.- 1.1 The relation... ~... (rel M).- 1.2 Finite projections.- 1.3 The dimension function.- 2 The Tomita - Takesaki Theory.- 2.1 Noncommutative integration.- 2.2 The GNS construction.- 2.3 The Tomita-Takesaki theorem (for states).- 2.4 Weights and generalized Hilbert algebras.- 2.5 The KMS boundary condition.- 2.6 The Radon-Nikodym theorem and conditional expectations.- 3 The Connes Classification of Type III Factors.- 3.1 The unitary cocycle theorem.- 3.2 The Arveson spectrum of an action.- 3.3 The Connes spectrum of an action.- 3.4 Alternative descriptions of ?(M).- 4 Crossed-Products.- 4.1 Discrete crossed-products.- 4.2 The modular operator for a discrete crossed-product.- 4.3 Examples of factors.- 4.4 Continuous crossed-products and Takesakis duality theorem.- 4.5 The structure of properly infinite von Neumann algebras.- Appendix: Topological Groups.- Notes.

On majorization and Schur products

Abstract Suppose A, D1,…,Dm are n × n matrices where A is self-adjoint, and let X = Σ m k = 1 D k AD ∗ k . It is shown that if ΣD k D ∗ k = ΣD ∗ k D k = I , then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D1,…,Dm, a result is obtained in terms of weak majorization. If each Dk is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if bii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.

Functional analysis : spectral theory

Normed spaces - vector spaces, normed spaces, linear operators, the Hahn-Banach theorem, completeness, some topological considerations Hilbert spaces - inner product spaces, some preliminaries, orthonormal bases, the adjoint operator, strong and weak convergence C*-algebras - Banach algebras, Gelfand-Naimark theory, commutative C*-algebras, representations of C*-algebras, the Hahn-Hellinger theorem some operator theory - the spectral theorem, polar decomposition, compact operators, Fredholm operators and index unbounded operators - closed operators, symmetric and self-adjoint operators, spectral theorem and polar decomposition. Appendices: some linear algebra transfinite considerations topological spaces compactness measure and integration the Stone-Weierstrass theorem the Riesz representation theorem.

The planar algebra associated to a Kac algebra

We obtain (two equivalent) presentations — in terms of generators and relations — of the planar algebra associated with the subfactor corresponding to (an outer action on a factor by) a finite-dimensional Kac algebra. One of the relations shows that the antipode of the Kac algebra agrees with the ‘rotation on 2-boxes’.

FROM SUBFACTOR PLANAR ALGEBRAS TO SUBFACTORS

We present a purely planar algebraic proof of the main result of a paper of Guionnet–Jones–Shlaykhtenko which constructs an extremal subfactor from a subfactor planar algebra whose standard invariant is given by that planar algebra.

Distance between normal operators

Lidskii and Wielandt have proved independently that if A and B are selfadjoint operators on an n-dimensional space H, with eigenvalues {α k } n k=l and {β k } n k=1 respectively (counting multiplicity), then, ||A-B||≥min σeS n ||diag (α k -β σ(k) )|| for any unitarily invariant norm on L(H). In this note an example is given to show that this result is no longer true if A and B are only required to be normal (even unitary). It is also shown that the above inequality holds in the operator norm, if A is selfadjoint and B is skew-self-adjoint.

On permutations, convex hulls, and normal operators

Abstract A spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite-dimensional space. This is used to prove the following: if A and B are normal operators on an n -dimensional complex Hilbert space H with eigenvalues given by α 1 ,…,α n and β 1 ,…, β n respectively, and if A − B is also normal, then ‖A − B‖ ⩽ max σ ϵ S n ‖ diag (α k −β σ(k) )‖ for any unitarily invariant norm on L ( H ).

On hypergroups of matrices

After recalling the definition and some basic properties of finite hypergroups—a notion introduced in a recent paper by one of the authors—several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: ‘let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a 1…a n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then Ak =Pk (A) is a non-negative matrix, where pk denotes the characteristic polynomial of the top k× k principal submatrix of A, for k=1,…,n. The matrices Ak as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) ai =0 for all ior (ii) ai =0 for i<n and an =1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.

An extremal property of the permanent and the determinant

Abstract Given an n × n matrix A , define the n ! × n ! matrix A , with rows and columns indexed by the permutation group S n , with (σ, τ) entry Π n i =1 a τ( i ), σ( i ) . It is shown that if A is positive semidefinite, then det A is the smallest eigenvalue of A ; it is conjectured that per A is the largest eigenvalue of A , and the conjecture proved for n ⩽3. Several known and some unknown inequalities are derived as consequences.

Actions of Finite Hypergroups

This paper is concerned with actions of finite hypergroups on sets. After introducing the definitions in the first section, we use the notion of ‘maximal actions’ to characterise those hypergroups which arise from association schemes, introduce the natural sub-class of *-actions of a hypergroup and introduce a geometric condition for the existence of *-actions of a Hermitian hypergroup. Following an insightful suggestion of Eiichi Bannai we obtain an example of the surprising phenomenon of a 3-element hypergroup with infinitely many pairwise inequivalent irreducible *-actions.