Herbert Halpern

Matrix norm inequalities and the relative Dixmier property

AbstractIf x is a selfadjoint matrix with zero diagonal and non-negative entries, then there exists a decomposition of the identity into k diagonal orthogonal projections {pm} for which

Unitary implementation of automorphism groups on von Neumann algebras

\parallel \sum p_m xp_m \parallel \leqslant (1/k)\parallel x\parallel

Integral decomposition of functionals on *-algebras

From this follows that all bounded matrices with non-negative entries satisfy the relative Dixmier property or, equivalently, the Kadison Singer extension property. This inequality fails for large Hadamard matrices. However a similar inequality holds for all matrices with respect to the Hilbert-Schmidt norm with constant k−1/2 and for Hadamard matrices with respect to the Schatten 4-norm with constant 21/4k−1/2.

A generalized dual for a *-algebra

We first prove that in a sigma-finite von Neumann factor M, a positive element

Spectral representations for abelian automorphism groups of von Neumann algebras

a

Irreducible module homomorphisms of a von Neumann algebra into its center

with properly infinite range projection R_a is a linear combination of projections with positive coefficients if and only if the essential norm ||a||_e with respect to the closed two-sided ideal J(M) generated by the finite projections of M does not vanish. Then we show that if ||a||_e>1, then a is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra. Secondly, we provide a necessary condition for a positive operator a to be a finite sum of projections in terms of the principal ideals generated by the excess part a_+:=(a-I)\chi_a(1,\infty) and the defect part a_-:= (I-a)\chi_a(0, 1) of a; this result appears to be new also for B(H). Thirdly, we prove that in a type II_1 factor a sufficient condition for a positive diagonalizable operators to be a finite sum of projections is that \tau(a_+)- \tau(a_-)>0.

Module homomorphisms of a von Neumann algebra into its center

A necessary and sufficient continuity condition is obtained in order that a topological group of automorphisms of a semi-finite von Neumann algebra in standard form is unitarily implemented. The methods used are extended to the study of unitary implementation for a general von Neumann algebra of those automorphism groups that commute with the one-parameter modular automorphism group.

ESSENTIAL CENTRAL SPECTRUM AND RANGE FOR ELEMENTS OF A VON NEUMANN ALGEBRA

1. Introduction. An element C in a von Neumann algebra stf is said to be a commutator in stf if there are elements A and B in stf such that C=AB — BA. For finite homogeneous discrete algebras and for properly infinite factor algebras the set of commutators has been completely described [l]-[5], [10]. In each of these special cases any element C is a commutator modulo a central element depending on C. In this paper we show that given any element C in a properly infinite von Neumann algebra stf there is an element C0 in the center of stf depending on C such that C— C0 is a commutator in stf. The element C0 is an arbitrary element in the intersection criTc of the center with the uniform closure of the convex hull of {U*CU | U unitary in ?tf} [6, III, §5]. We then present a few facts about those elements C such that 0 e Xc or what is the same as far as determining commutators is concerned about those elements C such that OeJf^-ics for some invertible S in stf. 2. Commutators. Let stf be a C*-algebra with identity and let / be a closed two-sided ideal in stf. The image of the element A e stf in the factor algebra stf(T) = stf/1 under the canonical homomorphism of stf onto stf/I will be denoted by A(T). If £ is a maximal ideal of the center of stf, the smallest closed two-sided ideal in stf containing £ is denoted by [{]. For simplicity we write A([Q) as A(t). The set of maximal (respectively, primitive) ideals of stf with the hull-kernel topology is called the strong structure space (respectively, structure space) of stf. If stf is a von Neumann algebra, then the strong structure space M(stf) of stf is homeomorphic with the spectrum of the center S of stf under the map M —> M C\ J? [13]. This means M(stf) is extremely disconnected. Proposition 1. Let stf be a properly infinite von Neumann algebra and let A be a fixed element of stf. The function M -> ||^(M)|| of the strong structure space M (stf) of stf into the real numbers is continuous. Proof. For every «?Owe know that the set X={M e M (stf) \ \\A(M)\\ ^«} is closed. If /=p| X, then \\A(I)\\^a [8, Lemma 1.9] and so M(M)|| ^a for every MeM(stf) containing /. Thus X={M e M (stf) | I<=M}.