A trace inequality of John von Neumann

The principal aim of this note is to establish an effectively self-contained proof of J. von Neumann's inequality \(\left| {tr\left( {AB} \right)} \right| \leqslant \sum\limits_{r = 1}^n {\varrho _r \sigma _r } \) , whereA, B are any complexn×n matrices with singular values ?₁?...?? _n , σ₁?...?σ _n respectively.     

On von Neumann regular rings,III

This note is a natural sequel to [8] and [9]. Further characteristic properties of arbitrary von Neumann regular rings and strongly regular rings are given in terms of annihilators and simple modules. A prime ring with certain annihilator conditions is shown to be primitive (this is related to the following problem ofKaplansky: Are prime regular rings primitive?). Necessary and sufficient conditions for leftq-rings to be regular are also considered: For example, a leftq-ring is regular iff every simple rightA-module is flat. A sufficient condition is given for a leftqc-ring to be a uniserial, strongly left and strongly rightqc, left and rightq-ring. One of the main results ofJain, Mohamed andSingh onq-rings [5, Theorem 2.13] is generalised. Finally, it is shown that a prime left continuous ring either has zero socle or is primitive, left self-injective regular.     

On von Neumann regular rings,XIII

Summary Generalizations of projectivity and quasi-injectivity, calledC-projectivity andIC-injectivity, are introduced to study von Neumann regular rings, continuous and self-injetive regular rings. Conditions for non-reduced ideals to contain non-trivial central idempotents are considered.

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Riassunto Vengono introdotte delle generalizzazioni delle proiettività e delle quasi-iniettività detteC-proiettività eIC-iniettività per studiare gli anelli regolari di von Neumann, anelli continui e regolari auto-iniettivi. Sono inoltre considerate condizioni a<nchè ideali non ridotti contengano idempotenti centrali non banali.

     

Constrained von Neumann Inequalities

An equivalent formulation of the von Neumann inequality states that the backward shift S* on ?₂ is extremal, in the sense that if T is a Hilbert space contraction, then ‖p(T)‖?‖p(S*)‖ for each polynomial p. We discuss several results of the following type: if T is a Hilbert space contraction satisfying some constraints, then S* restricted to a suitable invariant subspace is an extremal operator. Several operator radii are used instead of the operator norm. Applications to inequalities of coefficients of rational functions positive on the torus are given.     

von Neumann regular modules

In this paper, we introduce von Neumann regular modules and give many characterizations of von Neumann regular modules. Further, we investigate the relations between von Neumann regular modules and other classical modules. Finally, we characterize Noetherian von Neumann regular modules.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Directed Graphs, von Neumann Algebras, and Index

In this paper, we assign index numbers to finite directed graphs. Motivated by the indices of Jones and Watatani (from operator algebra theory), we introduce and compute a new graph-theoretical index, and consider the connection with Watatani’s extended Jones index. Starting with an inclusion of finite directed graphs, we show that there is a natural subgroupoid inclusion, and then a tower of von Neumann algebras. In particular, each step in the tower having the same index number, under certain normalization.     

On von Neumann regular f-algebras

It is shown that for a large class of f-algebras, von Neumann regularity and -lateral completeness are equivalent notions.     

Enhancements to the von Neumann trace inequality

Upper trace bounds for the product of two n × n complex matrices are presented. The real component of the trace inequality is tighter than von Neumann’s inequality, and the imaginary component is new.     

Operator valued weights in von Neumann algebras,II

Let M be a von Neumann algebra, and N a sub von Neumann algebra of M. We prove that if φ and ψ are n.f.s. weights on N and M respectively, such that σ_t^ψ extends σ_t^φ, then there is a unique n.f.s. operator valued weight T from M to N, such that ψ = φ ° T. Moreover we generalize the notion of modular automorphism groups associated with conditional expectations to operator valued weights.     

Representations,weights, and expectations in von Neumann algebras

Translated from Issledovaniya po Prikladnoi Matematike, No. 10, pp. 67–72, 1984.     

Nest-subalgebras of von Neumann algebras

The structure of a certain class of separably acting reflexive operator algebras is investigated for which the nest algebras of J. Ringrose can be considered prototypes. To a fixed von Neumann algebra and a complete nest of projections contained therein one associates the algebra of all operators in the von Neumann algebra which leave every member of the nest invariant. A generalization of the Ringrose criterion for inclusion in the Jacobson radical of a nest algebra is given for this more general class of algebras. Further properties of the radical are studied.     

Quasicrystals,aperiodic order,and groupoid von Neumann algebras

We introduce tight binding operators for quasicrystals that are parametrized by Delone sets. These operators can be regarded in a natural operator algebra framework that encodes the long range aperiodic order. This algebraic point of view allows us to study spectral theoretic properties. In particular, the integrated density of states of the tight binding operators is related to a canonical trace on the associated von Neumann algebra. To cite this article: D. Lenz, P. Stollmann, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1131–1136.     

Preduals of von Neumann Algebras

Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra A is the third dual of some Banach space, then the space A is isometrically isomorphic to the second dual of some von Neumann algebra A and the von Neumann algebra A is uniquely determined by its enveloping von Neumann algebra (up to von Neumann algebra isomorphism) and is the unique second predual of A (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders.