Compact operators in type III\(_{\lambda}\) and type III\(_0\) factors
type: Item
instance of: scholarly article
title: Compact operators in type III\(_{\lambda}\) and type III\(_0\) factors
zbMATH Open document ID: 0591.46053
DOI: 10.1007/BF01451405
author: Herbert Halpern
author: Victor Kaftal
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164053
review text: For semifinite von Neumann algebras N, the two-sided ideal I(N) generated by the finite projections and its norm closure J(N) can be viewed as the generalization of the ideals of finite rank and compact operators on a Hilbert space. Analogous notions of finiteness and compactness are defined by the authors in the purely infinite case. The approach consists in using the semifiniteness of the crossed product of the algebra by the modular action of a given weight. More precisely, if M is a type \(III_{\lambda}\) factor \((0\leq \lambda <1)\), \(M=N\otimes Z\) where N is of type \(II_{\infty}\). If E is the conditional expectation \(M\to N\), then \[ I(M)=span\{x\in M^+;\quad E(x)\in I(N)\}\quad and\quad J(M)=\overline{I(M)}. \] They are respectively hereditary \({}^*\)- subalgebra and hereditary \(C^*\)-subalgebra of M, and J(M) is the hereditary class defined by J(N). Then the elements of J(M) verify the classical conditions verified by compacts operators on a Hilbert space. Similarly, the \(C_ p(M)\) spaces are introduced, generalizing the trace class operators when \(p=1\), with \(I(M)\subset C_ p(M)\subset J(M)\), and their relation with the \(L_ p(M)\) spaces defined by Haagerup is described.
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 3949044
zbMATH Keywords: semifinite von Neumann algebras
zbMATH Keywords: two-sided ideal
zbMATH Keywords: ideals of finite rank and compact operators on a Hilbert space
zbMATH Keywords: semifiniteness of the crossed product of the algebra by the
zbMATH Keywords: modular action of a given weight
zbMATH Keywords: conditional expectation
zbMATH Keywords: hereditary \(C^*\)-subalgebra
zbMATH Keywords: trace class operators
zbMATH Keywords: semifiniteness of the crossed product of the algebra by the modular action of a given weight
reviewed by: G. Loupias
MaRDI profile type: Publication
cites work: Fredholm theories in von Neumann algebras. I
cites work: Abelian subalgebras of von Neumann algebras
cites work: Une classification des facteurs de type ${\rm III}$
cites work: Q3861964
cites work: Q5723435
cites work: Relative Weak Convergence in Semifinite von Neumann Algebras
cites work: Spectral projections of \(L^ 1\) operators in type \(III_{\lambda}\) von Neumann algebras
cites work: Index theory in von Neumann algebras
cites work: Convergence of Fourier Series in Discrete Crossed Products of von Neumann Algebras
cites work: Q3205815
cites work: Q3213694
cites work: ON A CLASS OF OPERATORS IN VON NEUMANN ALGEBRAS WITH SEGAL MEASURE ON THE PROJECTORS
cites work: Q3876492
OpenAlex ID: W2998899426
description: scientific article; zbMATH DE number 3949044