On homomorphisms of an orthogonally decomposable Hilbert space
type: Item
instance of: scholarly article
title: On homomorphisms of an orthogonally decomposable Hilbert space
zbMATH Open document ID: 0609.47049
DOI: 10.1016/0022-1236(86)90103-5
author: T. B. Dang
author: Sadayuki Yamamuro
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: Let M be a von Neumann algebra on a Hilbert space H with a cyclic separating vector. \(H^ J\), the real part of H, is then an ordered Hilbert space with respect to the natural cone \(H^+\). On this space, we investigate the properties of the non-commutative counterpart of the lattice homomorphism, which is called an o.d. homomorphism and is defined as a continuous linear operator \(\phi\) on H such that \(\phi (H^+)\subset H^+\) and \((\phi (\xi),\phi (\eta))=0\) whenever \(\xi \in H^+\), \(\eta \in H^+\), and \((\xi,\eta)=0\). Firstly a characterization of such operators is given. Secondly, we discuss the duals of o.d. homomorphisms and when they are normal. Thirdly, we consider semigroups of o.d. homomorphisms.
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 3985993
zbMATH Keywords: orthogonally decomposable Hilbert space
zbMATH Keywords: Tomita-Takesaki's theory
zbMATH Keywords: Kato's equality
zbMATH Keywords: von Neumann algebra on a Hilbert space
zbMATH Keywords: cyclic separating vector
zbMATH Keywords: real part
zbMATH Keywords: ordered Hilbert space
zbMATH Keywords: non-commutative counterpart of the lattice homomorphism
zbMATH Keywords: duals of o.d. homomorphisms
zbMATH Keywords: semigroups of o.d. homomorphisms
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90103-5
OpenAlex ID: W2080702417
cites work: Kato's equality and spectral decompositon for positive \(C_ 0\)-groups
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cites work: On orthogonally decomposable ordered Banach spaces
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description: scientific article; zbMATH DE number 3985993