The multi-form generalised Gleason theorem
type: Item
instance of: scholarly article
title: The multi-form generalised Gleason theorem
zbMATH Open document ID: 0939.46041
author: Oliver Rudolph
author: J. D. Maitland Wright
published in: Communications in Mathematical Physics
publication date: 2000-06-18T00:00:00Z
review text: The authors prove the following theorem: Theorem. Let \(X\) be a Banach space. Let \(k\) be a positive integer. For \(j=1,2,\dots,k\) let \(A_j\) be a von Neumann algebra with no direct summand of Type \(I_2\). Let \(m\) be an \(X\)-valued \(k\)-measure on \(P(A_1)\times P(A_2)\times \cdots \times P(A_k)\) whose values are bounded in norm by a constant \(K\). Then there exists a unique \(f\) in \text{Lin}\((A_1, A_2,\dots,A_k;X)\) such that \(f(p_1, p_2,\dots,p_k)=m(p_1, p_2,\dots,p_k)\) for all projections \(p_1, p_2,\dots,p_k\) in, respectively \(A_1, A_2,\dots,A_k\). Also \(\|f\|\leq 4^kK\). Here \text{Lin}\((A_1, A_2,\dots,A_k;X)\) denotes the Banach space of all \(X\)-valued \(k\)-linear forms. It is claimed that the result is a necessary tool for the study of decoherence in quantum theory.
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46G25
zbMATH DE Number: 1269792
zbMATH Keywords: Gleason theorem
zbMATH Keywords: decoherence
zbMATH Keywords: bi-measure
zbMATH Keywords: Banach space of all \(X\)-valued \(k\)-linear forms
reviewed by: Chandra Shekhar Sharma
MaRDI profile type: Publication
full work available at URL: s002200050491
OpenAlex ID: W2068974509
DOI: 10.1007/S002200050491
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description: scientific article; zbMATH DE number 1269792