Spectral scale of selfadjoint operators and trace inequalities
type: Item
instance of: scholarly article
title: Spectral scale of selfadjoint operators and trace inequalities
zbMATH Open document ID: 0655.47032
DOI: 10.1016/0022-247X(85)90176-3
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: In a quantum system the entropy of a state can be expressed as -tr(\(\rho\) log \(\rho)\), where \(\rho\) is a density matrix and tr is the trace. This entropy function is concave in \(\rho\). J. von Neumann pointed out the reason for this concavity is that the real-valued function \(t\mapsto t \log t\) is concave and the linear function \(\rho\) \(\mapsto tr f(\rho)\) is concave for any concave function f. In this paper, motivated by the notion of the generalized singular values due to \textit{T. Fack} [J. Operator Theory 7, No.2, 307-333 (1982; Zbl 0493.46052)], the author inroduces the spectral scale of a selfadjoint element in a finite von Neumann algebra and generalizes the aforementioned result of von Neumann to C *-algebras with a tracial state. Let R be a finite von Neumann algebra with a fixed faithful normal normaized trace \(\tau\). For any selfadjoint operator A affiliated to R the spectral scale is defined as \(\lambda_ t(A)=\min [s:\tau (P_ A(s,\infty))\leq t]\), \(t\in [0,1]\), where \(\int tdP_ A(t)\) is the spectral resolution of A. It is clear that \(\lambda_ t(A)\) is continuous from the right and nonincreasing in t. The scale \(\lambda\).(A):[0,1]\(\to {\mathbb{R}}\) is also measure preserving if [0,1] is endowed with the Lebesgue measure and the Borel measure \(H\mapsto \tau (P_ A(H))\) is considered on \({\mathbb{R}}\). Thus, for any bounded Borel function f of \({\mathbb{R}}\) we have \(\tau (f(A))=\int^{1}_{0}f(\lambda_ t(A))dt\). This property together with the following minimax characterization paves the way for the two main results in this paper, which will be stated immediately after the following: \(\lambda_ t(A)=\inf_{e}\sup_{s}\{<As,s>:s\in H\), H the Hilbert space A is acting on, \(\| s\| =1\), \(es=e\}\), where inf is taken over all the projections e such that \(\tau (e^{\perp})\leq t.\) Result 1 (Theorem 2): If f:\({\mathbb{R}}\to {\mathbb{R}}\) is a monotone function then \(a\mapsto \tau (f(a))\) is monotone on \(R_{S.A.}\). Result…
Mathematics Subject Classification ID: 47C15
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 4069643
zbMATH Keywords: entropy of a state
zbMATH Keywords: density matrix
zbMATH Keywords: trace
zbMATH Keywords: concavity
zbMATH Keywords: generalized singular values
zbMATH Keywords: spectral scale of a selfadjoint element in a finite von Neumann
zbMATH Keywords: C *-algebras with a tracial state
zbMATH Keywords: tracial positive linear maps
author: Dénes Petz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4069643