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Generalisation of Michelson contrast for operators and its properties

Sami Abdullah Abed, Irina Nikolaeva, Andrei Novikov

Abstract

In this article we consider generalisation of Michelson contrast for positive operators of countably decomposable $W^*$-algebras and find a few several of its properties.

Generalisation of Michelson contrast for operators and its properties

Abstract

In this article we consider generalisation of Michelson contrast for positive operators of countably decomposable -algebras and find a few several of its properties.

Paper Structure

This paper contains 9 sections, 22 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Centrality and invertibility in $B(H)^+$
  4. The centrality measure
  5. Applications
  6. Gardner's inequality inspired simulation
  7. $L_1$ equality violation
  8. Data Avaialabilty Statement
  9. Discussion

Key Result

Lemma 1

Let $\mathcal{A}$ be a unital $C^*$-algebra with unit $\mathbf{1}$ and its element $x(\in \mathcal{A})$ be hermitian (i.e. $x=x^*$), then

Figures (3)

  • Figure 1: $x=\Delta(X)$, $y=\Delta(Y)$, with $X\in\mathbb{M}^+_n(\mathbb{R}), Y\in \mathbb{M}_n(\mathbb{R}), \|X\|=\|Y\|=1$ and $z=|\mathrm{Tr}(XY)|-\mathrm{Tr}(X|Y|)$. The left column is a 3D scatter plot, the middle column is a plot of $z$ vs. $x$ and the right column is $z$ vs. $y$, the upper row is for $n=2$ and the lower row is for $n=3$.
  • Figure 2: The scatter plots above are visualising results of simulations with $x=\Delta(X)$, $y=\Delta(Y)$, with $X\in\mathbb{M}_n(\mathbb{R}), Y\in \mathbb{M}_n(\mathbb{R}), \|X\|=\|Y\|=1$ and $z=|\mathrm{Tr}(XY)|-\mathrm{Tr}(|X||Y|)$. The left column is a 3D scatter plot, the middle column is a plot of $z$ vs. $x$ and the right column is $z$ vs. $y$. The rows correspond for $2$, $3$, $4$ and $5$-dimentional simulations respectively.
  • Figure 3: The scatter plots above are visualising results of simulations with $x=\Delta(|X|)$, $y=\Delta(Y)$, with $X\in\mathbb{M}_n^{sa}(\mathbb{R}), Y\in \mathbb{M}_n^+(\mathbb{R}), \|X\|=\|Y\|=1$ and $z=\mathrm{Tr}(Y|X|Y)|-\mathrm{Tr}(|YXY|)$. The left column is a 3D scatter plot, the middle column is a plot of $z$ vs. $x$ and the right column is $z$ vs. $y$. The rows correspond for $2$, $3$, $4$ and $5$-dimentional simulations respectively.

Theorems & Definitions (45)

  • Lemma 1: Dixmier
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • ...and 35 more