Characterizations of Jordan *-isomorphisms of C-algebras by weighted geometric mean related operations and quantities

Fadil Chabbabi, Mostafa Mbekhta, Lajos Molnár

Research output: Article

Abstract

In this paper we consider three operations on positive definite cones of C-algebras which are related to weighted geometric means and appear in the formulas defining various versions of quantum Rényi relative entropy. We show how Jordan *-isomorphisms between C-algebras can be characterized by the preservation of the norms of products under those operations or by the preservation of the operations themselves. We also obtain conditions for the commutativity of the underlying algebras by showing that we have that property if one of the quantities under considerations can be transformed by a surjective map to another different such quantity.

Original languageEnglish
Pages (from-to)364-390
Number of pages27
JournalLinear Algebra and Its Applications
Volume588
DOIs
Publication statusPublished - márc. 1 2020

Fingerprint

Jordan Isomorphism
Geometric mean
Algebra
C*-algebra
Preservation
Quantum Entropy
Relative Entropy
Commutativity
Positive definite
Cones
Cone
Entropy
Norm

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

Cite this

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abstract = "In this paper we consider three operations on positive definite cones of C⁎-algebras which are related to weighted geometric means and appear in the formulas defining various versions of quantum R{\'e}nyi relative entropy. We show how Jordan *-isomorphisms between C⁎-algebras can be characterized by the preservation of the norms of products under those operations or by the preservation of the operations themselves. We also obtain conditions for the commutativity of the underlying algebras by showing that we have that property if one of the quantities under considerations can be transformed by a surjective map to another different such quantity.",
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AU - Chabbabi, Fadil

AU - Mbekhta, Mostafa

AU - Molnár, Lajos

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AB - In this paper we consider three operations on positive definite cones of C⁎-algebras which are related to weighted geometric means and appear in the formulas defining various versions of quantum Rényi relative entropy. We show how Jordan *-isomorphisms between C⁎-algebras can be characterized by the preservation of the norms of products under those operations or by the preservation of the operations themselves. We also obtain conditions for the commutativity of the underlying algebras by showing that we have that property if one of the quantities under considerations can be transformed by a surjective map to another different such quantity.

KW - C-algebra

KW - Jordan -isomorphism

KW - Operator means

KW - Positive definite cone

KW - Preservers

KW - Weighted geometric means

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