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Contractive linear preservers of absolutely compatible pairs between \(\hbox {C}^*\)-algebras

  • Nabin K. Jana
  • Anil K. Karn
  • Antonio M. PeraltaEmail author
Original Paper
  • 45 Downloads

Abstract

Let a and b be elements in the closed ball of a unital C\(^*\)-algebra A (if A is not unital we consider its natural unitization). We shall say that a and b are domain (respectively, range) absolutely compatible (\(a\triangle _d b\), respectively, \(a\triangle _r b\), in short) if \(\Big | |a| -|b| \Big | + \Big | 1-|a|-|b| \Big | =1\) (respectively, \(\Big | |a^*| -|b^*| \Big | + \Big | 1-|a^*|-|b^*| \Big | =1\)), where \(|a|^2= a^* a\). We shall say that a and b are absolutely compatible (\(a\triangle b\) in short) if they are both range and domain absolutely compatible. In general, \(a\triangle _d b\) (respectively, \(a\triangle _r b\) and \(a\triangle b\)) is strictly weaker than \(ab^*=0 \) (respectively, \(a^* b =0\) and \(a\perp b\)). Let \(T: A\rightarrow B\) be a non-expansive bounded linear mapping between C\(^*\)-algebras. We prove that if T preserves domain absolutely compatible elements (i.e., \(a\triangle _d b\Rightarrow T(a)\triangle _d T(b)\)) then T is a triple homomorphism. A similar statement is proved when T preserves range absolutely compatible elements. It is finally shown that T is a triple homomorphism if, and only if, T preserves absolutely compatible elements.

Keywords

Absolute compatibility Commutativity \(\hbox {C}^{*}\)-algebra von Neumann algebra Projection Partial isometry Linear absolutely compatible preservers 

Mathematics Subject Classification

Primary 46L10 Secondary 46B40 46L05 

Notes

Acknowledgements

We would like to thank the Referee for her/his valuable suggestions in a detailed report, and for sharing with us the result in Lemma 3.6. Third author partially supported by Junta de Andalucía Grant FQM375.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.School of Mathematical ScienceNational Institute of Science Education and Research, HBNIBhubaneswarIndia
  2. 2.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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