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Orthograph related to mutual strong Birkhoff–James orthogonality in \(C^*\)-algebras

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Abstract

We study the relation of mutual strong Birkhoff–James orthogonality in two classical \(C^*\)-algebras: the \(C^*\)-algebra \({\mathbb {B}}(H)\) of all bounded linear operators on a complex Hilbert space H and the commutative, possibly nonunital, \(C^*\)-algebra. With the help of the induced graph it is shown that this relation alone can characterize right invertible elements. Moreover, in the case of commutative unital \(C^*\)-algebras, it can detect the existence of a point with a countable local basis in the corresponding compact Hausdorff space.

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Acknowledgements

The authors thank the anonymous referees for useful comments and suggestions that helped improve the presentation from the original manuscript. Ljiljana Arambašić and Rajna Rajić were fully supported by the Croatian Science Foundation under the project IP-2016-06-1046. Alexander Guterman and Svetlana Zhilina were fully supported by RSF grant 17-11-01124. Bojan Kuzma acknowledges the financial support from the Slovenian Research Agency, ARRS (research core funding No. P1-0222 and No. P1-0285).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Ljiljana Arambašić

  2. Department of Mathematics and Mechanics, Lomonosov Moscow State University, Moscow, 119991, Russia

    Alexander Guterman & Svetlana Zhilina

  3. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    Alexander Guterman, Bojan Kuzma & Svetlana Zhilina

  4. Moscow Institute of Physics and Technology, Dolgoprudny, 141701, Russia

    Alexander Guterman & Svetlana Zhilina

  5. University of Primorska, Glagoljaška 8, 6000, Koper, Slovenia

    Bojan Kuzma

  6. IMFM, Jadranska 19, 1000, Ljubljana, Slovenia

    Bojan Kuzma

  7. Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb, Pierottijeva 6, 10000, Zagreb, Croatia

    Rajna Rajić

Authors
  1. Ljiljana Arambašić
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  2. Alexander Guterman
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  3. Bojan Kuzma
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  4. Rajna Rajić
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  5. Svetlana Zhilina
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Correspondence to Ljiljana Arambašić.

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Communicated by Michael Frank.

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Arambašić, L., Guterman, A., Kuzma, B. et al. Orthograph related to mutual strong Birkhoff–James orthogonality in \(C^*\)-algebras. Banach J. Math. Anal. 14, 1751–1772 (2020). https://doi.org/10.1007/s43037-020-00074-x

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  • DOI: https://doi.org/10.1007/s43037-020-00074-x

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