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Quasi-units as orthogonal projections

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Abstract

The notion of quasi-unit has been introduced by Yosida in unital Riesz spaces. Later on, a fruitful potential-theoretic generalization was obtained by Arsove and Leutwiler. Due to the work of Eriksson and Leutwiler, this notion also turned out to be an effective tool by investigating the extreme structure of operator segments. This paper has multiple purposes which are interwoven, and are intended to be equally important. On the one hand, we identify quasi-units as orthogonal projections acting on an appropriate auxiliary Hilbert space. As projections form a lattice and are extremal points of the effect algebra, we conclude the same properties for quasi-units. Our second aim is to apply these results for nonnegative sesquilinear forms. Constructing an order-preserving bijection between operator and form segments, we provide a characterization of being extremal in the convexity sense, and we give a necessary and sufficient condition for the existence of the greatest lower bound of two forms. Closing the paper we revisit some statements by using the machinery developed by Hassi, Sebestyén, and de Snoo. It will turn out that quasi-units are exactly the closed elements with respect to the antitone Galois connection induced by parallel addition and subtraction.

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Authors and Affiliations

  1. Department of Applied Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c., Budapest, H-1117, Hungary

    Zsigmond Tarcsay

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reál-tanoda u. 13-15., Budapest, H-1053, Hungary

    Tamás Titkos

  3. BBS University of Applied Sciences, Alkotmány u. 9., Budapest, H-1054, Hungary

    Tamás Titkos

Authors
  1. Zsigmond Tarcsay
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  2. Tamás Titkos
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Correspondence to Zsigmond Tarcsay.

Additional information

Dedicated to Professor Zoltán Sebestyén on the occasion of his 75th birthday

Communicated by L. Molnár

Supported by the Hungarian Ministry of Human Capacities, NTP-NFTÖ-17.

Supported by the Hungarian National Research, Development and Innovation Office - NKFIH (PD128374), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-18-4-BGE-3 New National Excellence Program of the Ministry of Human Capacities.

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Tarcsay, Z., Titkos, T. Quasi-units as orthogonal projections. ActaSci.Math. 84, 413–430 (2018). https://doi.org/10.14232/actasm-017-088-z

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  • DOI: https://doi.org/10.14232/actasm-017-088-z

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