Absolutely compatible pair of elements in a von Neumann algebra-II

Abstract

Let A be a unital C\(^*\)-algebra with unity \(1_A\). A pair of elements \(0 \le a, b \le 1_A\) in A is said to be absolutely compatible if, \(\vert a - b \vert + \vert 1_A - a - b \vert = 1_A.\) In this paper we provide a complete description of absolutely compatible pair of strict elements in a von Neumann algebra. The end form of such a pair has a striking resemblance with that of a ‘generic pair’ of projections on a complex Hilbert space introduced by Halmos.

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Acknowledgements

The author is grateful to Antonio M. Peralta for introducing him the notion of ‘strict’ elements. The author is also grateful to the referees for their valuable suggestions.

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Correspondence to Anil Kumar Karn.

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Karn, A.K. Absolutely compatible pair of elements in a von Neumann algebra-II. RACSAM 114, 153 (2020). https://doi.org/10.1007/s13398-020-00883-7

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Keywords

  • von Neumann algebras
  • Absolute compatibility
  • Strict element
  • Generic pair of projections

Mathematics Subject Classification

  • Primary 46L10
  • Secondary 46B40