Taiwanese Journal of Mathematics

Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant

Marcell Gaál, Gergő Nagy, and Patricia Szokol

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In this paper we explore the structure of certain surjective generalized isometries (which are transformations that leave any given member of a large class of generalized distance measures invariant) of the set of positive invertible elements in a finite von Neumann factor with unit Fuglede-Kadison determinant. We conclude that any such map originates from either an algebra $^*$-isomorphism or an algebra $^*$-antiisomorphism of the underlying operator algebra.

Article information

Taiwanese J. Math., Volume 23, Number 6 (2019), 1423-1433.

Received: 17 October 2018
Revised: 22 February 2019
Accepted: 25 February 2019
First available in Project Euclid: 8 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L40: Automorphisms
Secondary: 47L30: Abstract operator algebras on Hilbert spaces

isometries von Neumann algebras positive definite operators Fuglede-Kadison determinant totally geodesic submanifolds


Gaál, Marcell; Nagy, Gergő; Szokol, Patricia. Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant. Taiwanese J. Math. 23 (2019), no. 6, 1423--1433. doi:10.11650/tjm/190205. https://projecteuclid.org/euclid.twjm/1552013881

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  • W. N. Anderson and G. E. Trapp, Operator means and electrical networks, Proc. 1980 IEEE International Symposium on Circuits and Systems (1980), 523–527.
  • R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer-Verlag, New York, 1997.
  • P. T. Fletcher and S. Joshi, Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors, Lecture Notes in Comput. Sci. 3117 (2004), 87–98.
  • B. Fuglede and R. V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55 (1952), 520–530.
  • O. Hatori, Examples and applications of generalized gyrovector spaces, Results Math. 71 (2017), no. 1-2, 295–317.
  • R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras II: Advanced Theory, Graduate Studies in Mathematics 16, American Mathematical Society, Providence, RI, 1986.
  • C. F. Manara and M. Marchi, On a class of reflection geometries, Istit. Lombardo Accad. Sci. Lett. Rend. A 125 (1991), no. 2, 203–217.
  • M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl. 26 (2005), no. 3, 735–747.
  • L. Molnár, General Mazur-Ulam type theorems and some applications, in: Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, 311–342, Oper. Theory Adv. Appl. 250, Birkhäuser, Cham, 2015.
  • L. Molnár and P. Szokol, Transformations on positive definite matrices preserving generalized distance measures, Linear Algebra Appl. 466 (2015), 141–159.
  • L. Molnár and D. Virosztek, Continuous Jordan triple endomorphisms of $\mathbb{P}_2$, J. Math. Anal. Appl. 438 (2016), no. 2, 828–839.
  • G. K. Pedersen, $C^*$-algebras and Their Automorphism Groups, London Mathematical Society Monographs 14, Academic Press, London, 1979.
  • B. Simon, Trace Ideals and Their Applications, Second edition, Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, 2005.