Taiwanese Journal of Mathematics

Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant

Marcell Gaál, Gergő Nagy, and Patricia Szokol

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

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

Source
Taiwanese J. Math., Advance publication (2019), 11 pages.

Dates
First available in Project Euclid: 8 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1552013881

Digital Object Identifier
doi:10.11650/tjm/190205

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

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

Citation

Gaál, Marcell; Nagy, Gergő; Szokol, Patricia. Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant. Taiwanese J. Math., advance publication, 8 March 2019. doi:10.11650/tjm/190205. https://projecteuclid.org/euclid.twjm/1552013881


Export citation

References

  • 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.