Abstract
In this paper we consider some important non Kubo–Ando means on positive definite cones of \(C^*\)-algebras and investigate their relations to the usual (Löwner) order. We study two basic questions. First, when, on the positive definite cone of a \(C^*\)-algebra, does such a mean \(\sigma \) satisfy the inequality \(A\le A \sigma B \le B\) for any pair A, B of elements with \(A\le B\). This requirement seems to be the most basic one a mean should fulfill even for numbers. Actually, we study stronger forms of that question and obtain results characterizing central elements and commutativity of \(C^*\)-algebras. Second, we investigate monotonicity properties of those means with respect to the usual order which also turn to be closely related to the commutativity of the underlying algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29, 328–347 (2006)
Audenaert, K.M.R., Hiai, F.: On matrix inequalities between the power means: counterexamples. Linear Algebra Appl. 439, 1590–1604 (2013)
Beneduci, R., Molnár, L.: On the standard K-loop structure of positive invertible elements in a \(C^*\)-algebra. J. Math. Anal. Appl. 420, 551–562 (2014)
Bhagwat, K.V., Subramanian, R.: Inequalities between means of positive operators. Math. Proc. Camb. Philos. Soc. 83, 393–401 (1978)
Bhatia, R.: Positive Definite Matrices. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2007)
Bhatia, R., Gaubert, S., Jain, T.: Matrix versions of the Hellinger distance. Lett. Math. Phys. 109, 1777–1804 (2019)
Bhatia, R., Jain, T., Lim, Y.: On the Bures–Wasserstein distance between positive definite matrices. Expo. Math. 37, 165–191 (2019)
Bhatia, R., Jain, T., Lim, Y.: Inequalities for the Wasserstein mean of positive definite matrices. Linear Algebra Appl. 576, 108–123 (2019)
Busch, P., Gudder, S.P.: Effects as functions on projective Hilbert space. Lett. Math. Phys. 47, 329–337 (1999)
Dumitru, R., Franco, J.A.: The Rényi power means of matrices. Linear Algebra Appl. 607, 45–57 (2020)
Fiedler, M., Pták, V.: A new positive definite geometric mean of two positive definite matrices. Linear Algebra Appl. 251, 1–20 (1997)
Hiai, F., Petz, D.: Introduction to Matrix Analysis and Applications. Springer, New York (2014)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. I. Academic Press, Cambridge (1983)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, vol. II. Academic Press, Cambridge (1986)
Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)
Molnár, L.: Maps preserving the geometric mean of positive operators. Proc. Am. Math. Soc. 137, 1763–1770 (2009)
Molnár, L.: Maps preserving the harmonic mean or the parallel sum of positive operators. Linear Algebra Appl. 430, 3058–3065 (2009)
Molnár, L.: Maps preserving general means of positive operators. Electron. J. Linear Algebra 22, 864–874 (2011)
Molnár, L.: A characterization of central elements in \(C^*\)-algebras. Bull. Aust. Math. Soc. 95, 138–143 (2017)
Molnár, L.: Quantum Rényi relative entropies: their symmetries and their essential difference. J. Funct. Anal. 277, 3098–3130 (2019)
Molnár, L.: Maps on positive definite cones of \(C^*\)-algebras preserving the Wasserstein mean. Proc. Am. Math. Soc. 150, 1209–1221 (2022)
Nagy, G.: Characterizations of centrality in \(C^*\)-algebras via local monotonicity and local additivity of functions. Integral Equ. Oper. Theory 91, Paper No. 28 (2019)
Sturm, J.: Similarity and other spectral relations for symmetric cones. Linear Algebra Appl. 312, 135–154 (2000)
Virosztek, D.: Connections between centrality and local monotonicity of certain functions on \(C^*\)-algebras. J. Math. Anal. Appl. 453, 221–226 (2017)
Virosztek, D.: Characterizations of centrality by local convexity of certain functions on \(C^*\)-algebras. In: The Diversity and Beauty of Applied Operator Theory, Operator Theory: Advances and Applications, vol 268. Birkhäuser/Springer, Cham, pp 487–494 (2018)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author states that there is no conflict of interest. No datasets were generated or analysed during the current study.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author acknowledges support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund and financed under the TKP2021-NVA funding scheme, Project no. TKP2021-NVA-09. The research has also been supported by the National Research, Development and Innovation Office of Hungary, NKFIH, Grant No. K134944.
Rights and permissions
About this article
Cite this article
Molnár, L. On Certain Order Properties of Non Kubo–Ando Means in Operator Algebras. Integr. Equ. Oper. Theory 94, 25 (2022). https://doi.org/10.1007/s00020-022-02702-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00020-022-02702-7