Abstract
We examine under which assumptions on a positive normal functional φ on a von Neumann algebra, \({\cal M}\) and a Borel measurable function f: R + → R with f(0) = 0 the subadditivity inequality φ (f(A+B)) ≤ φ(f(A))+φ (f (B)) holds true for all positive operators A, B in \({\cal M}\). A corresponding characterization of tracial functionals among positive normal functionals on a von Neumann algebra is presented.
Similar content being viewed by others
References
L.G. Brown H. Kosaki (1990) ArticleTitleJensen’s inequality in semi-finite von Neumann algebras J. Operator Theory 23 3–19
J. Dixmier (1969) Les algèbres d’ opérateurs dans l’espace Hilbertien (algèbres de von Neumann) 2e édition Gauthier-Villars Paris
T. Fack H. Kosaki (1986) ArticleTitleGeneralized s-numbers of τ-measurable operators Pacific J. Math 123 269–300
L.T. Gardner (1979) ArticleTitleAn inequality characterizes the trace Canad.J. Math 31 1322–1328
Kadison R.V., Ringrose J.R. Fundamentals of the Theory of Operator Algebras, Vols I, II, Academic Press, 1983, 1986.
G.K. Pedersen E. Størmer (1982) ArticleTitleTraces on Jordan algebras Canad. J. Math 34 370–373
D. Petz J. Zemánek (1988) ArticleTitleCharacterizations of the Trace Linear Algebra Appl 111 43–52 Occurrence Handle10.1016/0024-3795(88)90050-X
Stolyarov, A.I., Tikhonov, O.E. and Sherstnev, A.N.: Characterization of normal traces on von Neumann algebras by inequalities for the modulus, (Russian) Mat. Zametki 72 (2002), 448–454; English translation in Math. Notes 72 (2002), 411–416.
M. Takesaki (1979) Theory of operator algebras I Springer-Verlag Berlin-Heidelberg-New York
Tikhonov O.E. (1987) Convex functions and inequalities for a trace, in Constructive theory of functions and functional analysis, No. 6 (Russian), Kazan State University, Kazan, 1987, pp. 77–82.
O.E. Tikhonov (1998) ArticleTitleOn matrix-subadditive functions and a relevant trace inequaity Linear Multilinear Algebra 44 25–28
Author information
Authors and Affiliations
Corresponding author
Additional information
O.E. Tikhonov - Supported by the Russian Foundation for Basic Research (grant no. 01-01-00129) and the scientific program Universities of Russia – Basic Research (grant no. UR.04.01.061).
Rights and permissions
About this article
Cite this article
Tikhonov, O.E. Subadditivity Inequalities in von Neumann Algebras and Characterization of Tracial Functionals. Positivity 9, 259–264 (2005). https://doi.org/10.1007/s11117-005-2711-1
Issue Date:
DOI: https://doi.org/10.1007/s11117-005-2711-1
Keywords
- algebra of matrices
- functional calculus
- positive normal functional
- subadditivity inequality
- tracial functional
- von Neumann algebra