Abstract
It is proved that the inequality
characterizes tracial functionals among all positive normal functionals \(\varphi\) on a von Neumann algebra \(\mathcal{A}\). This strengthens the L. T. Gardner’s characterization (1979). As a consequence, a criterion for commutativity of von Neumann algebras is obtained. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra via this inequality. Every faithful normal semifinite trace \(\varphi\) on a von Neumann algebra \(\mathcal{A}\) satisfies this relation. Let \(|||\cdot|||\) be a unitarily invariant norm on a unital \(C^{*}\)-algebra \(\mathcal{A}\). Then \(|||A|||\leq|||A+\textrm{i}B|||\) for all \(A\in\mathcal{A}^{+}\) and \(B\in\mathcal{A}^{\text{sa}}\).
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REFERENCES
S. A. Abed, ‘‘An inequality for projections and convex functions,’’ Lobachevskii J. Math. 39 (9), 1287–1292 (2018).
A. M. Bikchentaev, ‘‘Commutation of projections and characterization of traces on von Neumann algebras. III,’’ Int. J. Theor. Phys. 54, 4482–4493 (2015).
A. M. Bikchentaev, ‘‘Inequality for a trace on a unital \(C^{*}\)-algebra,’’ Math. Notes 99, 487–491 (2016).
A. M. Bikchentaev, ‘‘Differences of idempotents in \(C^{*}\)-algebras,’’ Sib. Math. J. 58, 183–189 (2017).
A. M. Bikchentaev, ‘‘Differences of idempotents in \(C^{*}\)-algebras and the quantum Hall effect,’’ Theor. Math. Phys. 195, 557–562 (2018).
A. M. Bikchentaev and A. N. Sherstnev, ‘‘Studies on noncommutative measure theory in Kazan University (1968–2018),’’ Int. J. Theor. Phys. 60, 585–596 (2021).
A. M. Bikchentaev, ‘‘Trace and differences of idempotents in \(C^{*}\)-algebras,’’ Math. Notes 105, 641–648 (2019).
L. T. Gardner, ‘‘An inequality characterizes the trace,’’ Canad. J. Math. 31, 1322–1328 (1979).
D. T. Hoa and O. E. Tikhonov, ‘‘Weighted monotonicity inequalities for traces on operator algebras,’’ Math. Notes 88, 177–182 (2010).
D. T. Hoa, H. Osaka, and H. M. Toan, ‘‘On generalized Powers-Størmer’s inequality,’’ Linear Algebra Appl. 438, 242–249 (2013).
D. Petz and J. Zemánek, ‘‘Characterizations of the trace,’’ Linear Algebra Appl. 111, 43–52 (1988).
M. Takesaki, Theory of Operator Algebras (Springer, Berlin, 2002), Vol. 1.
S. M. Manjegani, ‘‘Inequalities in operator algebras,’’ PhD Thesis (Regina Univ., Regina, Canada, 2004).
M. Ruskai, ‘‘Inequalities for traces on von Neumann algebras,’’ Commun. Math. Phys. 26, 280–289 (1972).
H. Araki, ‘‘On an inequality of Lieb and Thirring,’’ Lett. Math. Phys. 19, 167–170 (1990).
B. Simon, Trace Ideals and their Applications, 2nd ed., Vol. 120 of Mathematical Surveys and Monographs (Am. Math. Soc., Providence, RI, 2005).
A. M. Bikchentaev and O. E. Tikhonov, ‘‘Characterization of the trace by Young’s inequality,’’ J. Inequal. Pure Appl. Math. 6 (2), 49 (2005).
A. M. Bikchentaev and O. E. Tikhonov, ‘‘Characterization of the trace by monotonicity inequalities,’’ Linear Algebra Appl. 422, 274–278 (2007).
A. M. Bikchentaev, ‘‘Commutativity of projections and characterization of traces on von Neumann algebras,’’ Sib. Math. J. 51, 971–977 (2010).
A. M. Bikchentaev, ‘‘Commutation of projections and trace characterization on von Neumann algebras. II,’’ Math. Notes 89, 461–471 (2011).
A. M. Bikchentaev, ‘‘The Peierls–Bogoliubov inequality in \(C^{*}\)-algebras and characterization of tracial functionals,’’ Lobachevskii J. Math. 32 (3), 175–179 (2011).
A. M. Bikchentaev, ‘‘Commutativity of operators and characterization of traces on \(C^{*}\)-algebras,’’ Dokl. Math. 87, 79–82 (2013).
K. Cho and T. Sano, ‘‘Young’s inequality and trace,’’ Linear Algebra Appl. 431, 1218–1222 (2009).
R. Bhatia, Matrix Analysis, Vol. 169 of Graduate Texts in Mathematics (Springer, New York, 1997).
B. Blackadar, ‘‘Operator algebras, theory of \(C^{*}\)-algebras and von Neumann algebras,’’ in Encyclopaedia of Mathematical Sciences, Vol. 122: Operator Algebras and Non-commutative Geometry, III (Springer, Berlin, 2006).
M. Takesaki, Theory of Operator Algebras II, Vol. 125 of Encyclopaedia Math. Sci. (Springer, New York, 2003).
H. Upmeier, ‘‘Automorphism groups of Jordan \(C^{*}\)-algebras,’’ Math. Z. 176, 21–34 (1981).
Sh. A. Ayupov, Classification and Representation of Ordered Jordan Algebras (Fan, Tashkent, 1986) [in Russian].
O. E. Tikhonov, ‘‘Subadditivity inequalities in von Neumann algebras and characterization of tracial functionals,’’ Positivity 9, 259–264 (2005).
C. A. Akemann, J. Anderson, and G. K. Pedersen, ‘‘Triangle inequalities in operator algebras,’’ Linear Multilinear Algebra 11, 167–178 (1982).
A. I. Stolyarov, O. E. Tikhonov, and A. N. Sherstnev, ‘‘Characterization of normal traces on von Neumann algebras by inequalities for the modulus,’’ Math. Notes 72, 411–416 (2002).
A. M. Bikchentaev, ‘‘On a property of \(L_{p}\)-spaces on semifinite von Neumann algebras,’’ Math. Notes 64, 159–163 (1998).
A. M. Bikchentaev, ‘‘Metrics on projections of the von Neumann algebra associated with tracial functionals,’’ Sib. Math. J. 60, 952–956 (2019).
A. M. Bikchentaev and S. A. Abed, ‘‘Projections and traces on von Neumann algebras,’’ Lobachevskii J. Math. 40 (9), 1260–1267 (2019).
A. M. Bikchentaev, ‘‘Inequalities for determinants and characterization of the trace,’’ Sib. Math. J. 61, 248–254 (2020).
D. T. Hoa and O. E. Tikhonov, ‘‘Weighted trace inequalities of monotonicity,’’ Lobachevskii J. Math. 26, 63–67 (2007).
An. An. Novikov and O. E. Tikhonov, ‘‘Characterization of central elements of operator algebras by inequalities,’’ Lobachevskii J. Math. 36 (2), 208–210 (2015).
T. Fack and H. Kosaki, ‘‘Generalized \(s\)-numbers of \(\tau\)-measurable operators,’’ Pacif. J. Math. 123, 269–300 (1986).
S. Lord, F. Sukochev, and D. Zanin, Singular Traces. Theory and Applications, Vol. 46 of De Gruyter Studies in Mathematics (De Gruyter, Berlin, 2013).
F. A. Sukochev and V. I. Chilin ‘‘The triangle inequality for operators that are measurable with respect to Hardy–Littlewood order,’’ Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, No. 4, 44–50 (1988).
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The work was carried out as part of the development program of the Scientific and Educational Mathematical Center of the Volga Federal District, agreement no. 075-02-2020-1478.
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Alhasan, H., Fawwaz, K. Characterization of Tracial Functionals on Von Neumann Algebras. Lobachevskii J Math 42, 2273–2279 (2021). https://doi.org/10.1134/S1995080221100024
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DOI: https://doi.org/10.1134/S1995080221100024