Quasi-linear functionals on locally compact spaces
Confluentes Mathematici, Tome 13 (2021) no. 1, pp. 3-34.

This paper combines new and known results in a single convenient source for anyone interested in learning about quasi-linear functionals on locally compact spaces. We define singly generated subalgebras in different settings and study signed and positive quasi-linear functionals. Quasi-linear functionals are, in general, nonlinear, but linear on singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on C c (X), for bounded quasi-linear functionals on C 0 (X) on a locally compact space, and for quasi-linear functionals on C(X) on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also “isometric” when topological measures are finite. We present many properties of quasi-linear functionals and give an explicit example of a quasi-linear functional on 2 . Results of the paper will be helpful for further study and application of quasi-linear functionals in different areas of mathematics, including symplectic geometry.

Reçu le :
Accepté le :
Accepté après révision le :
Publié le :
DOI : https://doi.org/10.5802/cml.69
Classification : 46E27,  46G99,  28A25,  28C15
Mots clés : quasi-linear functional, signed quasi-linear functional, singly generated subalgebra, topological measure, symplectic quasi-state
@article{CML_2021__13_1_3_0,
     author = {Svetlana V. Butler},
     title = {Quasi-linear functionals on locally compact spaces},
     journal = {Confluentes Mathematici},
     pages = {3--34},
     publisher = {Institut Camille Jordan},
     volume = {13},
     number = {1},
     year = {2021},
     doi = {10.5802/cml.69},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.69/}
}
Svetlana V. Butler. Quasi-linear functionals on locally compact spaces. Confluentes Mathematici, Tome 13 (2021) no. 1, pp. 3-34. doi : 10.5802/cml.69. https://cml.centre-mersenne.org/articles/10.5802/cml.69/

[1] J. Aarnes. Physical States on C*-algebra, Acta Math., 122:161–172, 1969. | Article | MR 247482 | Zbl 0183.14203

[2] —. Quasi-states on C * -algebras, Trans. Amer. Math. Soc., 149:601–625, 1970. | Article | MR 282602 | Zbl 0212.15403

[3] —. Quasi-states and quasi-measures, Adv. Math., 86(1):41–67, 1991. | Article | MR 1097027 | Zbl 0744.46052

[4] —. Pure quasi-states and extremal quasi-measures, Math. Ann., 295:575–588, 1993. | Article | MR 1214949 | Zbl 0791.46028

[5] J. Aarnes and A. Rustad. Probability and quasi-measures–a new interpretation, Math. Scand., 85(2):278–284, 1999. | Article | MR 1724240 | Zbl 0967.28014

[6] C. Akemann and S. Newberger. Physical states on C*-algebra, Proc. Amer. Math. Soc., 40(2):500, 1973. | Article | MR 318860 | Zbl 0272.46037

[7] V. Bogachev. Measure Theory, vol. 1. Regular and Chaotic Dynamics, Izhevsk 2003, English transl., Springer-Verlag, Berlin, 2007. | Article

[8] M. Borman. Symplectic reductions of quasi-morphisms and quasi-states, J. Symplectic Geom., 10(2):225–246, 2012. | Article | MR 2926996 | Zbl 1266.53069

[9] L. Buhovsky, M. Entov, and L. Polterovich. Poisson brackets and symplectic invariants, Selecta Math. (N. S.), 18:89–157, 2012. | Article | MR 2891862 | Zbl 1242.53099

[10] S. Butler. Density in the space of topological measures, Fund. Math., 174:239–251, 2002. | Article | MR 1925001 | Zbl 1027.28017

[11] —. q-Functions and extreme topological measures, J. Math. Anal. Appl., 307:465–479, 2005. | Article | MR 2142438 | Zbl 1074.28007

[12] —. Extreme topological measures, Fund. Math., 192:141–153, 2006. | Article | MR 2283756 | Zbl 1116.28011

[13] —. Ways of obtaining topological measures on locally compact spaces, Bull. Irkutsk State Univ. Series “Mathematics”, 25:33–45, 2018. | Article | MR 3861945 | Zbl 1409.28005

[14] —. Signed topological measures on locally compact spaces, Anal. Math., 45:757–773, 2019. | Article | MR 4042929 | Zbl 1449.28014

[15] —. Non-linear functionals, deficient topological measures, and representation theorems on locally compact spaces, Banach J. Math. Anal., 14(3):674–706, 2020. | Article | MR 4123306 | Zbl 1460.46008

[16] —. Integration with respect to deficient topological measures on locally compact spaces, Math. Slovaca, 70(5):1113–1134, 2020. | Article | MR 4156812

[17] —. Deficient topological measures on locally compact spaces, Math. Nachr., 294(6): 1115–1133, 2021. | Article | MR 4288487

[18] —. Weak convergence of topological measures. J. Theor. Prob., 24/04/2021. | Article

[19] —. Semisolid sets and topological measures, preprint. arXiv: 2103.09401

[20] —. Repeated quasi-integration on locally compact spaces, Positivity, to appear. arXiv:1902.06901

[21] D. Denneberg. Non-additive measure and integral. Kluwer, 1994. | Article | Zbl 0826.28002

[22] A. Dickstein and F. Zapolsky. Approximation of quasi-states on manifolds, J. Appl. and Comput. Topol., 3:221–248, 2019. | Article | MR 3996955 | Zbl 07103893

[23] J. Dugundji. Topology. Allyn and Bacon, 1966.

[24] M. Entov. Quasi-morphisms and quasi-states in symplectic topology, Proceedings of the International Congress of Mathematicians, Seoul, 1147–1171, 2014. | Zbl 1373.53116

[25] M. Entov and L. Polterovich. Calabi Quasimorphism and Quantum Homology, Int. Math. Res. Not., 30:1635–1676, 2003. | Article | Zbl 1047.53055

[26] —. Quasi-states and symplectic intersections, Comm. Math. Helv., 81:75–99, 2006. | Article | MR 2208798 | Zbl 1096.53052

[27] —. Symplectic Quasi-states and Semi-simplicity of Quantum Homology, in Toric Topology (eds. M. Harada, Y. Karshon, M. Masuda and T. Panov), Contemporary Mathematics, AMS, 460: 47–70, 2008. | Article | Zbl 1146.53066

[28] —, Lie quasi-states, J. Lie Theory, 19:613–637, 2009. | Zbl 1182.53075

[29] —. C 0 -rigidity of Poisson brackets, Contemp. Math., 512: 25–32, 2010. | Article | Zbl 1197.53115

[30] M. Entov, L. Polterovich, and D. Rosen. Poisson Brackets, Quasi-states and Simplectic integrators, Discrete Contin. Dyn. Syst., 28(4):1455–1468, 2010. | Article | MR 2679719 | Zbl 1200.53068

[31] M. Entov, L. Polterovich, and F. Zapolsky. Quasi-morphisms and the Poisson Bracket, Pure and Appl. Math. Q, 3(4) (Special issue : In honor of Gregory Margulis, part 1 of 2):1037–1055, 2007. | Article | MR 2402596 | Zbl 1143.53070

[32] —. An “Anti-Gleason” Phenomen and Simultaeous Measurements in Classical Mechanics, Found. Phys., 37:1306–1316, 2007. | Article | Zbl 1129.81007

[33] A. Gleason. Measures on the closed subspaces of a Hilbert space, J. Math. Mech., 6: 885–893, 1957. | Article | MR 96113 | Zbl 0078.28803

[34] D. Grubb. Signed Quasi-measures, Trans. Amer. Math. Soc., 349(3):1081–1089, 1997. | Article | MR 1407700 | Zbl 0876.28017

[35] —. Lectures on quasi-measures and quasi-linear functionals on compact spaces, unpublished, 1998.

[36] —. Signed Quasi-measures and Dimension Theory, Proc. Amer. Math. Soc., 128(4):1105-1108, 2000. | Article | MR 1636950 | Zbl 0942.28011

[37] E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer-Verlag, 1965. | Article | Zbl 0137.03202

[38] R. Kadison. Transformation of states in operator theory and dynamics, Topology, 3:177–198, 1965. | Article | MR 169073 | Zbl 0129.08705

[39] S. Lanzat. Quasi-morphisms and Symplectic Quasi-states for convex Symplectic Manifolds, Int. Math Res. Not., 2013(23):5321–5365, 2013. | Article | MR 3142258 | Zbl 1329.53119

[40] G. Mackey. Quantum mechanics and Hilbert space, Amer. Math. Monthly, 64:45–57, 1957. | Article | MR 96112 | Zbl 0137.23805

[41] —. The Mathematical Foundations of Quantum Mechanics. Benjamin, 1963.

[42] A. Monzner and F. Zapolsky. A comparison of symplectic homogenization and Calabi quasi-states, J. Topol. Anal, 3(3):243–263, 2011. | Article | MR 2831264 | Zbl 1235.28002

[43] L. Polterovich and D. Rosen. Function theory on symplectic manifolds. AMS, 2014. | Article | Zbl 1310.53002

[44] A. Rustad. Unbounded quasi-integrals, Proc. Amer. Math. Soc., 129(1):165–172, 2000. | Article | MR 1694879 | Zbl 0957.28004

[45] D. Shakmatov. Linearity of quasi-states on Commutative C * algebras of stable rank 1, unpublished.

[46] J. von Neumann. Mathematical Foundations of Quantum Mechanics. Princeton University Press, 1955. Translation of Mathematische Grundlagen der Quantenmechanik Springer, 1932. | Zbl 58.0929.06

[47] R. Wheeler. Quasi-measures and dimension theory, Topology Appl., 66:75–92, 1995. | Article | MR 1357876 | Zbl 0842.28005

[48] F. Zapolsky. Isotopy-invariant topological measures on closed orientable surfaces of higher genus, Math. Z., 270:133–143, 2012. | Article | MR 2875825 | Zbl 1272.28015

Cité par document(s). Sources :