Received:

Accepted:

Revised after acceptance:

Published online:

DOI:
10.5802/cml.69

Accepted:

Revised after acceptance:

Published online:

Classification:
46E27, 46G99, 28A25, 28C15

Keywords: quasi-linear functional, signed quasi-linear functional, singly generated subalgebra, topological measure, symplectic quasi-state

Keywords: quasi-linear functional, signed quasi-linear functional, singly generated subalgebra, topological measure, symplectic quasi-state

Author's affiliations:

Svetlana V. Butler ^{1}

License: CC-BY-NC-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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/} }

TY - JOUR AU - Svetlana V. Butler TI - Quasi-linear functionals on locally compact spaces JO - Confluentes Mathematici PY - 2021 SP - 3 EP - 34 VL - 13 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.69/ DO - 10.5802/cml.69 LA - en ID - CML_2021__13_1_3_0 ER -

Svetlana V. Butler. Quasi-linear functionals on locally compact spaces. Confluentes Mathematici, Volume 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. | DOI | MR | Zbl

[2] —. Quasi-states on ${C}^{*}$-algebras, Trans. Amer. Math. Soc., 149:601–625, 1970. | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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. | DOI | Zbl

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

[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

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

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

[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. | DOI | Zbl

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

[29] —. ${C}^{0}$-rigidity of Poisson brackets, Contemp. Math., 512: 25–32, 2010. | DOI | Zbl

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

[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. | DOI | MR | Zbl

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

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

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

[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. | DOI | MR | Zbl

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

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

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

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

[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. | DOI | MR | Zbl

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

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

[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

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

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

*Cited by Sources: *