# CONFLUENTES MATHEMATICI

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 ${C}_{c}\left(X\right)$, for bounded quasi-linear functionals on ${C}_{0}\left(X\right)$ on a locally compact space, and for quasi-linear functionals on $C\left(X\right)$ 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 ${ℝ}^{2}$. Results of the paper will be helpful for further study and application of quasi-linear functionals in different areas of mathematics, including symplectic geometry.

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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/

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