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 ${\mathbb{R}}^{2}$. 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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Keywords: quasi-linear functional, signed quasi-linear functional, singly generated subalgebra, topological measure, symplectic quasi-state

Svetlana V. Butler ^{1}

@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/

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