Quasi-linear functionals on locally compact spaces

Svetlana V. Butler

doi:10.5802/cml.69

Svetlana V. Butler ¹

¹ Department of Mathematics, University of California Santa Barbara, 552 University Rd., Isla Vista, CA 93117, USA

Confluentes Mathematici, Volume 13 (2021) no. 1, pp. 3-34.

Abstract

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 $ℝ^{2}$ . Results of the paper will be helpful for further study and application of quasi-linear functionals in different areas of mathematics, including symplectic geometry.

Received: 2019-09-02
Accepted: 2021-01-03
Revised after acceptance: 2021-02-28
Published online: 2021-10-20

DOI: 10.5802/cml.69

Classification: 46E27, 46G99, 28A25, 28C15
Keywords: quasi-linear functional, signed quasi-linear functional, singly generated subalgebra, topological measure, symplectic quasi-state

Author's affiliations:

Svetlana V. Butler ¹

¹ Department of Mathematics, University of California Santa Barbara, 552 University Rd., Isla Vista, CA 93117, USA

License:

CC-BY-NC-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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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