Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-06T09:53:44.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Krivine’s Function Calculus and Bochner Integration

Part of: Topological linear spaces and related structures Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  15 October 2018

V. G. Troitsky  and
M. S. Türer
V. G. Troitsky
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada Email: [email protected]
M. S. Türer
Affiliation:
Department of Mathematics and Computer Science, İstanbul Kültür University, Bakırköy 34156, İstanbul, Turkey Email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that Krivine’s Function Calculus is compatible with integration. Let $(\unicode[STIX]{x1D6FA},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ be a finite measure space, $X$ a Banach lattice, $\mathbf{x}\in X^{n}$, and $f:\mathbb{R}^{n}\times \unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ a function such that $f(\cdot ,\unicode[STIX]{x1D714})$ is continuous and positively homogeneous for every $\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}$, and $f(\mathbf{s},\cdot )$ is integrable for every $\mathbf{s}\in \mathbb{R}^{n}$. Put $F(\mathbf{s})=\int f(\mathbf{s},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$ and define $F(\mathbf{x})$ and $f(\mathbf{x},\unicode[STIX]{x1D714})$ via Krivine’s Function Calculus. We prove that under certain natural assumptions $F(\mathbf{x})=\int f(\mathbf{x},\unicode[STIX]{x1D714})\,d\unicode[STIX]{x1D707}(\unicode[STIX]{x1D714})$, where the right hand side is a Bochner integral.

Keywords

Banach latticefunction calculusBochner integral

MSC classification

Primary: 46B42: Banach lattices
Secondary: 46A40: Ordered topological linear spaces, vector lattices
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 663 - 669
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The first author was supported by an NSERC grant.

References

Abramovich, Y. A. and Aliprantis, C. D., An invitation to operator theory . Graduate Studies in Mathematics, 50, American Mathematical Society, Providence, RI, 2002.Google Scholar
Aliprantis, C. D. and Burkinshaw, O., Positive operators . Springer, Dordrecht, 2006. Reprint of the 1985 original.Google Scholar
Davis, W. J., Garling, D. J. H., and Tomczak-Jaegermann, N., The complex convexity of quasinormed linear spaces . J. Funct. Anal. 55(1984), no. 1, 110–150.Google Scholar
Diestel, J. and Uhl, J. J. Jr., Vector measures . Mathematical Surveys, 15, American Mathematical Society, Providence, RI, 1977.Google Scholar
Kalton, N. J., Hermitian operators on complex Banach lattices and a problem of Garth Dales . J. Lond. Math. Soc. (2) 86(2012), no. 3, 641–656.Google Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. II . Function spaces, Springer-Verlag, Berlin, 1979.Google Scholar