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On traces of Bochner representable operators on the space of bounded measurable functions
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 11 January 2024, pp. 224-235
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- Article
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Let Σ be a σ-algebra of subsets of a set Ω and
$B(\Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let 
$\tau(B(\Sigma),ca(\Sigma))$ denote the natural Mackey topology on 
$B(\Sigma)$. It is shown that a linear operator T from 
$B(\Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space 
$(B(\Sigma),\tau(B(\Sigma),ca(\Sigma)))$ and the Banach space E. We derive a formula for the trace of a Bochner representable operator 
$T:B({\cal B} o)\rightarrow B({\cal B} o)$ generated by a function 
$f\in L^1({\cal B} o, C(\Omega))$, where Ω is a compact Hausdorff space.
