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A note on the Mackey dual of $ C(K)$

Author: Dave Wilkins
Journal: Proc. Amer. Math. Soc. 118 (1993), 89-92
MSC: Primary 46E25; Secondary 46A20, 46J10, 54C35
MathSciNet review: 1128731
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Abstract: Let $ K$ be a compact metric space, and let $ \tau $ denote the Mackey topology on $ M(K)$ with respect to the $ \langle C(K),M(K)\rangle $ duality. That is, $ \tau $ is the topology of uniform convergence on the weakly compact subsets of $ C(K)$. Just as for the weak$ ^{\ast}$ topology, the dual space of $ (M(K),\tau )$ is $ C(K)$. However, $ \tau $ is very different from weak$ ^{\ast}$. Indeed, it is obvious that if $ \{ {x_n}\} $ is a sequence converging to $ x$ in $ K$, then $ \delta ({x_n})$ converges to $ \delta (x)$ in the weak$ ^{\ast}$ topology, yet Kirk has shown (Pacific J. Math. 45 (1973), 543-554) that $ \{ \delta (x)\vert x \in K\} $ is closed and discrete in the Mackey topology. We obtain a further result along these lines: For each $ A \subset K$ set $ \Delta A = \{ \delta (x) - \delta (y)\vert x \ne y,x,y \in A\} $. Let $ \mathcal{D}$ denote the totality of all subsets $ A$ of $ K$ with the property that $ 0 \in {\overline {\Delta A} ^\tau }$. Then a closed set is in $ \mathcal{D}$ iff it is uncountable. Alternatively stated, a closed subset $ A$ of $ K$ is countable if and only if there is a weakly compact subset $ L$ of $ C(K)$ such that for every pair $ x,\;y \in A,\;x \ne y$, there is an $ h \in L$ with $ \vert h(x) - h(y)\vert \geqslant 1$.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1993 American Mathematical Society

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