Re: variants of Pontryagin duality
- From: Kevin Buzzard <buzzard@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: 10 Nov 2008 18:08:31 -0500
John Baez <baez@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Suppose A is an abelian locally compact Hausdorff topological
group, and let
A' = hom(A,C*)
be the set of continuous homomorphisms from A to the invertible
complex numbers. A' is an abelian group, and it becomes a
topological group with the compact-open topology.
Is A' again locally compact Hausdorff?
If A is an infinite direct sum of copies of the integers, with the
discrete topology, then I think that the Pontrjagin dual of A is
an infinite direct product of copies of U(1), with the product topology,
but I think that hom(A,C^*) with the compact-open topology turns out to be an
infinite product of copies of C^*, with the product topology, and this
won't be locally compact---an infinite product of locally compact
spaces is, I think, only locally compact when all but a finite number
are compact.
Another question: if A is second-countable, is A'?
(I don't even know this for U(1) replacing C*
Me neither.
Kevin
.
- References:
- variants of Pontryagin duality
- From: John Baez
- variants of Pontryagin duality
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