Posted on 13th February 2021
I've added a short document to my Mathematical writeups collection. Thanks to Yemon Choi for a conversation which motivated me to write this up, and to Philip Spain who alerted me to the work of Harry Dowson.
It would be interesting to have a natural Banach Algebra example. By this, I mean a naturally occurring example of a dual Banach Algebra \( M \) and a weak\( ^\ast \)-dense subalgebra \( A\subseteq M \) such that there is a unit vector in \( M \) which is not the weak\( ^\ast \)-limit of any bounded net in \( A \).
What do I mean by "natural" here? Some possible examples:
Let \( E \) be a reflexive Banach space with the approximation property. Then it's known that \( \mathcal B(E) \) is the bidual of \( \mathcal K(E) \). Thus \( \mathcal K(E) \) is weak\( ^\ast \)-dense in \( \mathcal B(E) \), but in this case, we do have an (isometric) version of Kaplansky Density, because biduals satisfy Goldstine's Theorem. I still think this example is a little interesting: if \( E \) had the metric approximation property, given \( T\in\mathcal B(E) \) it is easy to write down a net in \( \mathcal K(E) \) which converges to \( T \) and is bounded by \( \|T\| \). If \( E \) merely has the approximation property then this seems less clear to me, though it is ensured by the general theory. Update: This was a little naive: if \( E \) is reflexive then it goes back to Grothendieck that \( E \) having the AP already implies that \( E \) has the MAP. So really there is no mystery here.
Let \( G \) be a locally compact group, and consider \( PM_p(G) \) which by definition is the weak\( ^\ast \) closure of \( PF_P(G) \) inside \( \mathcal B(L^p(G)) \). Excepting when \( p=2 \) I do not know if we have Kaplansky Density here, though if \( G \) has a suitable approximation-like property (amenability, or a weakening) then I believe one can show this: I don't know if this is explicitly in the literature.