# Counter-examples to Kaplansky Density

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.

• 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.