# Biduals of Banach algebras

For a Banach algebra $A$ the bidual $A^{**}$ may be given two natural products called the Arens products. By local reflexivity, there is an ultrafilter $U$ so that $A^{**}$ embeds into the ultrapower $A^U$ isometrically via some map $h$. This map has a one-sided inverse $\sigma\colon A^U\to A^{**}$ given by $\sigma([x_i])=w^*-\lim x_i$.

Is this map an algebra homomorphism for the Arens products in $A^{**}$?

• Although it's not needed for your actual question, you might want to look at Daws's paper arxiv.org/abs/0708.4029 if you haven't already done so – Yemon Choi Apr 12 '17 at 17:27

In general, no: for certain $A$ one can get non-zero $c\in A$ and sequences $(a_n)$ and $(b_n)$ in $A$ that converge weakly to zero, such that $a_n b_n=c$ for all $n$; these will show that $\ker\sigma$ is not even closed under multiplication, let alone an ideal.
We can arrange for $A$ to be Arens regular, so that $A^{**}$ is even a dual Banach algebra.
(As a loose analogy: one should suspect that the question has a negative answer just because $\lim_i \lim_j a_{ij}$ can be very different from $\lim_n a_{nn}$ for a doubly-indexed sequence of scalars.)
Here is one example, which I probably learned from conversations with Matt Daws; there should be many others with similar behaviour. I use $\ell_2$ to denote $\ell_2({\bf N})$. Take $A= K(\ell_2)$, let $a_n$ be the operator that sends $\delta_n$ to $\delta_1$ and all other basis vectors to $0$; and let $b_n$ be the operator that sends $\delta_1$ to $\delta_n$ and all other basis vectors to $0$. Clearly $a_nb_n$ is the projection onto the span of $\delta_1$, for all $n$.
It only remains to justify the claim that $a_n \stackrel{w}{\longrightarrow} 0$ and $b_n \stackrel{w}{\longrightarrow} 0$. By trace duality every functional on $A$ is of the form $x\mapsto {\rm Tr}(yx)$ for some trace class operator $y$. A direct calculation shows that ${\rm Tr}(ya_n) = \langle y\delta_1, \delta_n\rangle\to 0$ as $n\to\infty$, and likewise ${\rm Tr}(yb_n)=\langle y\delta_n,\delta_1 \rangle = \langle \delta_n, y^*\delta_1\rangle \to 0$.