Regarding the double dual of a Banach algebra

Question

Let $A$ be a commutative unital Banach algebra. Then is its Double dual $A$**, with the Aren's product also commutative? If not, then can you give an example?

Answer 1

No, it is often noncommutative.

The standard example is the convolution Banach algebra $\ell_1(\mathbb{Z})$. Its bidual, $\ell_\infty(\mathbb{Z})^\ast$ is the algebra $M(\beta \mathbb{Z})$ of measures on the Stone-Čech compactification of the integers. If you consider elements of $M(\beta \mathbb{Z})$, $p\in \overline{ \mathbb{N}}\setminus \mathbb{N}$ and $q\in \overline{ \mathbb{-N}}\setminus \mathbb{-N}$, then you can write $$p=\lim_\alpha \delta_{n_\alpha}\qquad q=\lim_\beta \delta_{-m_\beta},$$ where $n_\alpha$ and $m_\beta$ are all positive integers and $\delta_n$ stands for the point-mass measure at $n$. If you now compute $$ p{\scriptsize\square} q=\lim_\alpha\lim_\beta \delta_{n_\alpha-m_\beta}\in \overline{-\mathbb{N}},$$ while $$ q{\scriptsize\square} p=\lim_\beta\lim_\alpha \delta_{n_\alpha-m_\beta}\in \overline{\mathbb{N}}.$$ Recall finally that disjoint subsets of a (discrete) set have disjoint closures in the Stone-Čech compactification and you will have that $p{\scriptsize\square} q\neq q{\scriptsize\square} p$.

This can be extended much further as the convolution algebra $L_1(G)$ is not commutative for any infinite locally compact Abelian group and can be linked to Arens-regularity of Banach algebras (Banach algebras for which both Arens products coincide).

It can be interesting to note that $\ell_1(\mathbb{Z})$ with pointwise multiplication is commutative. All this can be found in the first pages of T. W. Palmer's book: Banach Algebras and the General Theory of *-Algebras: Volume 1.