Arens regularity of $\mathrm{BV}(\mathbb{R})$

Question

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of functions $\mathbb{R} \to \mathbb{C}$ which are continuous and of bounded variation.

Is $\BV(\mathbb{R})$ Arens regular?

Here's how far I've gotten until now:

• I have not seen the question mentioned in any of the literature so far. In particular, it doesn't seem to be in Palmer's book.

• Since $C^*$-algebras are Arens regular, it would be enough to realize $\BV(\mathbb{R})$ as a closed subalgebra of a $C^*$-algebra, or equivalently to represent it on a Hilbert space.

• Characterizing the dual space $\BV(\mathbb{R}^n)^*$ is an open problem, but I'm not sure if this includes the case $n = 1$. If so, I'm hoping that my question might still have a known answer that doesn't depend on a characterization of the dual first.

Answer 1

Writing this in a hurry on my way to a ritual English social cohesion exercise, so apologies for the lack of formatting and the sketchy nature of claims and references.

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I think it might be "known" to experts that this Banach algebra is not Arens regular, without it being written down anywhere. My reasoning is that there should be a surjective continuous homomorphism from $\mathrm{BV}(\Bbb R)$ to $\mathrm{bv}(\Bbb N)$, given by restriction of functions from $\Bbb R$ to $\Bbb N$. Now Arens regularity passes to quotients, so if $\mathrm{BV}(\Bbb R)$ were Arens regular then $\mathrm{bv}(\Bbb N)$ would be Arens regular. But $\mathrm{bv}(\Bbb N)$ is isomorphic to the unitization of the convolution algebra $A=\ell^1({\mathbb N}_\min)$ where ${\mathbb N}_\min$ is the semigroup obtained by equipping ${\mathbb N}$ with $\min$ as the product, and it is known that $A$ fails to be Arens regular. I don't know what the earliest reference for this last fact is: a more general construction for weighted analogues of $\ell^1({\mathbb N}_\min)$ can be found in the recent PhD thesis of M. E. Celorrio, see Prop 3.3.1, but the unweighted case has certainly been known for a long time.

On a side note: Arens regularity is much weaker than being isomorphic to a closed subalgebra of $B(H)$ for some Hilbert space $H$, so while it is possible to prove Arens regularity by embedding as a closed subalgebra of $B(H)$ this is usually asking too much.