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.

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 BV(R) to bv(N), given by restriction of functions from R to N. Now Arens regularity passes to quotients, so if BV(R) were Arens regular then bv(N) would be Arens regular. But bv(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.

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.

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.

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.

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 BV(R) to bv(N), given by restriction of functions from R to N. Now Arens regularity passes to quotients, so if BV(R) were Arens regular then bv(N) would be Arens regular. But bv(N) is isomorphic to 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.

In fact, I have a strong suspicion that there is no isomorphism of Banach spaces (= bicontinuous linear bijection) between BV(R) and any Cstar algebra, but I would need to check details and look up results. One possible way to show this would be to invoke the result that the Banach-space-dual of any Cstar algebra has cotype 2 (Tomczak-Jaegermann) while BV(R) should, if memory serves rightly, be isomorphic as a Banach space to the Banach-space dual of $C_0(R)$, so that $BV(R)^\ast$ would be isomorphic to a $C(K)$-space and hence definitely not have cotype 2.

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.

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 BV(R) to bv(N), given by restriction of functions from R to N. Now Arens regularity passes to quotients, so if BV(R) were Arens regular then bv(N) would be Arens regular. But bv(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.