Examples of a Banach space with an algebra structure having only left continuity There is a theorem (see for example, Rudin's Functional Analysis, theorem 10.2 ) that if $A$ is a Banach space with an algebra structure, such that both left and right multiplication are continuous, then $A$ has a renorming such that $A$ is a Banach algebra. 
He also provides an example where the lack of completeness causes this to fail. 
I'm looking to construct an example where $A$ is an algebra, as well as a Banach space, such that only left multiplication is continuous. 
Any thoughts? Is this possible?
 A: Let $X$ be a Banach space, $f:X\to K$ an unbounded linear functional. Define the multiplication on $X$ by
$$
x\cdot y := f(y)x.
$$
Then $(x,y)\mapsto x\cdot y$ is continuous in $x$ for fixed $y$, but discontinous in $y$ for fixed $x\ne0$.
A: 
Arens, Richard. The adjoint of a bilinear operation. Proc. Amer. Math. Soc. 2, (1951). 839–848.

If you take the second dual $\text{A}^{**}$ of a Banach space $\text{A}$ equipped with the first (or second) Arens multiplication (see here or the aforementioned paper for a definition), then $\text{A}^{**}$ with the weak$^*$-topology is continuous with respect to only one of the variables, but not necessarily the second variable in general.
For example, let $\text{A}$ be $\text{L}^1(\mathbb{R})$ with convolution product. Then the set of all elements in $\text{A}^{**}$ continuous with the other variable is only the elements of $\text{A}$ embedded in $\text{A}^{**}$.
Please see the following here.

Lau, Anthony To-Ming; Losert, Viktor. On the second conjugate algebra of $\text{L}^1(\text{G})$ of a locally compact group. J. London Math. Soc. (2) 37 (1988), no. 3, 464–470. 

