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?