TITLE: Is there an algebraic homomorphism between two Banach algebras which is not continuous?
QUESTION [17 upvotes]: According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces.
Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative linear map?
Or, is there some obstruction, ensuring that every algebraic homomorphism between two Banach algebras is continuous? (I know that this is true for $*$-homomorphisms between C*-algebras.)

REPLY [3 votes]: Let $X$ be a Banach space and denote by $B(X)$ the Banach algebra of all bounded linear operators on $X$. It is well-known that if $X$ is isomorphic to $X\oplus X$, then any homomorphism from $B(X)$ into any Banach algebra is continous. Charles Read constructed however an example of a space $X_R$ such that $B(X_R)$ admits a discontinuous homomorphism. 

C. J. Read, Discontinuous derivations on the algebra of bounded operators
  on a Banach space, J. London Math. Soc. 40 (1989), 305–326.

A recent PhD thesis by Skillicorn contains a synthesis of the subject (in particular Lemma 1.3.3 therein gives you description of such homomorphism).

R. Skillicorn, Discontinuous Homomorphisms from
  Banach Algebras of Operators, PhD thesis, Lancaster 2016.

You may also like the paper

R. Skillicorn, The uniqueness-of-norm problem for Calkin algebras,
  Math. Proc. R. Ir. Acad. 115 A (2015), 145–152.

which describes an example of discontinuous homomorphism on $B(X) / K(X)$, the Calkin algebra of $X$.