Is there an algebraic homomorphism between two Banach algebras which is not continuous?

"There are" examples of discontinuous homomorphisms between Banach algebras. However, the quotes are there because the question is independent of the usual axioms of set theory. I quote from the introduction to W. Hugh Woodin, "A discontinuous homomorphism from $C(X)$ without CH", J. London Math. Soc. (2) 48 (1993), no. 2, 299-315, MR1231717:

Suppose that $X$ is an infinite compact Hausdorff space and let $C(X)$ be the algebra of continuous real-valued functions on $X$. Then $C(X)$ is a commutative Banach algebra relative to the supnorm: ${}\|f\|= \sup\{f(p)\mid p\in X\}$. A well-known question of I. Kaplansky posed around 1947 asks whether every algebra homomorphism of $C(X)$ into a Banach algebra $B$ is necessarily continuous.

There is a discontinuous homomorphism of $C(X)$ if and only if there is a discontinuous homomorphism of $C(X, {\mathbb C})$, the $C^*$-algebra of continuous complex-valued functions on $X$. We prefer to deal with the real case; some of the references adopt the complex view. The question is now known to be independent of the axioms of set theory, ZFC. H. G. Dales [1] and J. Esterle [3] independently constructed discontinuous homomorphisms of $C(X)$ for any infinite space $X$ assuming the Continuum Hypothesis, CH. About the same time R. Solovay [4] proved that it is relatively consistent with ZFC that every homomorphism of $C(X)$ for any space X is necessarily continuous. Solovay's result was improved [5] fairly soon thereafter, to obtain the relative consistency with ZFC + Martin's Axiom (ZFC + MA) that every homomorphism of $C(X)$ for any space $X$ is continuous. We refer the reader to [2] for an exposition of the latter result concerning MA, historical points and related results.

After these results several questions remained. This paper is concerned with the question of whether the existence of a discontinuous homomorphism of $C(X)$ is possible given the failure of the Continuum Hypothesis.

The references listed above are:

1. H. G. Dales, "A discontinuous homomorphism from $C(X)$" Amer. J. Math. 101 (1979) 647-734. MR533196
2. H. G. Dales and H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Notes 115 (University Press, Cambridge, 1987). MR942216 and Review by M. Alain Louveau in the Bulletin of the AMS
3. J. R. Esterle, "Injection de semi-groupes divisibles dans des algebres de convolution et construction d'homomorphismes discontinus de $C(K)$" Proc. London Math. Soc. (3) 36 (1978) 59-85. MR482218
4. R. Solovay, personal communication, unpublished.
5. H. Woodin, Set theory and discontinuous homomorphisms from Banach algebras, Ph.D. Thesis, University of California at Berkeley, 1984.

I believe Solovay's original construction is simply superseded by Hugh's result, as described in his thesis. In any case, reference 2 is particularly good to learn about this fascinating topic.

Shortly after, in Stevo Todorcevic, Partition Problems in Topology (Contemporary Mathematics), American Mathematical Society (January 1989), MR980949, it is shown that automatic continuity of homomorphisms between Banach algebras is a theorem of ZFC + PFA. PFA is the proper forcing axiom, a strengthening of Martin's axiom.

There is also a much more recent reference: H. Garth Dales, Banach Algebras and Automatic Continuity (London Mathematical Society Monographs New Series), Oxford University Press, USA (May 17, 2001). MR1816726 and Review by George Willis in the Bulletin of the LMS.

Added: (T.B.) An excellent first introduction to automatic continuity is H.G. Dales's contribution (Part I) to Dales et. al. Introduction to Banach algebras, operators, and harmonic analysis, LMS Student Texts, Cambridge University Press (2003) electronic version (2009), MR2060440

You can construct a "silly" example of a discontinuous homomorphism as follows: Let $\omega: E \to \mathbb{C}$ be any discontinuous linear functional on a Banach space. (Assuming the Axiom of Choice, this exists.) Turn both $E$ and $\mathbb{C}$ into Banach algebras $E_0$ and $\mathbb{C}_0$ by defining the product of any two elements to be zero. N. B. that this is not the standard Banach algebra structure on $\mathbb{C}$, of course. The map $\omega$ is then a discontinuous homomorphism from $E_0$ to $\mathbb{C}_0$.

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 $X$.