Introduction: The study of automatic continuity for homomorphisms (multiplicative, linear maps) between Banach algebras has a long history. In 1960, Bade and Curtis [1] proved the existence of discontinuous homomorphisms between commutative Banach algebras in ZFC, showing that automatic continuity results can only be achieved by imposing additional hypotheses on the algebras
,
or on the map
.
A natural class of Banach algebras are unital, commutative C*-algebras, that is, the algebras of continuous functions on a compact, Hausdorff space
. Using that every character (that is, homomorphism
) of a Banach algebra
is contractive, it follows that every homomorphism
is contractive. The situation is completely different if instead we assume that the domain is
: In 1976 Dales and Esterle [2] independently produced discontinuous homomorphisms
for every infinite, compact, Hausdorff space
, assuming the continuum hypothesis (CH). This suggested to consider the axiom „no discontinuous homomorphisms“
(NDH) Every homomorphism is continuous (for every compact, Hausdorff space
and every Banach algebra
).
Cuntz [3] proved that continuity of linear maps out of a C*-algebra reduces to the commutative case: A linear map from a C*-algebra
to a Banach space
is continuous if and only if the restriction of
to every commutative sub-C*-algebra is continuous. It follows that (NDH) is equivalent to:
(NDH‘) Every homomorphism from a C*-algebra to a Banach algebra is continuous.
There are models of ZFC (for example, Solovay’s model assuming an inaccessible cardinal, or Woodin’s model obtained by forcing Martin’s Axiom) in which (NDH) holds [4]. Thus, (NDH) is independent of ZFC: it fails under (CH) by Esterle and Dales, but holds in some models of ZFC.
This raises the question if one can impose further assumptions to show automatic continuity. A positive result was obtained by Sakai [5], who showed that every surjective homomorphism between C*-algebras is automatically continuous; see Lemma 4.1.12 in [5]. The following open questions ask if the result of Sakai can be generalized by removing the assumption on the target algebra (Question 1), by removing the assumption on the map (Question 2) or by relaxing the assumptions on target algebra and map (Question 3).
Question 1: Is every surjective homomorphism
from a C*-algebra
onto a Banach algebra
automatically continuous?
Question 2: Is every homomorphism
between C*-algebras automatically continuous?
Question 3: Is every homomorphism
with dense range from a C*-algebra
to a semisimple Banach algebra
automatically continuous?
The answers to all questions is known to be positive if is commutative: For Question 1 this was shown by Laursen (Theorem 4 in [6]), and for Questions 2 and 3 it follows using that characters are contractive. For further partial results see [7].
[1] Bade, Curtis. Homomorphisms of commutative Banach algebras. Am. J. Math. 82 (1960), 589-608.
[2] Dales, Esterle. Discontinuous homomorphisms from C(X). Bull. Am. Math. Soc. 83 (1977), 257-259.
[3] Cuntz. On the continuity of semi-norms on operator algebras. Math. Ann. 220 (1976), 171-183.
[4] Dales, Woodin. An introduction to independence for analysts. Cambridge University Press (1987).
[5] Sakai. C-algebras and W-algebras. Springer-Verlag (1971).
[6] Laursen. Continuity of homomorphisms from C*-algebras into commutative Banach algebras. J. Lond. Math. Soc., II. Ser. 36 (1987), 165-175.
[7] Runde. An epimorphism from a C*-algebra is continuous on the center of its domain. J. Reine Angew. Math. 439 (1993), 93-102.