Event Details

The structure of homomorphisms of algebras of operators on Banach spaces

Eidelheit’s classical theorem combined with B.E. Johnson’s famous automatic continuity result asserts the following: Let X and Y be Banach spaces and let φ : B(X) → B(Y) be a surjective algebra homomorphism. Then the following hold:

1. The map φ is continuous.
2. If φ is injective then X and Y are isomorphic as Banach spaces.

The following question naturally arises: Let X and Y be both infinite-dimensional Banach spaces and let φ : B(X) → B(Y) be a surjective algebra-homomorphism. Is φ automatically injective? It is easy to see that for a 'very nice' class of Banach spaces (for example for l_p -spaces) the answer is positive. In our talk we shall present methods which allow us to extend the range of positive examples, including l_∞ and the Banach spaces (⊕_n=1^∞ l₂ )_co and (⊕_n=1^∞ l₂ )_l1 studied by Bourgain, Casazza, Lindenstrauss, Tzafriri and later by Laustsen, Loy, Read, Schlumprecht and Zsak. In the other direction, we will show that the answer to the question above is negative: For any separable, reflexive Banach space X there is a Banach space Y_X and a surjective algebra-homomorphism ψ : B(Y_X) → B(X) which is not injective.