Analysis and Probability Seminar: Bence Horvath

Wednesday 24 October 2018, 2:00pm to 3:00pm

Venue

PSC A54 LT - View Map

Open to

Postgraduates, Staff, Undergraduates

Registration

Registration not required - just turn up

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 lp -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 YX and a surjective algebra-homomorphism ψ : B(YX) → B(X) which is not injective.

Speaker

Bence Horvath Lancaster University

Contact Details

Name Dirk Zeindler
Email

d.zeindler@lancaster.ac.uk

Telephone number

+44 1524 593644