Jesus M.F. Castillo :
The structure that subspaces and quotients of Banach spaces may have
The contents of the lectures could be described as "basic elements of
homological algebra and their applications to (quasi)-Banach space theory,
starting with the algebra, ending with the topology."
Thus, we would like to touch the following topics (depending on time and
preferences of the audience we can stress some and/or omit some):
- Basic elements of homological algebra, with applications. Here we
describe exact sequences (subspaces and quotients), the 3-lemma (equivalence
of exact sequences), functors and natural transformations (interpolation
methods, Ext,...); the pull-back and their applications
(Johnson-Lindenstrauss spaces and the 3-space problem for WCG spaces,
counterexamples to the 3-space problem for the Dunford-Pettis property and
the problem of duals, twisted sums of R and a topological vector space...);
the push-out and their applications (locally convex twisted sums, derived
spaces,...). The long homology sequence with some applications (3-space
property of K-spaces, some invariants for quasi-Banach spaces,...)
- The basic exact sequences and the possible exact sequences made with
classical Banach spaces (L_p and C(K) spaces).
- A theory of duality for twisted sums of Banach spaces and the
counterexample to the 3-space problem for the properties "to be isomorphic
to a dual space" and "to be complemented in its bidual".
- Twisted properties (exact sequences in which the subspace has a property
P and the quotient has another property Q). Counterexamples to the 3-space
problem for WCG spaces and the Plichko-Valdivia property. Sobczyk's theorem.
