54 research outputs found
Non-unitarisable representations and maximal symmetry
We investigate questions of maximal symmetry in Banach spaces and the
structure of certain bounded non-unitarisable groups on Hilbert space. In
particular, we provide structural information about bounded groups with an
essentially unique invariant complemented subspace. This is subsequently
combined with rigidity results for the unitary representation of
on , where is the countably infinite regular tree, to describe
the possible bounded subgroups of extending a well-known
non-unitarisable representation of .
As a related result, we also show that a transitive norm on a separable
Banach space must be strictly convex
Displaying Polish groups on separable Banach spaces
A display of a topological group G on a Banach space X is a topological
isomorphism of G with the isometry group Isom(X,||.||) for some equivalent norm
||.|| on X, where the latter group is equipped with the strong operator
topology.
Displays of Polish groups on separable real spaces are studied. It is proved
that any closed subgroup of the infinite symmetric group S_\infty containing a
non-trivial central involution admits a display on any of the classical spaces
c0, C([0,1]), lp and Lp for 1 <=p <\infty. Also, for any Polsih group G, there
exists a separable space X on which {-1,1} x G has a display.Comment: 27 page
Banach spaces without minimal subspaces
We prove three new dichotomies for Banach spaces \`a la W.T. Gowers'
dichotomies. The three dichotomies characterise respectively the spaces having
no minimal subspaces, having no subsequentially minimal basic sequences, and
having no subspaces crudely finitely representable in all of their subspaces.
We subsequently use these results to make progress on Gowers' program of
classifying Banach spaces by finding characteristic spaces present in every
space. Also, the results are used to embed any partial order of size
into the subspaces of any space without a minimal subspace ordered by
isomorphic embeddability
Some strongly bounded classes of Banach spaces
We show that the classes of separable reflexive Banach spaces and of spaces
with separable dual are strongly bounded. This gives a new proof of a recent
result of E. Odell and Th. Schlumprecht, asserting that there exists a
separable reflexive Banach space containing isomorphic copies of every
separable uniformly convex Banach spaces.Comment: 10 page
- …