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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
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.
We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder–Bernstein Index of any of these spaces is
. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder–Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.
The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of $c_0$ or $\ell_p$. An analogous problem concerning spreading models is also addressed.
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