Ferleger S.
BASES IN (UMD)-SPACES, APPLICATIONS TO THE NON-COMMUTATIVE SYMMETRIC SPACES
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ABSTRACT.  The present paper deals with the problems of existence of
Schauder
bases and unconditional finite dimensional decompositions (UFDD) in
some non-commutative symmetric spaces.  In order to solve these
problems we study bases in arbitrary (UMD)-spaces with strongly continuous
representations of compact abelian groups.  It turns out that under
certain conditions the eigenvectors of such representations form
bases in (UMD)-spaces while the eigenspaces form an unconditional
finite dimensional decompositions of the spaces. As an application, we
construct the first example of a Schauder basis in every operator $L_p$
space, $1<p<\infty$ associated with the hyperfinite von Neumann factor of
type $II_1.$