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Ondrej F. K. Kalenda

Publications and source records attributed to Ondrej F. K. Kalenda.

3 recordsLinked to original sources
arXiv2016

Preduals of JBW$^*$-triples are 1-Plichko spaces

We prove that the predual, $M_*$, of a JBW$^*$-triple $M$ is a 1-Plichko space (i.e. it admits a countably 1-norming Markushevich basis or, equivalently, it has a commutative 1-projectional skeleton), and obtain a natural description of the $Σ$-subspace of $M$. This generalizes and improves similar results for von Neumann algebras and JBW$^*$-algebras. Consequently, dual spaces of JB$^*$-triples also are 1-Plichko spaces. We also show that $M_*$ is weakly Lindelöf determined if and only if $M$ is $σ$-finite if and only if $M_*$ is weakly compactly generated. Moreover, contrary to the proof for JBW$^*$-algebras, our proof dispenses with the use of elementary submodels theory.

math.OA↗
arXiv2014

Rich families and elementary submodels

We compare two methods of proving separable reduction theorems in functional analysis -- the method of rich families and the method of elementary submodels. We show that any result proved using rich families holds also when formulated with elementary submodels and the converse is true in spaces with fundamental minimal system an in spaces of density $\aleph_1$. We do not know whether the converse is true in general. We apply our results to show that a projectional skeleton may be without loss of generality indexed by ranges of its projections.

math.FA↗