Pettis' lemma and topological properties of dual algebras (Q1095383)

Based on Pettis' lemma, the authors prove the following theorems: Let \({\mathcal A}\) be a weak* closed subspace of L(\({\mathcal H})\) (\({\mathcal H}\) being a complex Hilbert space) where the weak and weak* topologies agree. Then \({\mathcal A}\) has property \((A_{1/n}(r))\) for some n and r. Further, if \({\mathcal A}\) is a weak* closed subspace with property \((A_{1/m})\), then it has the property \((A_{1/2m}(r))\) for some r. Thus, \((A_{1/n})\) means that every [L] in \(Q_{{\mathcal A}}\) can be written as \(\sum^{h}_{i=1}[x_ i\otimes y_ i]\) (sum of operators of rank 1), \((A_{1/n}(r))\) means that this composition also fulfills \[ \sum^{n}_{n=1}\| x_ i\| \| y_ i\| \leq C\| [L]\| \] for any \(C>r\).