On the kernel of a Radon measure in a Banach space (Q1345662)

The author studies the notion of a kernel of a measure \(\mu\) in a Banach space introduced by \textit{S. Chevet} [Lect. Notes Math. 860, 51-84 (1981; Zbl 0458.60005)]. Namely, the kernel of a Radon measure \(\mu\) is called the space \({\mathcal H}_\mu= (X^*, \tau_\mu)^*\), where \(\tau_\mu\) is the topology of convergence in measure \(\mu\). The notion represents a generalization of the notion of the reproducing kernel of a Gaussian measure. The author shows that the set of nonsingular translations of \(\mu\) is a part of \({\mathcal H}_\mu\). The main results of the paper are: Theorem 1. There exists \((x_n)\in \ell_1 (X)\) such that \({\mathcal H}_\mu \subset \{\sum a_n x_n\): \((a_n)\in \ell_\infty\}\). Theorem 2. If \(X\) is of some cotype and possesses bounded approximation property, then for any \((x_n)\in \ell_1 (X)\) there exists a measure \(\mu\) such that \(\{\sum a_n x_n\): \((a_n)\in \ell_\infty\} \subset {\mathcal H}_\mu\): and conversely, the last property implies that \(X\) is of some cotype. Reviewer's remark. The first part of Theorem 1 seems to be most interesting. Unfortunately, the author only outlines the proof of it. We mean the part of the proof dealing with the coincidence of the topologies \(\tau_p (X^*, X)= \tau_1 (X^*, X)\), \(p<1\), for spaces of some cotype with BAP.