The non-commutative Yosida-Hewitt decomposition revisited (Q2844845)

Let \(\mathcal{M}\) be a semi-finite von Neumann algebra with a semi-finite faithful normal trace \(\tau\). Let \(E\) be a strongly symmetric space (satisfying a very mild additional condition) of \(\tau\)-measurable operators affiliated with \(\mathcal{M}\). (Several theorems in the paper under review are stated without assuming \(E\) to be norm-complete. Also, the authors point out that their terminology ``(strongly) symmetric'' differs from the one in some previous papers.) In the dual of \(E\), normality, complete additivity and singularity of a functional are defined in a natural way, for example, \(\phi\in E^*\) is normal if \(\phi(x_{\alpha})\to0\), whenever \(x_{\alpha}\searrow0\) in \(E\). The set of normal (of singular) elements of \(E^*\) is denoted by \(E_n^*\) (by \(E_s^*\)).NEWLINENEWLINE In the case \(E=\mathcal{M}\), Takesaki's classical generalization of the Yosida-Hewitt decomposition states that a functional \(\phi\in\mathcal{M}^*\) has a unique decomposition \(\phi=\phi_n+\phi_s\), where \(\phi_n\in{\mathcal{M}}_n^*\) and \(\phi_s\in{\mathcal{M}}_s^*\). Recall further that this decomposition gives rise to two bounded positive projections \(P_n\) and \(P_s=1-P_n\) from \(\mathcal{M}^*\) onto \({\mathcal{M}}_n^*\) and onto \({\mathcal{M}}_s^*\), respectively. Moreover, it is known that an element of \(\mathcal{M}^*\) is normal iff it is completely additive and that \(\mathcal{M}_n^*\) identifies with \(L_1\) (\(=\mathcal{M}_*\)) via tracial duality (i.e., \(\phi\in{\mathcal{M}}_n^*\) iff it is of the form \(x\mapsto\tau(xy)\) for a (unique) \(y\in L_1\)). Finally, the decomposition is compatible with the module structure, more precisely, if \(\phi\) is normal (singular), then so are \(x\phi\) and \(\phi x\) for \(x\in \mathcal{M}\).NEWLINENEWLINE In the paper under review, the authors generalize all these facts to strongly symmetric spaces \(E\). To do so, the most important step consists in proving the following Yosida-Hewitt decomposition:NEWLINENEWLINEEvery \(\psi\in E^*\) can be written uniquely as \(\psi=\tau(\cdot y)+\psi_s\), where \(\tau(\cdot y)\in E_n^*\), \(\psi\in E_s^*\) and \(y\) is in the Köthe dual \(K^{\times}\); if \(\psi\) is positive, then so are \(y\) and \(\psi_s\).NEWLINENEWLINEIn [\textit{P. G. Dodds} et al., Positivity 9, No. 3, 457--484 (2005; Zbl 1123.46044)], such a decomposition has already been shown, but only for the smaller classes of strongly symmetric spaces containing \(\mathcal{M}\) and of fully symmetric spaces. The word ``revisited'' in the title does not only refer to this generalization from fully to strongly symmetric spaces, but also to the different approach which might be called `more intrinsic' or `more direct'; for example, the old proof uses the identification of \(E^{\times}\) with \(E_n^*\) via tracial duality, while the new one bears this identification (or rather the more difficult of its two directions) as a consequence. Other consequences are the identification ``normal = completely additive'', the compatibility with the module structure, the existence, linearity, positivity, boundedness and sequential weak\(^*\)-continuity of the Yosida-Hewitt projections \(P_n\) and \(P_s=1-P_n\) on \(E^*\).NEWLINENEWLINEGiven the smoothness of these generalizations from the case \(E=\mathcal{M}\) (see above) to arbitrary strongly symmetric \(E\), it is remarkable that, as the authors point out, it is an open question whether \(P_n\) and \(P_s\) are not only bounded but even contractive (although they are known to be contractive on the positive cone of \(E\)).NEWLINENEWLINEThe final section deals with the set \(E^{oc}=\{x\in E: |x|\geq x_{\alpha}\searrow0\Rightarrow\|x_{\alpha}|\searrow0\}\) of elements of order continuous norm. (\(E\) has order continuous norm iff \(E=E^{oc}\).) Some nice natural descriptions of \(E^{oc}\) are proven. For instance, \(E^{oc}\) turns out to be an order ideal such that \(E^{oc}={^{\perp}E_s^*}\). Moreover, \((E^{oc})^{\perp}=E_s^*\) iff \(E_s^*\) is weak\(^*\)-closed in \(E^*\). And \(E^{oc}=E^{an}\), where the latter space is the set of all elements having absolutely continuous norm (meaning that \(\|e_nxe_n\|\to0\), whenever \((e_n)\) is a sequence of projections of \(\mathcal{M}\) decreasing to \(0\)).