The Lebesgue decomposition of representable forms over algebras (Q2859487)

This article is concerned with noncommutative Lebesgue-Radon-Nikodým decompositions, an area of problems that have been studied for some time. Given a complex vector space \(\mathcal V\), a form \(\omega:\mathcal V\times\mathcal V\to\mathbb C\) is a (positive semidefinite) sesquilinear form. For such a form \(\omega\), a canonical construction provides a Hilbert space \((\mathcal H_\omega;\langle\cdot,\cdot\rangle_\omega)\) and a linear operator \(\pi:\mathcal V\to\mathcal H_\omega\) such that \(\omega(u,v)=\langle\pi(u),\pi(v)\rangle_\omega\) for all \(u,v\in\mathcal V\), and \(\pi\) has dense range in \(\mathcal H_\omega\). The main issue refers to the setting of two given forms \(\omega\) and \(\chi\) for which a decomposition \(\omega=\omega_{\text{reg}}+\omega_{\text{sing}}\) is searched, with additional requirements that \(\omega_{\text{reg}}\) shows a certain regularity with respect to \(\chi\), while \(\omega_{\text{sing}}\) is singular with respect to \(\chi\), in the sense that \(0\) is the only form that is majorized by both \(\chi\) and \(\omega_{\text{sing}}\). The regularity of \(\omega\) with respect to \(\chi\) can be characterized in different ways, e.g., in the sense of closability, that is, whenever a sequence \((x_n)_n\) in \(\mathcal V\) has the properties \(\omega(x_n-x_m,x_n-x_m)\to 0\) and \(\chi(x_n,x_n)\to 0\) as \(m,n\to 0\), then \(\omega(x_n,x_n)\to 0\), as \(n\to 0\). In the case that \(\mathcal V\) is an algebra, a condition on boundedness of the shift operators \(\mathcal V\ni u\to vu\in\mathcal V\) with respect to \(\omega\), for all \(v\in\mathcal V\), provides a representation of \(\mathcal V\) on \(\mathcal H_\omega\). This condition defines the class of representable forms. For representable forms, Lebesgue-Radon-Nikodým decompositions show certain additional properties. In the special case of \(*\)-algebras, representability is transferred to the regular and singular parts automatically.NEWLINENEWLINE Since the literature on noncommutative Lebesgue-Radon-Nikodým decompositions is so vast, some comments are necessary. The author, as well as the referee and the editor of this article, do not seem to be aware of the fact that most of the results in this article follow from long known facts. More precisely, given two forms \(\omega\leq\chi\) on the same complex vector space \(\mathcal V\), the identical linear map \(\mathcal V\to\mathcal V\) can be lifted to a contraction \(J_{\chi,\omega}:\mathcal H_\chi\to\mathcal H_\omega\) which allows us to define \(D_\chi(\omega)=J_{\chi,\omega}^* J_{\chi,\omega}\), a positive contractive operator in the Hilbert space \(\mathcal H_\chi\). One calls \(D_{\chi}(\omega)\) the Radon-Nikodým derivative of \(\omega\) with respect to \(\chi\), following \textit{W. B. Arveson} [Acta Math. 123, 141--224 (1969; Zbl 0194.15701)]. It was observed, e.g., by \textit{A. Gheondea} and \textit{A. S. Kavruk} [J. Math. Phys. 50, No. 2, 022102, 29 p. (2009; Zbl 1202.46064)] that the Radon-Nikodým derivative provides an affine and order preserving isomorphism and hence, that the theory of Lebesgue-Radon-Nikodým derivatives developed by \textit{T. Ando} [Acta Sci. Math. 38, 253--260 (1976; Zbl 0337.47011)] provides answers to most of the questions raised in the article under review.