Radon--Nikodým theorem for biweights and regular biweights on partial *-algebras (Q2569661)

A partial *-algebra \(A\) is a vector space, taken to be over \(\mathbb{C}\), with involution and with muliplication defined over a fixed set of pairs of elements; multiplication is not necessarily associative. This should be useful for quantum physics and for justification one would like to have a representation of the algebra as linear (possibly unbounded) operators on a Hilbert space related to the state, and so being defined only on suitable subspaces. For C*-algebras, a state is taken to be a positive linear functional, or, when dealing with unbounded operators, a weight [cf.\ \textit{F.\,Combes}, C.\ R.\ Acad.\ Sci., Paris, Sér.\ A 265, 340--343 (1967; Zbl 0154.38902)]. In recent articles by the first author and associates, the state for partial *-algebras has taken to be a biweight [cf.\ J.~Math.\ Anal.\ Appl.\ 242, No.\,2, 164--190 (2000; Zbl 0949.47056)], a development of the positive (involution-)invariant sesquilinear form of \textit{J.--P.\thinspace Antoine, A.\,Inoue} and \textit{C.\,Trapani} [Publ.\ Res.\ Inst.\ Math.\ Sci.\ 27, No.\,3, 399--430 (1991; Zbl 0749.47030)]. Bilinearity is not appropriate because for positivity one needs the form to be invariant under involution, and because multiplication could be only partially defined and need not be associative. As is usual, Gelfand--Naimark--Segal-type methods are used to construct representations associated with the sesquilinear forms. The present article contains considerably many extracts from earlier articles co-authored with the first author and it is not always clearly expressed. Fixing a complex Hilbert space \(\mathcal{H}\) and an everywhere dense subspace \(\mathcal{D}\) of \(\mathcal{H}\), the authors denote by \(\mathcal{L}\)\(^{\dag}\) the partial *-algebra, with respect to suitable operations, of all closable linear operators on \(\mathcal{H}\) whose domain is \(\mathcal{D}\) and the domain of their adjoints contains \(\mathcal{D}\). By *-representation of \(A\) the authors mean a *-homomorphism to \(\mathcal{L}\)\(^{\dag}\). They give definitions for representations to be closed, fully closed or selfadjoint. Let \(\phi\) denote a positive sesquilinear form on \(\mathcal{D}(\phi) \times \mathcal{D}(\phi)\) and let \(\mathcal{N}\phi)\) be the subspace of \(D(\phi)\) such that \(\phi(x,y) = 0\) for all \(y \in D(\phi)\). For a positive invariant sesquilinear form \(\phi\) on \(A\), there may be several cores, i.e., everywhere dense subspaces of \(\mathcal{H}\) which support the associated representation. Given a positive sesquilinear form on \(\mathcal{D} \times \mathcal{D}\), they construct the Hilbert space completion of \(D(\phi) / \mathcal{N}_{\phi}\) and the ground-state-vector \(\lambda_{\phi}(x) = x + \mathcal{N}_{\phi}\), with inner product \((\lambda_{\phi}(x),\lambda_{\phi}(y)) = \phi(x,y)\). A biweight for \(\phi\) is a positive sesqulinear form for which there exists at least one suitable core in \(\mathcal{D}\), which they denote by \(B(\phi)\). The authors can then construct the representation \(\pi_{\phi}^{B}\) satisfying \(\pi_{\phi}^{B}(a)\lambda_{\phi}(x) = \lambda_{\phi}(ax)\). In \S3, following \textit{A.\,Inoue} [J.~Oper.\ Theory 10, 77--86 (1983; Zbl 0517.47028)], they give definitions for a biweight to be to be absolutely continuous and to be singular with respect to a given biweight \(\phi\) and, following the ideas of \textit{S.\,P.\thinspace Gudder} [Pac.\ J.\ Math.\ 80, 141--149 (1979; Zbl 0406.46055)], they prove a Radon--Nykodým type theorem and a Lebesgue decomposition theorem decomposing a biweight into a sum of an absolutely continuous and a singular part. In \S4, they consider nets of positive sesquilinear forms and define a biweight to be regular if, essentially, it is the supremum of a net of of positive sesquilinear forms. They define singularity in terms of non-existence of a non-null positive sesquilinear form \(\psi\) such that \(\psi(x,x) \leq \phi (x,x)\) for all \(x\in A\), but they do not bother to relate this to the definition of singularity in \S3. Following \textit{A.\,Inoue} and \textit{H.\,Ogi} [J.~Math.\ Soc.\ Japan 50, No.\,1, 227--252 (1998; Zbl 0904.47037)], they prove a Lebesgue-type decomposition theorem decomposing a biweight \(\phi\) into the sum of a regular and a singular part, essentially when \(\pi_{\phi}^{B}\) is selfadjoint.