Local operator spaces, unbounded operators and multinormed \(C^*\)-algebras
The study of `local operator spaces' was initiated by \textit{E. Effros} and \textit{C. Webster} [NATO ASI Ser., Ser.~C, Math.\ Phys.\ Sci.\ 495, 163--207 (1997; Zbl 0892.46065)] as a concept related to locally convex topological vector spaces in a way that is similar to the relation between operator spaces and normed spaces, and motivated by a desire to consider linear spaces of unbounded operators. For a linear space \(E\), a matrix seminorm \(p\) on \(E\) is a family of seminorms \(p^{(n)} : \mathbb M_n(E) \to [0, \infty)\) (where \(\mathbb M_n(E)\) denotes the \(n \times n\) matrices with entries in \(E\)) satisfying (sub)additivity with respect to direct sums and a matrix convexity condition with respect to multiplication by scalar matrices. Each such matrix seminorm gives rise to a seminorm \(p^{(\infty)} : \mathbb M(E) \to [0, \infty)\), where \(\mathbb M(E) = \bigcup_n \mathbb M_n(E)\), with \(\mathbb M_n(E)\) considered as contained in \(\mathbb M_{n+1}(E)\) by adding an extra row and column of zeros. An abstract local operator space is \(E\) together with a family of matrix seminorms determining a (Hausdorff) locally convex topology on \(\mathbb M(E)\). The concrete cases considered here arise from a Hilbert space \(H\) and a directed (upward) family \({\mathcal E} = \{ H_\alpha \}_{\alpha \in \Lambda}\) of closed subspaces \(H_\alpha \subset H\) such that the union \({\mathcal D} = \bigcup_\alpha H_\alpha\) is a dense (linear) subspace of \(H\). For \(L({\mathcal D})\) the algebra of linear transformations of \({\mathcal D}\), \(P_\alpha\) the orthogonal projection of \(H\) onto \(H_\alpha\), one may take \(E = C_{\mathcal E} ({\mathcal D}) = \{ T \in L({\mathcal D}) : T P_\alpha = P_\alpha T P_\alpha \in {\mathcal B}(H)\) \(\forall \alpha \in \Lambda \}\) and introduce matrix seminorms \(p_\alpha\) such that \(p_\alpha^{(1)}(T) = \| P_\alpha T P_\alpha\|\) and \(p_\alpha^{(n)}\) is related to \(p_\alpha^{(1)}\) in a way that is familiar from operator space theory. One of the main results is that an abstract local operator space can be represented concretely as a subspace of \(C_{\mathcal E} ({\mathcal D})\) for suitable \(\mathcal E\) and \(H\). There are also self-adjoint subspaces \(C^*_{\mathcal E} ({\mathcal D}) \subset C_{\mathcal E} ({\mathcal D})\) considered. \(C_{\mathcal E} ({\mathcal D})\) is a unital subalgebra of \(L({\mathcal D})\), while \(C^*_{\mathcal E} ({\mathcal D})\) is a \(*\)-algebra (referred to as a multinormed \(C^*\)-algebra, meaning that it is equipped with a family of seminorms, each satisfying familiar \(C^*\) conditions). A Stinespring representation is established for `local matrix contractions' \(\phi : A \to C^*_{\mathcal E} ({\mathcal D})\) which are also `local matrix positive' on a multinormed \(C^*\)-algebra \(A\). There is also a representation result for such multinormed \(C^*\)-algebras as \(C^*_{\mathcal E} ({\mathcal D})\). The final section considers injectivity.
scientific article; zbMATH DE number 5365131
scientific article