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Journal of Operator Theory

Volume 55, Issue 1, Winter 2006  pp. 91-116.

Upper regularization for extended self-adjoint operators

Authors Henri Comman
Author institution: Department of Mathematics, University of Santiago de Chile, Bernardo O'Higgins 3363 Santiago, Chile

Summary:  We show that the complete lattice of $\overline{\mathbb{R}}$-valued sup-preserving maps on a complete lattice $\mathcal{G}$ of projections of a von Neumann algebra $\mathcal{M}$, is isomorphic to some complete lattice $\mathcal{M}^{\mathcal{G}}_{\overline{\mathbb{R}}}$ of extended spectral families in $\mathcal{M}$, provided with the spectral order. We get various classes of (not necessarily densely defined) self-adjoint operators affiliated with $\mathcal{M}$ as conditionally complete lattices with completion $\mathcal{M}^{\mathcal{G}}_{\overline{\mathbb{R}}}$, extending the Olson's results. When $\mathcal{M}$ is the universal enveloping von Neumann algebra of a $C^*$-algebra $A$, and $\mathcal{G}$ the set of open projections, the elements of $\mathcal{M}^{\mathcal{G}}_{\overline{\mathbb{R}}}$ are said to be extended $q$-upper semicontinuous, generalizing the usual notions. The $q$-upper regularization map is defined using the spectral order, and characterized in terms of the above isomorphism. When $A$ is commutative with spectrum $X$, we give an isomorphism $\Pi$ of complete lattices from $\overline{\mathbb{R}}^{X}$ into the set of extended self-adjoint operators affiliated with $\mathcal{M}$. By means of $\Pi$, the above characterizations appear as generalizations of well-known properties of the upper regularization of $\overline{\mathbb{R}}$-valued functions on $X$. A noncommutative version of the Dini-Cartan's lemma is given. An application is sketched.

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