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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