# Journal of Operator Theory

Volume 78, Issue 2, Fall 2017 pp. 357-416.

Structure for regular inclusions. I**Authors**: David R. Pitts

**Author institution:**Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, U.S.A.

**Summary:**We give general structure theory for pairs $(\mathcal{C},\mathcal{D})$ of unital $C^*$-algebras where $\mathcal{D}$ is a regular and abelian $C^*$-subalgebra of $\mathcal{C}$. When $\mathcal{D}$ is maximal abelian in $\mathcal{C}$, we prove existence and uniqueness of a completely positive unital map $E$ of $\mathcal{C}$ into the injective envelope $I(\mathcal{D})$ of $\mathcal{D}$ such that $E|_\mathcal{D}=\mathrm{id}_\mathcal{D}$; $E$ is a useful replacement for a conditional expectation when no expectation exists. When $E$ is faithful, $(\mathcal{C},\mathcal{D})$ has numerous desirable properties: e.g.\ the linear span of the normalizers has a unique minimal $C^*$-norm; $\mathcal{D}$ norms $\mathcal{C}$; and isometric isomorphisms of norm-closed subalgebras lying between $\mathcal{D}$ and $\mathcal{C}$ extend uniquely to their generated $C^*$-algebras.

**DOI:**http://dx.doi.org/10.7900/jot.2016sep15.2128

**Keywords:**inclusions of $C^*$-algebras, pseudo-expectation, regular homomorphism

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