# STRUCTURE FOR REGULAR INCLUSIONS. I

@article{Pitts2012STRUCTUREFR, title={STRUCTURE FOR REGULAR INCLUSIONS. I}, author={David R. Pitts}, journal={arXiv: Operator Algebras}, year={2012} }

We study pairs (C,D) of unital C*-algebras where D is a regular abelian C*-subalgebra of C. When D is a MASA in C, we prove the existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D whose restriction to D is the identity on D. We show that the left kernel of E, L(C,D), is the unique closed two-sided ideal of C maximal with respect to having trivial intersection with D. When L(C,D)=0, we show the MASA D norms C. We apply these results to extend…

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## References

SHOWING 1-10 OF 50 REFERENCES

Pseudo-Diagonals and Uniqueness Theorems

- Mathematics
- 2013

We examine a certain type of abelian C*-subalgebra that allows one to give a unified treatment of two uniqueness theorems: for graph C*algebras and for certain reduced crossed products. This note is…

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

- Mathematics
- 2006

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A 1 and A 2 are operator algebras, then any bounded epimorphism of A 1 onto A 2…

Extensions of states of C* -algebras, II

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 1982

Synopsis Let A be a unital C*-algebra and let B be an abelian C*-subalgebra containing the identity of A. For any pure state h of B let Fh be the set of states of A which restrict to h on B.…

INJECTIVE HULLS OF C* ALGEBRAS

- Mathematics
- 2010

Introduction. Our aim is to apply category concepts to the study of commutative C* algebras. In particular we shall study injective hulls. This appears to have been overlooked in the literature,…

Inverse semigroups and combinatorial C*-algebras

- Mathematics
- 2007

Abstract.We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the…

Algebraic isomorphisms of limit algebras

- Mathematics
- 2000

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an…

Cartan Subalgebras in $C^*$-Algebras

- MathematicsIrish Mathematical Society Bulletin
- 2008

According to J. Feldman and C. Moore's well- known theorem on Cartan subalgebras, a variant of the group measure space construction gives an equivalence of categories between twisted countable…

Subalgebras of Graph C*-algebras

- Mathematics
- 2008

I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras L G , associated with the “Fock space” of a graph G,…

Injectivity and projectivity in analysis and topology

- Mathematics
- 2007

We give new proofs of many injectivity results in analysis that make more careful use of the duality between abelian C*-algebras and topological spaces. We then extend many of these ideas to…