Mathematics, Department of
Date of this Version
4-7-2012
Citation
2012 Author
Abstract
We study pairs (C,D) of unital C∗-algebras where D is an abelian C∗-subalgebra of C which is regular in C in the sense that the span of {v 2 C : vDv∗ [ v∗Dv D} is dense in 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 in the sense of Pop-Sinclair-Smith. We apply these results to significantly extend existing results in the literature on isometric isomorphisms of norm-closed subalgebras which lie between D and C.
The map E can be used as a substitute for a conditional expectation in the construction of coordinates for C relative to D. We show that coordinate constructions of Kumjian and Renault which relied upon the existence of a faithful conditional expectation may partially be extended to settings where no conditional expectation exists.
As an example, we consider the situation in which C is the reduced crossed product of a unital abelian C∗-algebra D by an arbitrary discrete group acting as automorphisms of D. We charac- terize when the relative commutant Dc of D in C is abelian in terms of the dynamics of the action of and show that when Dc is abelian, L(C,Dc) = (0). This setting produces examples where no conditional expectation of C onto Dc exists.
In general, pure states of D do not extend uniquely to states on C. However, when C is separable, and D is a regular MASA in C, we show the set of pure states on D with unique state extensions to C is dense in D. We introduce a new class of well behaved state extensions, the compatible states; we identify compatible states when D is a MASA in C in terms of groups constructed from local dynamics near an element 2 ˆD.
A particularly nice class of regular inclusions is the class of C∗-diagonals; each pair in this class has the extension property, and Kumjian has shown that coordinate systems for C∗-diagonals are particularly well behaved. We show that the pair (C,D) regularly embeds into a C∗-diagonal precisely when the intersection of the left kernels of the compatible states is trivial.
Comments
Journal of Operator Theory, 78(2),(2017), 357-416.