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We introduce the notion of a Cartan envelope for a regular inclusion (C,D). When a Cartan envelope exists, it is the unique, minimal Cartan pair into which (C,D) regularly embeds. We prove a Cartan envelope exists if and only if (C,D) has the unique faithful pseudo-expectation property and also give a characterization of the Cartan envelope using the ideal intersection property. For any covering inclusion, we construct a Hausdorff twisted groupoid using appropriate linear functionals and we give a description of the Cartan envelope for (C,D) in terms of a twist whose unit space is a set of states on C constructed using the unique pseudo-expectation. For a regular MASA inclusion, this twist differs from the Weyl twist; in this setting, we show that the Weyl twist is Hausdorff precisely when there exists a conditional expectation of C onto D. We show that a regular inclusion with the unique pseudo-expectation property is a covering inclusion and give other consequences of the unique pseudo-expectation property., 47 pages

Given an inclusion D $\subseteq$ C of unital C*-algebras, a unital completely positive linear map $\Phi$ of C into the injective envelope I(D) of D which extends the inclusion of D into I(D) is a pseudo-expectation. The set PsExp(C,D) of all pseudo-expectations is a convex set, and for abelian D, we prove a Krein-Milman type theorem showing that PsExp(C,D) can be recovered from its extreme points. When C is abelian, the extreme pseudo-expectations coincide with the homomorphisms of C into I(D) which extend the inclusion of D into I(D), and these are in bijective correspondence with the ideals of C which are maximal with respect to having trivial intersection with D. Natural classes of inclusions have a unique pseudo-expectation (e.g., when D is a regular MASA in C). Uniqueness of the pseudo-expectation implies interesting structural properties for the inclusion. For example, when D $\subseteq$ C $\subseteq$ B(H) are W*-algebras, uniqueness of the pseudo-expectation implies that D' $\cap$ C is the center of D; moreover, when H is separable and D is abelian, we characterize which W*-inclusions have the unique pseudo-expectation property. For general inclusions of C*-algebras with D abelian, we characterize the unique pseudo-expectation property in terms of order structure; and when C is abelian, we are able to give a topological description of the unique pseudo-expectation property. Applications include: a) if an inclusion D $\subseteq$ C has a unique pseudo-expectation $\Phi$ which is also faithful, then the C*-envelope of any operator space X with D $\subseteq$ X $\subseteq$ C is the C*-subalgebra of C generated by X; b) for many interesting classes of C*-inclusions, having a faithful unique pseudo-expectation implies that D norms C. We give examples to illustrate the theory, and conclude with several unresolved questions., Comment: 26 pages

In this note we use cohomological techniques to prove that if there is a linear map between two CSL algebras which is close to the identity, then the two CSL algebras are similar. We use our result to show that if 2f is a purely atomic, hyperreflexive CSL with uniform infinite multiplicity which satisfies the 4-cycle interpolation condition, then there are constants δ, C > 0 such that whenever Jt is another CSL such that d(Algi?, MgJt) < δ , then there is an invertible operator S such that S k\gS?S~ι = Alg^ and ||S|| US"1 II < 1 4- Cd(Alg&, 1. Introduction and preliminaries. In this paper, we consider two related types of perturbation questions for CSL algebras. The first deals with the problem of perturbing a linear isomorphism between two algebras to obtain an algebra isomorphism, and the second deals with the problem of deciding whether, for two such algebras, closeness implies isomorphism. Perturbation questions of this kind were considered by Kadison and Kastler for von Neumann algebras in [18] and further studied by many authors, including Christensen ([3, 4, 5]). Johnson ([16]) and Raeburn and Taylor ([21]) obtained results concerning perturbations of closed subalgebras of Banach algebras. Their results show that if $/ is a closed subalgebra of a Banach algebra 3S and certain cohomology groups for sf vanish, then any closed subalgebra of 3S "sufficiently close" to sf is actually isomorphic to stf . The nonself adjoint case was considered first for nest algebras by Lance in [19]. Perturbations of other nonself adjoint operator algebras were considered by Choi and Davidson ([2]), Davidson ([6]). In §2, we prove Theorem 3 which shows that if two CSL algebras s/\ and J^2 are linearly isomorphic via an isomorphism close to the identity, they they are actually spatially isomorphic via an isomorphism which is close to a unitary equivalence. In §3, we introduce the 4-cycle interpolation property, which is closely related to a lattice condition appearing in [12] and to the notion of interpolating lattice introduced in [7]. The main result of §3, Theorem 16, shows that if sfγ is a CSL algebra which is sufficiently close to a purely atomic

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra $A_n$. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a $*$-extendible representation $\sigma$. A $*$-extendible representation of $A_n$ is ``regular'' if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given $*$-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of $\sigma(A_n)$ is weak-$*$ dense in the unit ball of the associated free semigroup algebra if and only if $\sigma$ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts., Comment: 26 pages, prepared with LATeX2e, submitted to Journal of Functional Analysis