Unique Pseudo-Expectations for C∗C^*-Inclusions

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 page

Bimodules over Cartan MASAs in von Neumann Algebras, Norming Algebras, and Mercer's Theorem

In a 1991 paper, R. Mercer asserted that a Cartan bimodule isomorphism between Cartan bimodule algebras A_1 and A_2 extends uniquely to a normal *-isomorphism of the von Neumann algebras generated by A_1 and A_2 [13, Corollary 4.3]. Mercer's argument relied upon the Spectral Theorem for Bimodules of Muhly, Saito and Solel [15, Theorem 2.5]. Unfortunately, the arguments in the literature supporting [15, Theorem 2.5] contain gaps, and hence Mercer's proof is incomplete. In this paper, we use the outline in [16, Remark 2.17] to give a proof of Mercer's Theorem under the additional hypothesis that the given Cartan bimodule isomorphism is weak-* continuous. Unlike the arguments contained in [13, 15], we avoid the use of the Feldman-Moore machinery from [8]; as a consequence, our proof does not require the von Neumann algebras generated by the algebras A_i to have separable preduals. This point of view also yields some insights on the von Neumann subalgebras of a Cartan pair (M,D), for instance, a strengthening of a result of Aoi [1]. We also examine the relationship between various topologies on a von Neumann algebra M with a Cartan MASA D. This provides the necessary tools to parametrize the family of Bures-closed bimodules over a Cartan MASA in terms of projections in a certain abelian von Neumann algebra; this result may be viewed as a weaker form of the Spectral Theorem for Bimodules, and is a key ingredient in the proof of our version of Mercer's theorem. Our results lead to a notion of spectral synthesis for weak-* closed bimodules appropriate to our context, and we show that any von Neumann subalgebra of M which contains D is synthetic. We observe that a result of Sinclair and Smith shows that any Cartan MASA in a von Neumann algebra is norming in the sense of Pop, Sinclair and Smith.Comment: 21 pages, paper is a completely reworked and expanded version of an earlier preprint with a similar titl

One-Sided Projections on C*-algebras

In [BEZ] the notion of a complete one-sided M-ideal for an operator space X was introduced as a generalization of Alfsen and Effros' notion of an M-ideal for a Banach space [AE72]. In particular, various equivalent formulations of complete one-sided M-projections were given. In this paper, some sharper equivalent formulations are given in the special situation that $X = \mathcal{A}$, a $C^*$-algebra (in which case the complete left M-projections are simply left multiplication on $\mathcal{A}$ by a fixed orthogonal projection in $\mathcal{A}$ or its multiplier algebra). The proof of the first equivalence makes use of a technique which is of interest in its own right--a way of ``solving'' multi-linear equations in von Neumann algebras. This technique is also applied to show that preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals. In addition, we show that in a $C^*$-algebra, the intersection of finitely many complete one-sided M-summands need not be a complete one-sided M-summand, unlike the classical situation

Exotic Ideals in Free Transformation Group C∗C^*-Algebras

Let $\Gamma$ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space $X$ and let $C(X) \rtimes_\eta \Gamma$ be the completion of the convolution algebra $C_c(\Gamma,C(X))$ with respect to a $C^*$-norm $\eta$. A non-zero ideal $J \unlhd C(X) \rtimes_\eta \Gamma$ is exotic if $J \cap C(X) = \{0\}$. We show that exotic ideals are present whenever $\Gamma$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen. Using the Koopman representation and a recent theorem of Elek, we show that when $\Gamma$ is a countably-infinite group having property (T) and $X$ is the Cantor set, there exists a free and minimal action of $\Gamma$ on $X$ and a $C^*$-norm $\eta$ on $C_c(\Gamma, C(X))$ such that $C(X)\rtimes_\eta\Gamma$ contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen. The opaque and grey ideals in $C(X)\rtimes_\eta \Gamma$ have trivial intersection with $C(X)$, and a result from arXiv:1901.09683 shows they coincide when the action of $\Gamma$ is free, however the problem of whether these ideals can be non-zero was left unresolved. We present an example of a free action of $\Gamma$ on a compact Hausdorff space $X$ along with a $C^*$-norm $\eta$ for which these ideals are non-trivial, in particular, they are exotic ideals.Comment: Article is totally rewritten, reorganized, and has a new title (former title: "Exotic Ideals in Represented Free Transformation Groups") Includes some new results. 16 page

Norming in Discrete Crossed Products

Let $G \curvearrowright A$ be an action of a discrete group on a unital $C^*$-algebra by $*$-automorphisms. In this note, we give two sufficient dynamical conditions for the $C^*$-inclusion $A \subseteq A \rtimes_r G$ to be norming in the sense of Pop, Sinclair, and Smith. As a consequence of our results, when $A$ is separable or simple, the inclusion $A \subseteq A \rtimes_r G$ is norming provided it has a unique pseudo-expectation in the sense of Pitts.Comment: 27 pages, 2 figure