Structure for regular inclusions. II: Cartan envelopes, pseudo-expectations and twists

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Abstract

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.

Introduction

In their influential 1977 paper [8], Feldman and Moore showed that the collection of pairs (M,D) consisting of a Cartan MASA D in the separably acting von Neumann algebra M is, up to suitable notions of equivalence, equivalent to the family (R,σ) consisting of measured equivalence relations and 2-cocycles σ on R. The success of the Feldman-Moore program naturally led to attempts to find appropriate C-algebraic analogs. An early attempt was by Kumjian in 1986 [12], who introduced the notion of C-diagonals and proved a Feldman-Moore type result for them using suitable twists. However, Kumjian's setting was somewhat restrictive, and excluded several classes of desirable examples. In a 2008 paper, Renault [18] extended Kumjian's work. Renault gave a definition of a Cartan MASA D in a C-algebra C and gave a method for associating a twist (Σ,G) to each such pair (C,D). The philosophy is to loosely regard the passage from (C,D) to (Σ,G) as somewhat akin to “analysis” in harmonic analysis. It is of course an interesting problem to determine when the original regular inclusion can be reconstructed (“synthesized”) from (Σ,G). In [18], Renault shows the class of Cartan inclusions is, to use Leibnitz's immortal phrase, ‘the best of all possible worlds.’ Indeed, for any Cartan inclusion (C,D), the topologies on Σ and G are Hausdorff, and the associated twist (Σ,G) contains enough of the information about (C,D) to completely recover (C,D). More precisely, Renault shows that if G(0) is the unit space of G and Cr(Σ,G) denotes the reduced C-algebra of (Σ,G), then (Cr(Σ,G),C(G(0))) is a Cartan inclusion isomorphic to the original Cartan inclusion (C,D). Thus, for Cartan inclusions, both analysis and synthesis are possible. With his results, Renault makes a very convincing case that his definition of Cartan MASA for C-algebras is the appropriate analog of the Feldman-Moore notion of a Cartan MASA in a von Neumann algebra.

While Renault's notion of Cartan MASA appears in a wide variety of examples, there are also quite natural examples of regular MASA inclusions (C,D) which are not Cartan because they lack a conditional expectation of C onto D. A large class of examples of regular MASA inclusions with no conditional expectation which arise from crossed products of abelian C-algebras by discrete groups is constructed in [15, Section 6.1]. The lack of a conditional expectation leads to serious problems when one attempts to apply the Kumjian-Renault methods to coordinatize (C,D) using a twist. Indeed, Theorem 4.4 below shows that for a regular MASA inclusion (C,D), the associated Weyl groupoid G is Hausdorff if and only if there is a conditional expectation of C onto D. Thus we are confronted with the problem of whether a suitable coordinatization of such pairs (C,D) exists, and what that would mean. If one is willing to utilize non-Hausdorff twists, it is possible to obtain a Kumjian-Renault type characterization of a class of non-Cartan inclusions, and this was recently done in [7]. However, here we shall primarily be interested in Hausdorff twists.

One approach to analyzing a non-Cartan inclusion is to attempt to embed it into a Cartan inclusion. In [15, Theorem 5.7] we characterized when a regular inclusion (C,D) regularly embeds into a C-diagonal, or equivalently, when it embeds into a Cartan inclusion. Applying this result produces a Cartan pair (C1,D1) into which (C,D) embeds, but (C1,D1) is in general not closely related to the original pair (C,D).

To address this issue, we introduce the notion of a Cartan envelope for a regular inclusion (C,D), see Definition 5.1. This is the “smallest” Cartan pair (C1,D1) into which the original pair (C,D) can be regularly embedded. We show the Cartan envelope is unique when it exists, and that the image of C in C1 is dense in a suitable pointwise topology.

In [15], we introduced the notion of a pseudo-expectation for an inclusion (C,D). For some purposes, pseudo-expectations can be used as a replacement for a conditional expectation. The advantage of pseudo-expectations is that they always exist, and for regular MASA inclusions, are unique [15, Theorem 3.5]. Furthermore, a regular inclusion is a Cartan inclusion if and only if it has a unique pseudo-expectation which is actually a faithful conditional expectation (see Proposition 5.5(b) below). Thus, regular inclusions with a unique and faithful pseudo-expectation are a natural class of regular inclusions containing the Cartan inclusions. We do not know a characterization of those regular inclusions (C,D) for which the pseudo-expectation is unique.

The issue of existence of a Cartan envelope for (C,D) is addressed in Theorem 5.2: we characterize the regular inclusions (C,D) which have Cartan envelope as those which have a unique pseudo-expectation which is also faithful. We also characterize the existence of the Cartan envelope in terms of the ideal intersection property.

Suppose (C,D) is a regular inclusion having Cartan envelope (C1,D1). If (Σ1,G1) is the twist associated to (C1,D1), elements of Σ1 and G1 can be viewed as functions (non-linear in the case of G1) on C1, and by restricting these functions to the image of C under the embedding of (C,D) into (C1,D1), we obtain families of functions on C. These restriction mappings are both one-to-one. In this way, (Σ1,G1) may be thought of as a “weak-coordinatization” of (C,D), or as a weak form of “spectral analysis” for (C,D). Unsurprisingly, it is possible for two distinct regular inclusions to have the same Cartan envelope, so in general it is not possible to synthesize the original inclusion from a weak-coordinatization without further data. We give examples of this phenomenon in Example 5.30.

For a Cartan MASA D in a von Neumann algebra M, Aoi's theorem shows that D is also a Cartan MASA in any intermediate von Neumann subalgebra DNM, see [1, Theorem 1.1] or [4, Theorem 2.5.9] for an alternate approach which does not require separability of the predual. While Aoi's theorem is not true in full generality in the C-algebra setting, partial results are obtained in [2]. In a sense, Proposition 5.31 of the present paper complements these results: it gives a description of those regular subinclusions (C0,D0) of a given Cartan pair (C,D) which are “nearly intermediate” in the sense that D0 is an essential subalgebra of D.

In [15, Section 4], we introduced the notion of a compatible state for a regular inclusion (C,D). The restriction of any compatible state on C to D is a pure state on D, and when the regular inclusion (C,D) has enough compatible states to cover Dˆ, it is a covering inclusion. We define the notion of a compatible cover for Dˆ (see Definition 2.11) and Theorem 7.24 shows that associated to each compatible cover, there is a Hausdorff twist. When (C,D) has a Cartan envelope, Theorem 6.9 shows it has a minimal (necessarily compatible) cover, and by Corollary 7.31, the twist associated to the minimal cover is the twist for the Cartan envelope.

We now give an outline of the sections of the paper. Section 2 gives provides a reference for some notation and preliminary results. Section 3 is also a preliminary section, but deals with twists and reduced C-algebras associated to Hausdorff twists. Section 4 establishes our motivational result that the Weyl groupoid of a regular MASA inclusion is Hausdorff if and only if there is a conditional expectation.

Our main results are in Sections 5, 6, and 7. In Section 5 we introduce and describe the Cartan envelope. Theorem 5.2 shows uniqueness and minimality of the Cartan envelope and characterizes its existence in terms of essential inclusions and also the unique faithful pseudo-expectation property. Section 6 provides some interesting structural consequences of the unique pseudo-expectation property and we propose Conjecture 6.13 as possible characterizations for regular inclusions with the unique pseudo-expectation property. Example 6.10 provides a negative answer to [16, Question 5]. Finally, Section 7 contains a main result, Theorem 7.24, which associates a twist to each compatible cover for an inclusion (C,D). A consequence of this result is description of the twist associated to the Cartan envelope of a regular inclusion. The results of Section 7 are refinements and improvements of the results contained in Section 8 of our preprint [14].

Acknowledgments

We thank Jon Brown, Allan Donsig, Ruy Exel, Adam Fuller, Sarah Reznikoff and Vrej Zarikian for helpful conversations. We also thank the referee for several helpful suggestions and alerting us to a number of typographic errors.

Section snippets

General preliminaries

Throughout the paper, unless otherwise stated, all C-algebras will be assumed unital, and a C-subalgebra A of the C-algebra B will usually be assumed to contain the identity of B. We will often use the notation (B,A) to indicate that A is a unital C-subalgebra of the C-algebra B. For any C-algebra A, we let U(A) be the unitary group of A.

We recall some terminology and notation. For a Banach space A, we use A# for the Banach space dual; likewise when u:AB is a bounded linear mapping

Additional preliminaries: twists and their C-algebras

In this section, we collect some generalities on twists and the (reduced) C-algebras associated to them for use in Sections 4 and 7. Much of this material can be found in [18] or [19]. The description of groupoids is standard, and we include it for notational purposes. Associated to a twist are a line bundle and its conjugate, and a well-known construction associates a reduced C-algebra to each. These C-algebras are anti-isomorphic. This unsurprising fact is doubtless known, but because we

Conditional expectations and Hausdorff Weyl groupoids

The purpose of this section is to establish Theorem 4.4, which shows that the groupoid of germs G for the Weyl semigroup associated to the regular MASA inclusion (C,D) is Hausdorff if and only if there exists a (necessarily unique) conditional expectation E:CD. The fact that G is Hausdorff in the presence of a conditional expectation was shown by Renault in [18]. We include a sketch of an alternate argument establishing this fact in the proof of Theorem 4.4. To our knowledge, the converse is

Cartan envelopes

It follows from [15, Theorem 5.7] that a regular inclusion (C,D) regularly embeds into a Cartan pair (C1,D1) precisely when the ideal Rad(C,D)={xC:ρ(xx)=0ρS(C,D)} vanishes. In general, the construction given in the proof of [15, Theorem 5.7] produces a Cartan pair (C1,D1) having little connection with the original pair (C,D). An example of this behavior is the inclusion (C[0,1],CI), where the Cartan pair into which (C[0,1],CI) embeds is (C[0,1],C[0,1]). However in some cases, the image of C

Some consequences of the unique pseudo-expectation property

This section has two main purposes. One goal is to show that regular inclusions with the unique pseudo-expectation property are covering inclusions. This gives a class of regular inclusions for which the results of Section 7 can be used for descriptions of packages via twists. The second goal is to record some consequences of the unique pseudo-expectation property for regular inclusion with the hope that they may be useful in obtaining a characterization of the unique pseudo-expectation

Twists associated to a regular covering inclusion

Let (C,D) be an inclusion for which there exists a compatible cover FS(C,D) for Dˆ. The main result of this section, Theorem 7.24, shows that given this data, there exists a twist (Σ,G) and a regular ⁎-homomorphism θ of (C,D) into the inclusion (Cr(Σ,G),C(G(0))) such that F is identified with the unit space of G and θ(C) is dense in Cr(Σ) with respect to a pointwise topology. The kernel of θ is the ideal KF={xC:ρ(xx)=0 for all ρF}, and when this ideal vanishes, (Cr(Σ,G),C(G(0)),θ) is a

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    The author is grateful for the support of the University of Nebraska's NSF ADVANCE grant #0811250 in the completion of this paper. This work was also partially supported by a grant from the Simons Foundation (#316952 to David Pitts).

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