Structure for regular inclusions. II: Cartan envelopes, pseudo-expectations and twists
Introduction
In their influential 1977 paper [8], Feldman and Moore showed that the collection of pairs consisting of a Cartan MASA in the separably acting von Neumann algebra is, up to suitable notions of equivalence, equivalent to the family 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 -algebraic analogs. An early attempt was by Kumjian in 1986 [12], who introduced the notion of -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 in a -algebra and gave a method for associating a twist to each such pair . The philosophy is to loosely regard the passage from to 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 . 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 , the topologies on Σ and G are Hausdorff, and the associated twist contains enough of the information about to completely recover . More precisely, Renault shows that if is the unit space of G and denotes the reduced -algebra of , then is a Cartan inclusion isomorphic to the original Cartan inclusion . 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 -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 which are not Cartan because they lack a conditional expectation of onto . A large class of examples of regular MASA inclusions with no conditional expectation which arise from crossed products of abelian -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 using a twist. Indeed, Theorem 4.4 below shows that for a regular MASA inclusion , the associated Weyl groupoid G is Hausdorff if and only if there is a conditional expectation of onto . Thus we are confronted with the problem of whether a suitable coordinatization of such pairs 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 regularly embeds into a -diagonal, or equivalently, when it embeds into a Cartan inclusion. Applying this result produces a Cartan pair into which embeds, but is in general not closely related to the original pair .
To address this issue, we introduce the notion of a Cartan envelope for a regular inclusion , see Definition 5.1. This is the “smallest” Cartan pair into which the original pair can be regularly embedded. We show the Cartan envelope is unique when it exists, and that the image of in is dense in a suitable pointwise topology.
In [15], we introduced the notion of a pseudo-expectation for an inclusion . 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 for which the pseudo-expectation is unique.
The issue of existence of a Cartan envelope for is addressed in Theorem 5.2: we characterize the regular inclusions 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 is a regular inclusion having Cartan envelope . If is the twist associated to , elements of and can be viewed as functions (non-linear in the case of ) on , and by restricting these functions to the image of under the embedding of into , we obtain families of functions on . These restriction mappings are both one-to-one. In this way, may be thought of as a “weak-coordinatization” of , or as a weak form of “spectral analysis” for . 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 in a von Neumann algebra , Aoi's theorem shows that is also a Cartan MASA in any intermediate von Neumann subalgebra , 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 -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 of a given Cartan pair which are “nearly intermediate” in the sense that is an essential subalgebra of .
In [15, Section 4], we introduced the notion of a compatible state for a regular inclusion . The restriction of any compatible state on to is a pure state on , and when the regular inclusion has enough compatible states to cover , it is a covering inclusion. We define the notion of a compatible cover for (see Definition 2.11) and Theorem 7.24 shows that associated to each compatible cover, there is a Hausdorff twist. When 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 -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 . 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 -algebras will be assumed unital, and a -subalgebra of the -algebra will usually be assumed to contain the identity of . We will often use the notation to indicate that is a unital -subalgebra of the -algebra . For any -algebra , we let be the unitary group of .
We recall some terminology and notation. For a Banach space A, we use for the Banach space dual; likewise when is a bounded linear mapping
Additional preliminaries: twists and their -algebras
In this section, we collect some generalities on twists and the (reduced) -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 -algebra to each. These -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 is Hausdorff if and only if there exists a (necessarily unique) conditional expectation . 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 regularly embeds into a Cartan pair precisely when the ideal vanishes. In general, the construction given in the proof of [15, Theorem 5.7] produces a Cartan pair having little connection with the original pair . An example of this behavior is the inclusion , where the Cartan pair into which embeds is . However in some cases, the image of
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 be an inclusion for which there exists a compatible cover for . The main result of this section, Theorem 7.24, shows that given this data, there exists a twist and a regular ⁎-homomorphism θ of into the inclusion such that F is identified with the unit space of G and is dense in with respect to a pointwise topology. The kernel of θ is the ideal , and when this ideal vanishes, is a
References (22)
Unique conditional expectations for abelian -inclusions
J. Math. Anal. Appl.
(2017)A construction of equivalence subrelations for intermediate subalgebras
J. Math. Soc. Jpn.
(2003)- et al.
Intermediate -algebras of Cartan embeddings
Proc. Am. Math. Soc. Ser. B
(2021) - et al.
Graded -algebras and twisted groupoid -algebras
N.Y. J. Math.
(2021) - et al.
Bimodules over Cartan MASAs in von Neumann algebras, norming algebras, and Mercer's theorem
N.Y. J. Math.
(2013) - et al.
Coordinate systems and bounded isomorphisms
J. Oper. Theory
(2008) Inverse semigroups and combinatorial -algebras
Bull. Braz. Math. Soc. (N.S.)
(2008)- et al.
Characterizing groupoid -algebras of non-Hausdorff étale groupoids
- et al.
Ergodic equivalence relations, cohomology, and von Neumann algebras. II
Trans. Am. Math. Soc.
(1977) - et al.
Injectivity and projectivity in analysis and topology
Sci. China Math.
(2011)
Regular embeddings of -algebras in monotone complete -algebras
J. Math. Soc. Jpn.
Cited by (6)
Regular Ideals, Ideal Intersections, and Quotients
2024, Integral Equations and Operator TheoryCharacterizing Groupoid C*-algebras of Non-Hausdorff Étale Groupoids
2022, Lecture Notes in Mathematics