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Superstrings, Geometry, Topology, and C*-algebras, Volume 81This page intentionally left blank
American Mathematical Society Providence, Rhode Island
PURE MATHEMATICS Proceedings of Symposia in
Volume 81
Robert S. Doran Greg Friedman Jonathan Rosenberg Editors
with support from the National Science Foundation, Grant DMS-0735233
2000 Mathematics Subject Classification. Primary 81–06, 55–06, 46–06, 46L87, 81T30.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Library of Congress Cataloging-in-Publication Data
NSF-CBMS Conference on Topology, C∗-algebras, and String Duality (2009 : Texas Christian University)
Superstrings, geometry, topology, and C∗-algebras : NSF-CBMS Conference on Topology, C∗- algebras, and String Duality, May 18–22, 2009, Texas Christian University, Fort Worth, Texas / Robert S. Doran, Greg Friedman, Jonathan Rosenberg, editors.
p. cm. — (Proceedings of symposia in pure mathematics ; v. 81) Includes bibliographical references and index. ISBN 978-0-8218-4887-6 (alk. paper) 1. Algebraic topology—Congresses. 2. Quantum theory—Congresses. 3. Functional analysis—
Congresses. I. Doran, Robert S., 1937– II. Friedman, Greg, 1973– III. Rosenberg, J.(Jonathan), 1951– IV. Conference Board of the Mathematical Sciences. V. National Science Foundation (U.S.) VI. Title.
QA612.N74 2009 530.12—dc22
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10 9 8 7 6 5 4 3 2 1 15 14 13 12 11 10
Functoriality of Rieffel’s Generalised Fixed-Point Algebras for Proper Actions Astrid an Huef, Iain Raeburn, and Dana P. Williams 9
Twists of K-theory and TMF Matthew Ando, Andrew J. Blumberg, and David Gepner 27
Division Algebras and Supersymmetry I John C. Baez and John Huerta 65
K-homology and D-branes Paul Baum 81
Riemann-Roch and Index Formulae in Twisted K-theory Alan L. Carey and Bai-Ling Wang 95
Noncommutative Principal Torus Bundles via Parametrised Strict Deformation Quantization
Keith C. Hannabuss and Varghese Mathai 133
A Survey of Noncommutative Yang-Mills Theory for Quantum Heisenberg Manifolds
Sooran Kang 149
From Rational Homotopy to K-Theory for Continuous Trace Algebras John R. Klein, Claude L. Schochet, and Samuel B. Smith 165
Distances between Matrix Algebras that Converge to Coadjoint Orbits Marc A. Rieffel 173
Geometric and Topological Structures Related to M-branes Hisham Sati 181
Landau-Ginzburg Models, Gerbes, and Kuznetsov’s Homological Projective Duality
Eric Sharpe 237
The Conference Board of the Mathematical Sciences (CBMS) hosted a regional conference, funded by the National Science Foundation, during the week of May 18–22, 2009, entitled Topology, C*-algebras, and String Duality at Texas Christian University in Fort Worth, Texas. The principal lecturer was Jonathan Rosenberg of the University of Maryland, whose conference lectures have been published in Volume 111 of the CBMS’s Regional Conference Series in Mathematics. In addition to Professor Rosenberg’s lectures, the conference featured talks by fifteen other speakers on topics related to his lectures and the general theme of the conference. The purpose of this volume is to collect the contributions of these speakers and other participants. All papers have been carefully refereed and will not appear elsewhere. At first sight these papers, which are highly interdisciplinary, may appear unrelated. To provide direction and historical context for the reader, a technical introduction describing how the various papers fit together in a natural way has been written by Professor Rosenberg. It appears as the first article in the volume.
The editors express their sincere gratitude and thanks to the speakers for their beautiful talks and their willingness to spend many hours writing them up so that the results would be available to the larger scientific community. In addition, we ac- knowledge the hard work and help of the referees. We thank the Conference Board of the Mathematical Sciences and the National Science Foundation for their sup- port via NSF Grant DMS-0735233. We thank Sergei Gelfand, Christine Thivierge, and the dedicated staff at the American Mathematical Society for their efforts in publishing these proceedings. Finally, we thank Texas Christian University and all the participants who helped ensure a wonderfully successful conference.
Robert S. Doran Greg Friedman
Jonathan Rosenberg
1 . G re g F ri ed m a n
2 . J o n a th a n R o se n b er g
3 . R o b er t D o ra n
4 . S h il in
Y u
5 . M a tt h ew
A n d o 6 . H u ic h i H u a n g
7 . V a rg h es e M a th a i 8 . E ft o n P a rk
9 . R u th
G o rn et
1 0 . R is h n i R a tn a m
1 1 . S te fa n M en d ez -D
ie z 1 2 . S o o ra n K a n g
1 3 . P ed
1 4 . J a cq u es
D is tl er
1 5 . P h u C h u n g
1 6 . S eu n g h u n H o n g 1 7 . V a le n ti n D ea co n u
1 8 . D o ri n D u m it ra sc u
1 9 . A la n C a re y
2 0 . R eb
R ez a
A b el
S a ti
2 5 . J o n S jo g re n
26 . E ri c S h ar p e
27 . K en
28 . J oh
29 . M ic h ae l T se n g
30 . J o d y T ro u t 3 1 . P et er
B o u w k n eg t
32 . W an
35 . M ar k T o m fo rd e
36 . M ag n u s G off
en g 3 7 . C la u d e S ch o ch et
3 8 . B ru ce
D o ra n
3 9 . J o h n H u er ta
4 0 . J a co b S h o tw
el l
4 1 . D a n ie l P a p e
4 2 . J a m es
W es t 4 3 . J o n a th a n B lo ck
4 4 . L o re d a n a C iu rd a ri u
4 5 . L o re n S p ic e
4 6 . A n n a S p ic e
4 7 . D a n a W
il li a m s
4 8 . B ra x to n C o ll ie r
N o t p ic tu re d : P au
l B a u m , A le x a n d er
A . K a tz , S n ig d h ay an
M a h a n ta , S co tt
N o ll et
Conference Attendees
Loredana Ciurdariu University Politechnic of Timisoara
Braxton Collier University of Texas at Austin
Valentin Deaconu University of Nevada, Reno
Jacques Distler University of Texas at Austin
Robert Doran Texas Christian University
Bruce Doran Accenture
Daniel Freed University of Texas at Austin
Greg Friedman Texas Christian University
Magnus Goffeng Chalmers University of Technology and University of Gothenburg
Ruth Gornet University of Texas at Arlington
Pedram Hekmati Royal Institute of Technology
Nigel Higson Pennsylvania State University
Seunghun Hong Pennsylvania State University
Huichi Huang University at Buffalo
John Huerta University of California, Riverside
Sooran Kang University of Colorado at Boulder
Alexander A. Katz St. John’s University
Snigdhayan Mahanta Johns Hopkins University
Varghese Mathai University of Adelaide
Stefan Mendez-Diez University of Maryland
Scott Nollet Texas Christian University
Wang Qingyun Washington University in St. Louis
Rishni Ratnam Australian National University
Letty Reza University of Houston
Ken Richardson Texas Christian University
Marc A. Rieffel University of California, Berkeley
Jonathan Rosenberg University of Maryland
Hisham Sati Yale University
Eric Sharpe Virginia Tech
Jon Sjogren Air Force Office of Scientific Research
John Skukalek Pennsylvania State University
Anna Spice University of Michigan
Loren Spice University of Michigan
Mark Tomforde University of Houston
Jody Trout Dartmouth College
Dana Williams Dartmouth College
Conference Speakers
Matthew Ando Twisted Generalized Cohomology and Twisted Elliptic Cohomology
Paul Baum Equivariant K Homology
Jonathan Block Homological Mirror Symmetry and Noncommutative Geometry
Peter Bouwknegt The Geometry Behind Non-geometric Fluxes
Alan Carey Twisted Geometric Cycles
Jacques Distler Geometry and Topology of Orientifolds I
Dan Freed Geometry and Topology of Orientifolds II
Nigel Higson The Baum-Connes Conjecture and Parametrization of Representations
Sooran Kang The Yang-Mills Functional and Laplace’s Equation on Quantum Heisenberg manifolds
Varghese Mathai The Index of Projective Families of Elliptic Operators
Marc Rieffel Vector Bundles for “Matrix Algebras Converge to the Sphere”
Hisham Sati Fivebrane Structures in String Theory and M-theory
Claude Schochet An Update on the Unitary Group
Eric Sharpe GLSMs, Gerbes, and Kuznetsov’s Homological Projective Duality
Dana Williams Proper Actions on C*-algebras
Proceedings of Symposia in Pure Mathematics
Jonathan Rosenberg
Abstract. The papers in this volume are the outgrowth of an NSF-CBMS Regional Conference in the Mathematical Sciences, May 18–22, 2009, orga- nized by Robert Doran and Greg Friedman at Texas Christian University. This introduction explains the scientific rationale for the conference and some of the common themes in the papers.
During the week of May 18–22, 2009, Robert Doran and Greg Friedman orga- nized a wonderfully successful NSF-CBMS Regional Research Conference at Texas Christian University. I was the primary lecturer, and my lectures have now been published in [29]. However, Doran and Friedman also invited many other mathe- maticians and physicists to speak on topics related to my lectures. The papers in this volume are the outgrowth of their talks.
The subject of my lectures, and the general theme of the conference, was highly interdisciplinary, and had to do with the confluence of superstring theory, algebraic topology, and C∗-algebras. While with “20/20 hindsight” it seems clear that these subjects fit together in a natural way, the connections between them developed almost by accident.
Part of the history of these connections is explained in the introductions to [11] and [17]. The authors of [11] begin as follows:
Until recently the interplay between physics and mathematics fol- lowed a familiar pattern: physics provides problems and mathe- matics provides solutions to these problems. Of course at times this relationship has led to the development of new mathematics. . . . But physicists did not traditionally attack problems of pure mathematics.
This situation has drastically changed during the last 15 years. Physicists have formulated a number of striking conjectures (such as the existence of mirror symmetry) . . . . The basis of the physi- cists’ intuition is their belief that underlying quantum field theory
2010 Mathematics Subject Classification. Primary 81-06; Secondary 55-06, 46-06, 46L87, 81T30.
Key words and phrases. string theory, supersymmetry, D-brane, C∗-algebra, crossed product, topological K-theory, twisted K-theory, classifying space, noncommutative geometry, Landau- Ginzburg theory, Yang-Mills theory.
Partially supported by NSF grant DMS-0805003.
c©0000 (copyright holder)
c©2010 American Mathematical Society
and string theory is a (as yet undiscovered) self-consistent mathe- matical framework.
Of course this was written over 10 years ago. In the last 10 years, the same principle has been borne out time and again. As far as the subject matter of this volume is concerned, there are a few key developments from the last 35 years that one can point to, that played an essential role:
(1) The Baum-Douglas [2, 3] and Kasparov [19, 20] approaches to (respec- tively) topological and analytic K-homology, and the realization that these theories are naturally isomorphic.
(2) The “Second Superstring Revolution” around 1995. Geometric objects, known as D-branes, were shown to play a fundamental role in string the- ory, and as time went on, it was realized that they naturally carry vector bundles and topological charges (see for example [23, 31, 22, 32]), liv- ing in K-theory or K-homology (or still more complicated generalized homology theories).
(3) The development of Connes’ theory of “noncommutative differential ge- ometry,” epitomized by the book [9], and the gradual acceptance of non- commutative geometry as a natural tool in quantum field theories.
(4) The invention of “twisted K-theory,” and the realization that it has a natural realization in terms of continuous trace C∗-algebras (see [28, 1, 18]).
My own interest in combining string duality with topology and noncommutative geometry followed a rather circuitous route. A classical theorem of Grothendieck and Serre [15] computed the Brauer group BrC(X) for X a finite complex, and found that it is isomorphic to the torsion subgroup of H3(X,Z). In the 1970’s, Phil Green [14] worked out a more general theory of the Brauer group of C0(X), for X a locally compact Hausdorff space. Green had the idea to drop all technical con- ditions on X and to allow continuous-trace algebras with infinite fiber dimension, not just classical Azumaya algebras, so as to get an isomorphism of the Brauer group BrC0(X) with all of H3(X,Z), not just with its torsion subgroup. (When X is a finite complex, it doesn’t matter what cohomology theory one uses, but for general locally compact spaces, Cech cohomology is appropriate here.) Now it so happens that Donovan and Karoubi [12] had used classical Azumaya algebras to define twisted K-theory with torsion twistings, so Green’s idea of using more gen- eral continuous-trace algebras to replace Azumaya algebras made possible defining twisted K-theory of X with arbitrary twistings fromH3(X,Z). In [27, §6] I pointed this out and explained how to generalize the Atiyah-Hirzebruch spectral sequence to make this twisted K-theory somewhat computable. But for the most part, the idea just sat around for a while since nobody had any immediate use for it.
A number of years later, Raeburn and I [24] happened to study crossed products of continuous trace algebras by smooth actions, and we discovered the following interesting “reciprocity law” [24, Theorem 4.12]:
Theorem 1. Let p : X → Z be a principal T-bundle, where T = R/Z is the circle group. Also assume X and Z are second-countable, locally compact Haus- dorff, with finite homotopy type. Let H ∈ H3(X,Z) and let A = CT (X,H) be the corresponding stable continuous-trace algebra with Dixmier-Douady class H. Then the free action of T = R/Z on X lifts (in a unique way, up to exterior equivalence)
to an action of R on A inducing the given action of R/Z on A = X. The crossed product AR is again a stable continuous-trace algebra A# = CT (X#, H#), with p# : X# → Z again a principal T-bundle. Furthermore, the characteristic classes of p and p# are related to the Dixmier-Douady classes H and H# by
p!(H) = [p#], (p#)!(H #) = [p],
where p! and (p#)! are the Gysin maps of the circle bundles.
At the time, Raeburn and I regarded this entirely as a curiosity, and we certainly didn’t expect any physical applications. A bit later [28], I continued my studies of continuous-trace algebras and twisted K-theory, but I still didn’t expect any physical applications.
Much to my surprise, I discovered many years later that my studies of continu- ous-trace algebras and twisted K-theory were starting to show up in the physics literature in papers such as [7] and [4]. In fact, twisted K-theory seemed to be exactly the mathematical framework needed to studying D-brane charges in string theory. Not only that, but the “reciprocity law” of [24] for continuous-trace algebras associated to circle bundles also showed up in physics, as the recipe for topology change and H-flux change in T-duality [5, 6]. Since that time, there has been a fruitful continuing interaction between the subjects of string theory, topology, and C∗-algebras, an interaction that led to the organization of the CBMS conference at TCU in 2009.
With this as background, I can now explain how the various papers in this volume fit together. The papers of Baum and of Carey and Wang deal with D- brane charges in K-homology and twisted K-homology, a natural continuation of the combination of items (1), (2), and (4) on the list of key developments above (page 2). Baum’s paper deals with the extension to the twisted case of the Baum- Douglas approach to topological K-homology. While Baum does not go into the associated physics, D-branes in type II string theories come with precisely the structures he is discussing, and thus produce “topological charges” in the twisted K-homology of spacetime. The paper of Carey and Wang goes into more detail on the same subject, and also discusses a Riemann-Roch theorem in twisted K-theory. Carey and Wang explain how D-brane charges in twisted K-theory arise in both type II and type I string theories.
The papers of Ando and Sati deal with roughly the same theme as those of Baum and Carey-Wang, but in a somewhat generalized context. Ando explains (from the point of view of a stable homotopy theorist) how to construct twisted generalized cohomology theories in general, and then specializes to the cases of twisted K, twisted elliptic cohomology, and twisted TMF. TMF [16], topological modular forms, is a version of elliptic cohomology that seems to play an impor- tant role in M-theory, the “master” theory that gives rise (on reduction from 11 dimensions to 10) to the five superstring theories. Sati’s paper concentrates on the physics side of the same topic, and explains how the physics of M-branes (which play the same role in M-theory that the D-branes play in string theory) leads to twisted String and Fivebrane structures. (These are higher-dimensional analogues of Spin and Spinc structures.) Sati also discusses the kinds of orientation conditions that arise for branes in F-theory [30], a 12-dimensional theory that is supposed to reduce to M-theory in certain circumstances.
Two of the papers in this volume, by Kang and by Baez and Huerta, deal with Yang-Mills gauge theory and its connection with noncommutative geometry. The basic Yang-Mills action on a spacetime manifold M is (up to a scalar factor) − ∫ M
Tr(F ∧ F ), where F is the curvature of a connection on a principal G-bundle over M . Here G is some Lie group which depends on the details of the theory; for example, in the “standard model” of particle physics it is SU(3) × SU(2) × U(1). Physicists have known for some time [8] that in some circumstances one can make this action supersymmetric, by adding in a fermionic term of the form (again, up to a constant factor) ψ, /∂ψ, where ψ is a spinor field and /∂ is the Dirac operator. However, this only appears to work in three, four, six and ten dimensions. The paper of Baez and Huerta gives an explanation for this fact in terms of the fact that division algebras over R only occur in dimensions 1, 2, 4, and 8 (where one has the reals, complexes, quaternions, and octonions, respectively). Kang’s paper deals with noncommutative Yang-Mills in the sense of Connes and Rieffel [10], where the basic Yang-Mills action becomes −Tr({Θ,Θ}), where Θ is the curvature 2-form for a connection on a finitely generated projective module (the natural analogue of a vector bundle) over the smooth subalgebra of some C∗-algebra A. Connes and Rieffel took A to be Aθ, the irrational rotation algebra generated by two unitaries U and V with UV = e2πiθV U . Kang considers the somewhat more complicated case of the “quantum Heisenberg manifold” in the sense of Rieffel [25]; this is a deformation quantization of the algebra of functions on a Heisenberg nilmanifold.
Just to relate the papers of Baez-Huerta and Kang to the rest of the volume, it is perhaps worthwhile to explain how Yang-Mills and super-Yang-Mills are related to string theory. There are two interconnected ties between the two subjects. On the one hand, as we mentioned already, D-branes naturally carry certain Chan-Paton vector bundles; on these there is a natural Yang-Mills action. In addition, there is a duality, known as the AdS/CFT correspondence, between type IIB string theory on S5×AdS5 (AdS5 is anti-de Sitter space, a 5-dimensional Lorentz manifold with a metric of constant negative curvature) and 4-dimensional super-Yang-Mills on S4
in the large-N limit [21]. The paper of Sharpe deals with Landau-Ginzburg models, a class of mod-
els which were originally constructed to model superconductivity, but which have turned out to be extremely useful for superstring theory as well. A Landau- Ginzburg model in string theory describes propagation of strings on a noncompact spacetime (always a complex manifold) with a holomorphic superpotential W , of- ten having a degenerate critical point. One of the results explained in Sharpe’s paper is that A-twisted correlation functions in the Landau-Ginzburg model on
X = Tot(E∨ π−→ B), E → B a holomorphic vector bundle, with W = p · π∗s, p a tautological section of π∗E∨ and s a holomorphic section of E , should match correlation functions in the nonlinear sigma model on {s = 0}. Since the com- plex geometry of the Landau-Ginzburg spacetime is usually quite different from the one which the sigma model lives, sometimes one gets interesting relations in enumerative algebraic geometry which are hard to explain directly.
The papers of Hannabuss-Mathai, Reiffel, Klein-Schochet-Smith, and an Huef- Raeburn-Williams all deal with various aspects of C∗-algebraic noncommutative geometry. Several of them also have ties to quantum physics and to topology. Rief- fel’s paper gives explicit examples of sequences of matrix algebras with dimensions going to ∞ whose “proximity” in a rather precise but technical sense goes to 0.
This sort of calculation is motivated by the use of “matrix models” to approximate quantum field theories on spaces with complicated geometry.
The paper of Klein-Schochet-Smith computes the rational homotopy type of the group U(A) of unitary elements in the Azumaya algebra A of sections of a bundle of matrix algebras Mn over a compact space X. This turns out to be independent of what Azumaya algebra one chooses (so that one might as well take A = C(X)⊗Mn), basically because the Brauer group of C(X) is torsion, and the authors are only interested in rational information. This paper also computes the map πj(U(A))⊗Q → Kj(A)⊗Q; this gives explicit information on the stable range for rationalized topological K-theory of X. The paper of Hannabuss and Mathai deals with Rieffel’s theory of strict deformation quantization [26] and the theory of noncommutative principal bundles due to Echterhoff, Nest, and Oyono-Oyono [13]. The main theorem of this paper is that for every such bundle with a suitable smooth structure A∞(X), there is a principal torus bundle T → X and a corresponding strict deformation quantization σ of C∞
fibre(Y ) (the continuous functions on Y that are fibrewise smooth), so that A∞(X) ∼= C∞
fibre(Y )σ. Finally, the paper by an Huef, Raeburn, and Williams talks about functoriality
issues in the theories of C∗-crossed products and fixed-point algebras for proper actions. Issues like this come up when one tries to use C∗-algebraic noncommutative geometry to study the geometry of spacetime in various physical theories.
We hope the diversity of the papers in this volume will give the reader some idea of the breadth and vitality of the current interplay between superstring theory, geometry/topology, and noncommutative geometry.
I would like to thank Robert Doran and Greg Friedman again for their excel- lent work in organizing the conference. In addition, all three of us would like to thank the Conference Board of the Mathematical Sciences and the National Science Foundation for their financial support. NSF Grant DMS-0735233 supported the conference, and NSF Grant DMS-0602750 supported the entire Regional Confer- ence program. Finally, we would like to thank the American Mathematical Society for encouraging the publication of this volume in the Proceedings of Symposia in Pure Mathematics series.
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Department of Mathematics, University of Maryland, College Park, MD 20742-
Proceedings of Symposia in Pure Mathematics
Functoriality of Rieffel’s Generalised Fixed-Point Algebras for Proper Actions
Astrid an Huef, Iain Raeburn, and Dana P. Williams
Abstract. We consider two categories of C∗-algebras; in the first, the iso-
morphisms are ordinary isomorphisms, and in the second, the isomorphisms are Morita equivalences. We show how these two categories, and categories of dynamical systems based on them, crop up in a variety of C∗-algebraic con-
texts. We show that Rieffel’s construction of a fixed-point algebra for a proper action can be made into functors defined on these categories, and that his Morita equivalence then gives a natural isomorphism between these functors and crossed-product functors. There are interesting applications to nonabelian
duality for crossed products.
Let α be an action of a locally compact group G on a C∗-algebra A. In [38], Rieffel studied a class of proper actions for which there is a Morita equivalence between the reduced crossed product Aα,rG and a generalised fixed-point algebra Aα sitting inside the multiplier algebra M(A). Rieffel subsequently proved that α is proper whenever there is a free and proper G-space T and an equivariant embedding : C0(T ) → M(A) [39, Theorem 5.7]. In [15], inspired by previous work of Kaliszewski and Quigg [13], it was observed that Rieffel’s hypothesis says precisely that ((A,α), ) is an object in a comma category of dynamical systems. It therefore becomes possible to ask questions about the functoriality of Rieffel’s construction, and about the naturality of his Morita equivalence.
These questions have been tackled in several recent papers [15, 7, 8], which we believe contain some very interesting results. In particular, they have substantial applications to nonabelian duality for C∗-algebraic dynamical systems. However, these papers also contain a confusing array of categories and functors. So our goal here is to discuss the main categories and explain why people are interested in them. We will then review some of the main results of the papers [13, 15, 7, 8], and try to explain why we find them interesting.
In all the categories of interest to us, the objects are either C∗-algebras or dynamical systems involving actions or coactions of a fixed group on C∗-algebras.
2000 Mathematics Subject Classification. 46L55. This research was supported by the Australian Research Council and the Edward Shapiro
fund at Dartmouth College.
c©0000 (copyright holder)
c©2010 American Mathematical Society
But when we decide what morphisms to use, we have to make a choice, and what we choose depends on what sort of theorems we are interested in. Loosely speaking, we have to decide whether we want the isomorphisms in our category to be the usual isomorphisms of C∗-algebras, or to be Morita equivalences. We think that, once we have made that decision, there is a “correct” way to go forward.
We begin in §1 with a discussion of commutative C∗-algebras; since Morita equivalence does not preserve commutativity, it is clear that in this case we want isomorphisms to be the usual isomorphisms. However, even then we have to do something a little odd: we want the morphisms from A to B to be homomorphisms : A → M(B). Once we have the right category, we can see that operator algebraists have been implicitly working in this category for years. The motivating example for Kaliszewski and Quigg was a duality theory for dynamical systems due to Landstad [19], and our main motivation is, as we said above, to understand Rieffel’s proper actions. We discuss Landstad duality in §2. In §3, we discuss its analogue for crossed products by coactions, which is due to Quigg [30], and how this makes contact with Rieffel’s theory of proper actions.
We begin §4 by showing how the search for naturality results leads us to a dif- ferent category C* of C∗-algebras in which the morphisms are based on right-Hilbert bimodules. Categories of this kind have been around much longer, and [2, 3], for example, contain a detailed discussion of how imprimitivity theorems provide nat- ural isomorphisms between functors with values in C*. In §5, we discuss a theorem from [7] which says that Rieffel’s Morita equivalences give a natural isomorphism between a crossed-product functor and a fixed-point-algebra functor. This power- ful result implies, for example, that the version in [9] of Mansfield imprimitivity for arbitrary subgroups is natural. We finish with a brief survey of one of the main results of [8] which uses an approach based on Rieffel’s theory to establish induction-in-stages for crossed products by coactions.
1. The Category C*nd and Commutative C∗-algebras
In our first course in C∗-algebras, we learned that commutative unital C∗- algebras are basically the same things as compact topological spaces. To make this formal, we note that the assignment X → C(X) is the object map in a contravari- ant functor C from the category Cpct of compact Hausdorff spaces and continuous functions to the category CommC*1 of unital commutative C∗-algebras and unital homomorphisms (which for us are always ∗-preserving); the morphism C(f) asso- ciated to a continuous map f : X → Y sends a ∈ C(Y ) to a f ∈ C(X). Then the Gelfand-Naimark theorem implies that the functor C is an equivalence of cat- egories. (This result goes back to [25], and we will go into the details of what it means in the proof of Theorem 2 below.)
The Gelfand-Naimark theorem for non-unital algebras says that commutative C∗-algebras are basically the same things as locally compact topological spaces. However, it is not so easy to put this version in a categorical context, and in doing so we run into some important issues which are very relevant to problems involving crossed products and nonabelian duality. So we will discuss these issues now as motivation for our later choices.
There is no doubt what the analogue of the functor C does to objects: it takes a locally compact Hausdorff space X to the C∗-algebra C0(X) of continuous functions a : X → C which vanish at infinity. However, there is a problem with
morphisms: composing with a continuous function f : X → Y does not necessarily map C0(Y ) into C0(X). For example, consider the function f : R → R defined by f(x) = (1 + x2)−1: any function a ∈ C0(R) which is identically 1 on [0, 1] satisfies a f = 1, and hence a f does not vanish at infinity. One way out is to restrict attention to the category in which the morphisms from X to Y are the proper functions f : X → Y for which inverse images of compact sets are compact, and then on the C∗-algebra side one has to restrict attention to the homomorphisms : A → B such that the products (a)b span a dense subspace of B. In [28], Pedersen does exactly this, and calls these proper homomorphisms. It turns out, though, that there is a very satisfactory way to handle arbitrary continuous functions between locally compact spaces, in which we allow morphisms which take values in Cb(X).
A homomorphism of one C∗-algebra A into the multiplier algebra M(B) of another C∗-algebra B is called nondegenerate if
(A)B := span{(a)b : a ∈ A, b ∈ B} is all of B. (This notation is suggestive: the Cohen factorisation theorem says that everything in the closed span factors as (a)b.) We want to think of the nondegen- erate homomorphisms : A → M(B) as morphisms from A to B. Every nonde- generate homomorphism extends to a unital homomorphism : M(A) → M(B) (see [35, Corollary 2.51], for example); the extension has to satisfy (m)((a)b) = (ma)b, and hence the nondegeneracy implies that there is exactly one such exten- sion, and that it is strictly continuous.
The following fundamental proposition is implicit in a number of earlier works, including [43, 44], [41] and [13, §1].
Proposition 1. There is a category C*nd in which the objects are C∗-algebras, the morphisms from A to B are the nondegenerate homomorphisms from A to M(B), and the composition of : A → M(B) and ψ : B → M(C) is ψ := ψ . The isomorphisms in this category are the usual isomorphisms of C∗-algebras.
Proof. It is easy to check that the composition ψ : A → M(C) is non- degenerate, and hence defines a morphism in C*nd. Since ψ is a homomorphism
from M(A) to M(C) which extends ψ , it must be the unique extension ψ . Thus if θ : C → M(D) is another nondegenerate homomorphism, we have
θ (ψ ) = θ (ψ ) = θ (ψ ) = (θ ψ)
= (θ ψ) = (θ ψ) = (θ ψ) , and composition in C*nd is associative. The identity maps idA : A → A, viewed as homomorphisms into M(A), satisfy idA = idM(A), and hence have the properties one requires of the identity morphisms in C*nd. Thus C*nd is a category, as claimed.
For the last assertion, notice first that every isomorphism is trivially nonde- generate, and hence defines a morphism in C*nd, which is an isomorphism because it has an inverse. Conversely, suppose that : A → M(B) and ψ : B → M(A) are inverses of each other in C*nd, so that ψ = idA and ψ = idB. Using first the nondegeneracy of ψ and then the nondegeneracy of , we obtain
(A) = (ψ(B)A) = (ψ(B))(A) = B(A) = B.
Thus has range B, and since ψ|B = ψ, we have ψ = idA. The same arguments show that ψ = idB, so is an isomorphism in the usual sense.
If f : X → Y is a continuous map between locally compact spaces and a ∈ C0(Y ), then a f is a continuous bounded function which defines a multiplier of C0(X). For every b in the dense subalgebra Cc(X), we can choose a ∈ Cc(Y ) such that a = 1 on f(supp b), and then b = (a f)b, so C0(f) : a → a f is a
nondegenerate homomorphism from C0(Y ) to M(C0(X)); the extension C0(f) to Cb(X) = M(C0(X)) is again given by composition with f . We now have a functor C0 from the category LCpct of locally compact spaces and continuous maps to the full subcategory CommC*nd of C*nd whose objects are commutative C∗-algebras. This functor has the properties we expect:
Theorem 2. The contravariant functor C0 : LCpct → CommC*nd is an equiva- lence of categories.
Proof. To say that C0 is an equivalence means that there is a functor G : CommC*nd → LCpct such that C0 G and G C0 are naturally isomorphic to the identity functors. To verify that it is an equivalence, though, it suffices to show that every object in CommC*nd is isomorphic to one of the form C0(X), which is exactly what the Gelfand-Naimark theorem says, and that C0 is a bijection on each set Mor(X,Y ) of morphisms (see [21, page 91]). Injectivity is easy: since C0(Y ) separates points of Y , af = ag for all a ∈ C0(Y ) implies that f(x) = g(x) for all x ∈ X. For surjectivity, we suppose that : C0(Y ) → C0(X) is a nondegenerate homomorphism. Then for each x ∈ X, the composition εx with the evaluation map is a homomorphism from C0(Y ) to C, and the nondegeneracy of implies that εx is non-zero. Since ε : y → εy is a homeomorphism of Y onto the maximal ideal space of C0(Y ), there is a unique f(x) ∈ Y such that εx = εf(x), and
f = ε−1 ∗ ε is continuous. The equation εx = εf(x) then says precisely that = C0(f).
The result in [21, page 91] which we have just used in the proof of Theorem 2 is a little unnerving to analysts. (Well, to us, anyway.) Its proof, for example, makes carefree use of the axiom of choice. So it is perhaps reassuring that in the situation of Theorem 2, there is a relatively concrete inverse functor Δ which takes a commutative C∗-algebra A to its maximal ideal space Δ(A). (We say “relatively concrete” here because the axiom of choice is also used in the proof that the Gelfand transform is an isomorphism.) The argument on page 92 of [21] shows that, once we have chosen isomorphisms ηA : A → C(Δ(A)) for every commutative C∗-algebra A, there is exactly one way to extend Δ to a functor in such a way that
η := {ηA : A ∈ Obj(CommC*nd)}
is a natural isomorphism. If we choose ηA : A → C0(Δ(A)) to be the Gelfand transform, then the functor Δ takes a morphism : A → M(B) to the map Δ() : ω → ω . So we have the following naturality result:
Corollary 3. The Gelfand transforms {ηA : A ∈ Obj(CommC*nd)} form a natu- ral isomorphism between the identity functor on CommC*nd and the composition C0Δ.
Of course, modulo the existence of the isomorphisms ηA, which is the content of the (highly non-trivial) Gelfand-Naimark theorem, this result can be easily proved directly: we just need to check that for every morphism : A → M(B) the following
A ηA
2. Crossed Products and Landstad Duality
Although the category C*nd has only been studied recently, nondegenerate ho- momorphisms have been around for years. For example, the unitary representa- tions U : G → U(H) of a locally compact group G on a Hilbert space H are in one-to-one correspondence with the nondegenerate representations πU of the group algebras L1(G) or C∗(G) on H. In this context, “nondegenerate” usually means that the elements πU (a)h span a dense subspace of H, but this is equivalent to the nondegeneracy of πU as a homomorphism into B(H) = M(K(H)). More gener- ally, if u : G → UM(B) is a strictly continuous homomorphism into the unitary group of a multiplier algebra, then there is a unique nondegenerate homomorphism πu : C∗(G) → M(B), called the integrated form of u, from which we can recover u by composing with a canonical unitary representation kG : G → UM(C∗(G)). The composition here is taken in the spirit of the category C*nd: it is the composition in the usual sense of the extension of πu to M(C∗(G)) with kG. We say that kG is universal for unitary representations of G.
One application of this circle of ideas which will be particularly relevant here is the existence of the comultiplication δG on C∗(G), which is the integrated form of the unitary representation kG ⊗ kG : G → UM(C∗(G)⊗ C∗(G)). Thus δG is by definition a nondegenerate homomorphism of C∗(G) into M(C∗(G)⊗ C∗(G)). Its other crucial property is coassociativity: (δG ⊗ id) δG = (id⊗ δG) δG, where the compositions are interpreted as being those in the category C*nd.
Now suppose that α : G → AutA is an action of a locally compact group G on a C∗-algebra. Nondegeneracy is then built into the notion of covariant representation of the system: a covariant representation (π, u) of a dynamical system (A,G, α) in a multiplier algebraM(B) consists of a nondegenerate homomorphism π : A → M(B) and a strictly continuous homomorphism u : G → UM(B) such that π(αt(a)) = utπ(a)u
∗ t . The crossed product is then, either by definition [33] or by theorem
[42, 2.34–36], a C∗-algebra A α G which is generated (in a sense made precise in those references) by a universal covariant representation (iA, iG) of (A,G, α) in M(Aα G). Each covariant representation (π, u) in M(B) has an integrated form π u which is a nondegenerate homomorphism of A α G into M(B) such that π = (π u) iA and u = (π u) iG.
The crossed product Aα G carries a dual coaction α, which is the integrated form of iG ⊗ kG : G → UM((A α G) ⊗ C∗(G)). This is another nondegenerate homomorphism, and the crucial coaction identity (α⊗ id) α = (id⊗ δG) α again has to be interpreted in the category C*nd. (Makes you wonder how we ever managed without C*nd.)
There is another version of the crossed-product construction which can be more suitable for spatial arguments, and which is particularly important for the issues we discuss in this paper. For any representation π : A → B(Hπ), there is a regular representation (π, U) of (A,G, α) on L2(G,Hπ) such that (π(a)h)(r) =
π(α−1 r (a))(h(r)) and λsh(r) = h(s−1r) for h ∈ L2(G,Hπ). The reduced crossed
product Aα,rG is the quotient of AαG which has the property that every πλ factors through a representation of A α,r G, and then π λ is faithful whenever π is [42, §7.2]. The reduced crossed product is also generated by a canonical covariant representation (irA, i
r G), and the dual coaction α factors through a coaction
αn : Aα,r G → M((Aα,r G)⊗ C∗(G)) characterised by
(1) αn irA(a) = irA(a)⊗ 1 and αn irG(s) = irG(s)⊗ kG(s).
This coaction is called the normalisation of α, and is in particular normal in the sense that the canonical map jAG of Aα,rG into M((Aα,rG)αG) is injective (see Proposition A.61 of [3]).
Kaliszewski and Quigg’s motivation for working in the category C*nd came from the following characterisation of the C∗-algebras which arise as reduced crossed products.
Theorem 4 (Landstad, Kaliszewski-Quigg). Suppose that B is a C∗-algebra and G is a locally compact group. Then there is a dynamical system (A,G, α) such that B is isomorphic to A α,r G if and only if there is a morphism π : C∗(G) → M(B) in C*nd and a nondegenerate (see Remark 6 below) normal coaction δ : B → M(B ⊗ C∗(G)) such that
(2) (π ⊗ id) δG = δ π. In [13] the authors say that this result follows from a theorem of Landstad
[19], and it is certainly true that most of the hard work is done by Landstad’s result. But we think it is worth looking at the proof; those who are not interested in the subtleties of coactions should probably skip to the end of the proof below. We begin by stating Landstad’s theorem in modern terminology.
Theorem 5 (Landstad, 1979). Suppose that B is a C∗-algebra and G is a locally compact group. Then there is a dynamical system (A,G, α) such that B is isomorphic to Aα,rG if and only if there are a strictly continuous homomorphism u : G → UM(B) and a reduced coaction δ : B → M(B ⊗ C∗
r (G)) such that
(b) δ(B)(1⊗ C∗ r (G)) = B ⊗ C∗
r (G).
The “reduced coaction” appearing in Landstad’s theorem is required to have slightly different properties from the full coactions which we use elsewhere in this paper, and which are used in [3] and [13], for example. A reduced coaction on B is an injective nondegenerate homomorphism of B into M(B ⊗ C∗
r (G)) rather than M(B ⊗ C∗(G)), and it is required to be coassociative with respect to the comultiplication δrG on C∗
r (G).
Remark 6. Nowadays, the second condition (b) in Theorem 5 is usually ab- sorbed into the assertion that δ is a coaction. Everyone agrees that for δ to be a coaction δ(B)(1⊗C∗
r (G)) must be contained in B⊗C∗ r (G), and Landstad described
the requirement of equality as “nondegeneracy”, which in view of our emphasis on C*nd has turned out to be unfortunate terminology. Coactions of amenable or discrete groups are automatically nondegenerate in Landstad’s sense, and dual coactions are always nondegenerate. We therefore follow modern usage and assume that all coac- tions satisfy (b), or its analogue in the case of full coactions. (So (b) can now be deleted from Theorem 5 and the word “nondegenerate” from Theorem 4.)
Proof of Theorem 4. For B = Aα,r G, we take δ = αn and π = πirG . The
second equation in (1) implies that
(π ⊗ id) δG(kG(s)) = π ⊗ id(kG(s)⊗ kG(s)) = irG(s)⊗ kG(s)
= αn π(kG(s)) for all s ∈ G, which implies (2).
Now suppose that there exist π and δ as described. Then we define u := π krG, and consider the reduction δr of δ, which since δ is normal is just δr := (id⊗πλ)δ. Now we compute:
δr(us) = (id⊗ πλ) δ(π(krG(s))) = id⊗ πλ δ π(krG(s)) = id⊗ πλ π ⊗ id δG(krG(s)) = π ⊗ πλ(k
r G(s)⊗ kG(s))
= π krG(s)⊗ λs = us ⊗ λs.
Thus u and δr satisfy the hypotheses of Landstad’s theorem (Theorem 5), and we can deduce from it that B is isomorphic to a reduced crossed product.
Kaliszewski and Quigg then made two further crucial observations. First, they recognised that there is a category of coactions associated to C*: the objects in C*coactnd(G) consist of full coactions δ on C∗-algebras B, and the morphisms from (B, δ) to (C, ε) are nondegenerate homomorphisms : B → M(C) such that ( ⊗ id) δ = ε . Then (2) says that the homomorphism π in Corollary 4 is a morphism in C*coactnd(G) from (C∗(G), δG) to (B, δ). Second, they knew that for every object a and every subcategory D in a category C there is a comma category a ↓ D in which objects are morphisms f : a → x in C from a to objects in D, and the morphisms from (x, f) to (y, g) are morphisms h : x → y in D such that h f = g. Thus Landstad’s theorem identifies the reduced crossed products as the C∗-algebras which can be augmented with a coaction δ and a homomorphism π to form an object in the comma category (C∗(G), δG) ↓ C*coactnnd(G), where C*coact
n nd(G) is the full subcategory of normal coactions.
The main results in [13] concern crossed-product functors defined on the cat- egory C*actnd(G) whose objects are dynamical systems (A,G, α) and whose mor- phisms : (A,α) → (B, β) are nondegenerate homomorphisms : A → M(B) such that αs = βs for s ∈ G (where yet again the composition on the right is taken in C*nd). The following theorem is Theorem 4.1 of [13].
Theorem 7 (Kaliszewski-Quigg, 2009). There is a functor CPr from C*actnd(G) to the comma category (C∗(G), δG) ↓ C*coactnnd(G) which takes the object (A,α) to (Aα,r G, αr, irG), and this functor is an equivalence of categories.
Landstad’s theorem, in the form of Theorem 4, says that CPr is essentially surjective: every object in the comma category is isomorphic to one of the form CPr(A,α) = A α,r G. Thus Theorem 7 can be viewed as an extension of Land- stad’s theorem, and Kaliszewski and Quigg call it “categorical Landstad duality for actions”. They also obtain an analogous result for full crossed products.
3. Proper Actions and Landstad Duality for Coactions
Quigg’s version of Landstad duality for crossed products by coactions [30] is also easy to formulate in categories based on C*nd. Suppose that δ is a coaction of G on a C∗-algebra C, and let wG denote the function s → kG(s), viewed as a multiplier
of C0(G,C∗(G)). A covariant representation of (C, δ) in a multiplier algebra M(B) consists of nondegenerate homomorphisms π : C → M(B) and μ : C0(G) → M(B) such that
(π ⊗ id) δ(c) = μ⊗ id(wG)(π(c)⊗ 1)μ⊗ id(wG) ∗ for c ∈ C,
where, as should seem usual by now, the composition is interpreted in C*nd. The crossed product CδG is generated by a universal covariant representation (jC , jG) in M(C δ G), in the sense that products jC(c)jG(f) span a dense subspace of
C δ G. The crossed product carries a dual action δ such that δs(jC(c)jG(f)) = jC(c)jG(rts(f)), where rt is defined by rts(f)(t) = f(ts). Quigg’s theorem identifies the C∗-algebras which are isomorphic to crossed products by coactions.
Theorem 8 (Quigg, 1992). Suppose that G is a locally compact group and A is a C∗-algebra. There is a system (C, δ) such that A is isomorphic to C δ G if and only if there are a nondegenerate homomorphism : C0(G) → M(A) and an action α of G on A such that (A,α, ) is an object in the comma category (C0(G), rt) ↓ C*actnd(G).
When A = C δ G, we can take := jG and α := δ, and the hard bit is to prove the converse. This is done in [30, Theorem 3.3]. It is then natural to look for a “categorical Landstad duality for coactions” which parallels the results of [13]. However, triples (A,α, ) of the sort appearing in Theorem 8 had earlier (that is, before [13]) appeared in important work of Rieffel on proper actions, and it has proved very worthwhile to follow up this circle of ideas in Rieffel’s context. To explain this, we need to digress a little.
If α : G → AutA is an action of a compact abelian group, then information about the crossed product can be recovered from the fixed point algebra Aα, and, more generally, from the spectral subspaces
Aα(ω) := { a ∈ A : αs(a) = ω(s)a } for ω ∈ G.
A fundamental result of Kishimoto and Takai [16, Theorem 2] says that if the spectral subspaces are large in the sense that Aα(ω)∗Aα(ω) is dense in Aα for every
ω ∈ G, then A α G is Morita equivalent to Aα. There is as yet no completely satisfactory notion of a free action of a group on a C∗-algebra (see [29], for example), but having large spectral subspaces is one example of such a notion.
WhenG is locally compact, the fixed-point algebra is often trivial. For example, if rt is the action of G = Z on R by right translation, then f ∈ C0(R)
rt if and only if f is periodic with period 1, which since f vanishes at ∞ forces f to be identically zero. However, if the orbit space for an action is nice enough, then the algebra of continuous functions on the orbit space can be used as a substitute for the fixed- point algebra. A right action of a locally compact group G on a locally compact space T is called proper if the map (x, s) → (x, x ·s) : T ×G → T ×T is proper. The orbit space T/G for a proper action is always Hausdorff [42, Corollary 3.43], and a classical result of Green [5] says that if the action of G on T is free and proper, then C0(T )rt G is Morita equivalent to C0(T/G) (for this formulation of Green’s result see [42, Remark 4.12]). We want to think of C0(T/G) as a subalgebra of the multiplier algebra M(C0(G)) = Cb(T ) which is invariant under the extensions rts of the automorphisms rts.
In the past twenty-five years, many researchers have investigated analogues of free and proper actions for noncommutative C∗-algebras [34, 38, 4, 24, 10,
39, 11]. Here we are interested in the notion of proper action α : G → AutA introduced by Rieffel [38]. He assumes that there is an α-invariant subalgebra A0
of A with properties like those of the subalgebra Cc(T ) of C0(T ), and that there is an M(A)α-valued inner product on A0. The completion Z(A,α) of A0 in this inner product is a full Hilbert module over a subalgebra Aα of M(A)α, which Rieffel calls the generalised fixed-point algebra for α. The algebra K(Z(A,α)) of generalised compact operators on Z(A,G, α) sits naturally as an ideal E(α) in the reduced crossed product A α,r G [38, Theorem 1.5]. The action α is saturated when E(α) is all of the reduced crossed product. Thus when α is proper and saturated, A α,r G is Morita equivalent to Aα. Saturation is a freeness condition: if G acts properly on T , then rt : G → Aut(C0(T )) is proper with respect to Cc(G), and the action is saturated if and only if G acts freely [23, §3]. On the face of it, though, Rieffel’s bimodule Z(A,α) and the fixed-point algebra Aα depend on the choice of subalgebra A0, and it seems unlikely that Rieffel’s process is functorial.
The connection with our categories lies in a more recent theorem of Rieffel which identifies a large family of proper actions for which there is a canonical choice of the dense subalgebra A0 [39, Theorem 5.7].
Theorem 9 (Rieffel, 2004). Suppose that a locally compact group G acts freely and properly on the right of a locally compact space T , and (A,G, α) is a dynamical system such that there is a nondegenerate homomorphism : C0(T ) → M(A) satisfying rt = α (with composition in the sense of C*nd). Then α is proper and saturated with respect to the subalgebra A0 = (Cc(T ))A(Cc(T )).
Example 10. A closed subgroup H of a locally compact group G acts freely and properly on G, and hence we can apply Theorem 9 to the pair (T,G) = (G,H) and to the canonical map jG : C0(G) → M(C δ G). In this case, highly nontrivial
results of Mansfield [22] can be used to identify the fixed-point algebra (C δ G)δ
with the crossed product Cδ,r (G/H) by the homogeneous space [9, Remark 3.4]. (These crossed products were introduced in [1]; the relationship with the crossed product Cδ| (G/H) by the restricted coaction, which makes sense when H is nor- mal, is discussed in [1, Remark 2.2].) Then Theorem 3.1 of [9] shows that Rieffel’s
Morita equivalence between (C δ G) δ,r H and (C δ G)δ extends Mansfield’s
imprimitivity theorem for coactions to arbitary closed subgroups (as opposed to the amenable normal subgroups in Mansfield’s original theorem [22, Theorem 27] and the normal ones in [12]).
From our categorical point of view, the hypotheses on in Theorem 9 say pre- cisely that (A,α, ) := ((A,α), ) is an object in the comma category (C0(T ), rt) ↓ C*actnd(G). Then Rieffel’s theorem implies that (A,α, ) → Aα is a construction which takes objects in the comma category to objects in the category C*nd. One naturally asks: is this construction functorial? More precisely, is there an analo- gous construction on morphisms which makes (A,α, ) → Aα into a functor from (C0(T ), rt) ↓ C*actnd(G) to C*nd?
This question was answered in [15, §2] using a new construction of Rieffel’s generalised fixed-point algebra. The crucial ingredient is an averaging process E of Olesen and Pedersen [26, 27], which was subsequently developed by Quigg in [31, 32] and used extensively in his proof of Theorem 8. This averaging process E
makes sense on the dense subalgebra A0 = (Cc(T ))A(Cc(T )), and satisfies
(f)E((g)a(h)) =
∫ G
(f)αs((g)a(h)) ds for f, g, h ∈ Cc(T ) and a ∈ A;
the integral on the right has an unambiguous meaning because properness implies that s → f rts(g) has compact support. It is shown in [15, Proposition 2.4] that the closure of E(A0) is a C∗-subalgebra of M(A), which we denote by Fix(A,α, ) to emphasise all the data involved in the construction. It is shown in [15, Propo- sition 3.1] that Fix(A,α, ) and Rieffel’s Aα are exactly the same subalgebra of M(A). If σ : (A,α, ) → (B, β, ψ) is a morphism in the comma category, so that in particular σ is a nondegenerate homomorphism from A to M(B), then the ex- tension σ maps Fix(A,α, ) into M(Fix(B, β, ψ)), and is nondegenerate. (This is Proposition 2.6 of [15]; a gap in the proof of nondegeneracy is filled in Corollary 2.3 of [8].)
Theorem 11 (Kaliszewski-Quigg-Raeburn, 2008). Suppose that a locally com- pact group G acts properly on the right of a locally compact space T . Then the assignments (A,α, ) → Fix(A,α, ) and σ → σ|Fix(A,α,) form a functor from (C0(T ), rt) ↓ C*actnd(G) to C*nd.
To return to the setting of Quigg-Landstad duality, we take (T,G) = (G,G) in this theorem. This gives us a functor Fix from (C0(G), rt) ↓ C*actnd(G) to C*nd. Because the fixed-point algebra Fix(A,α, ) is defined using the same averaging process E as Quigg used in [30, §3], Fix(A,α, ) is the same as the algebra C constructed by Quigg (unfortunately for us, he called it B). So Quigg proves in [30] that
δA(c) = ⊗ πλ(wG)(c⊗ 1)⊗ πλ(wG) ∗
defines a reduced coaction of G on C = Fix(A,α, ), and that A is isomorphic to the crossed product C δA G. An examination of the proof of [31, Theorem 4.7] shows that the similar formula
δfA(c) = ⊗ id(wG)(c⊗ 1)⊗ id(wG) ∗
defines the unique full coaction with reduction δA. The argument on page 2960 of [15] shows that this construction respects morphisms, so that Fix extends to a functor FixG from (C0(G), rt) ↓ C*actnd(G) to C*coact
n nd(G). The following very
satisfactory “categorical Landstad duality for coactions” is Corollary 4.3 of [15].
Theorem 12 (Kaliszewski-Quigg-Raeburn, 2008). Let G be a locally com-
pact group. Then (C, δ) → (C δ G, δ, jG) and π → π id form a functor from C*coact
n nd(G) to (C0(G), rt) ↓ C*actnd(G). This functor is an equivalence of cate-
gories with quasi-inverse FixG.
In fact, this is a much more satisfying theorem than its analogue for actions because we have a specific construction of a quasi-inverse. We would be interested to see an analogous process for Fixing over coactions.
4. Naturality
Now that we have a functorial version Fix of Rieffel’s generalised fixed-point algebra, we remember that the main point of Rieffel’s paper [38] was to construct a Morita equivalence between Aα = Fix(A,α, ) and the reduced crossed product
RCP(A,α, ) := A α,r G. This equivalence is implemented by an (A α,r G) – Fix(A,α, ) imprimitivity bimodule Z(A,α, ). There is another category C* of C∗-algebras in which the isomorphisms are given by imprimitivity bimodules, so it makes sense to ask whether these isomorphisms are natural. Of course, before discussing this problem, we need to be clear about what the category C* is.
If A and B are C∗-algebras, then a right-Hilbert A –B bimodule is a right Hilbert B-module X which is also a left A-module via a nondegenerate homomor- phism of A into the algebra L(X) of bounded adjointable operators on X. (These are sometimes called A –B correspondences.) The objects in C* are C∗-algebras, and the morphisms from A to B are the isomorphism classes [X] of full right- Hilbert A –B bimodules. Every nondegenerate homomorphism : A → M(B) gives a right-Hilbert bimodule: view B as a right Hilbert B-module over itself with b1 , b2B := b∗1b2, and define the action of A by a · b := (a)b. We denote the isomorphism class of this bimodule by []. In [2], it is shown that [] = [ψ] if and only if there exists u ∈ UM(B) such that ψ = (Adu) , so we are not just adding more morphisms to C*nd, we are also slightly changing the morphisms we already have.
If AXB and BYC are right Hilbert bimodules, then we define the composition using the internal tensor product: [Y ][X] := [X ⊗B Y ]. The identity morphism 1A on A is [AAA] = [idA]. Now we can see why we have had to take isomorphism classes of bimodules as our morphisms: the bimodule A⊗AX representing [X]1A = [X][idA] is only isomorphic to X. A similar subtlety arises when checking that composition of morphisms is associative. The details are in [2, Proposition 2.4]. In [2, Proposition 2.6], it is shown that the isomorphisms from A to B in C* are the classes [X] in which X is an imprimitivity bimodule, so that X also carries a left inner product Ax , y such that Ax , y · z = x · y , zB . Similar results were obtained independently by Landsman [17, 18] and by Schweizer [40], and a slightly more general category in which the bimodules are not required to be full as right Hilbert modules was considered in [3].
Theorem 3.2 of [15] says that, for every nondegenerate homomorphism σ : A → M(B), the diagram
(3) Aα,r G [Z(A,α,)]
commutes in C*, which means that
Z(A,α, )⊗Fix(A,α,) Fix(B, β, ψ) and (B β,r G)⊗Bβ,rG Z(B, β, ψ)
are isomorphic as right-Hilbert (Aα,r G) – Fix(B, β, ψ) bimodules. Thus Rieffel’s bimodules (or rather, the morphisms in C* which they determine) implement a nat- ural isomorphism between the functors RCP and Fix from (C0(T ), rt) ↓ C*actnd(G) to C*.
This naturality theorem certainly has interesting applications to nonabelian duality, where it gives naturality for the extension in [9] of Mansfield’s imprimi- tity theorem to closed subgroups (see [15, Theorem 6.2]). However, it is slightly
unsatisfactory: we were forced to change the target category from C*nd to C* be- cause the bimodules Z do not define morphisms in C*nd, but in the diagram (3) we have not fully committed to the change. Our goal in [7] was to find versions of the same functors defined on a category built from C* — that is, ones in which the morphisms are implemented by bimodules — to establish that Rieffel’s Morita equivalence gives a natural isomorphism between these functors, and to apply the results to nonabelian duality. We will describe our progress in the next section.
5. Upgrading to C*
Proposition 3.3 of [2] says that for every locally compact group G, there is a category C*act(G) whose objects are dynamical systems (A,α) = (A,G, α) and whose morphisms are obtained by adding actions to the morphisms of C*. Formally, if (A,α) and (B, β) are objects in C*(G) and AXB is a right-Hilbert bimodule, then an action of G on a right-Hilbert bimodule X is a strongly continuous homomor- phism u of G into the linear isomorphisms of X such that
us(a · x · b) = αs(a) · us(x) · βs(b) and us(x) , us(y)B = βs
( x , yB
) ,
and the morphisms in C*act(G) are isomorphism classes of pairs (X,u). Next we consider a free and proper action of G on a locally compact space
T and look for an analogue of the comma category for the system (C0(T ), rt). The objects are easy: to ensure that Fix is defined on objects, we need to insist that every system (A,α) is equipped with a nondegenerate homomorphism : C0(T ) → M(A) which is rt –α equivariant. We choose to use the semi-comma category C*act(G, (C0(T ), rt)) in which the objects are triples (A,α, ), and the morphisms from (A,α, ) to (B, β, ψ) are just the morphisms from (A,α) to (B, β) in C*act(G). In [7, Remark 2.4] we have discussed our reasons for adding the maps to our objects and then ignoring them in our morphisms, and the discussion below of how we Fix morphisms should help convince sceptics that this is appropriate.
We know how to Fix objects in the semi-comma category C*act(G, (C0(T ), rt)), and we need to describe how to Fix a morphism [X,u] from (A,α, ) to (B, β, ψ). We begin by factoring the morphism [X] in C* as the composition [K(X)XB][κA] of the isomorphism associated to the imprimitivity bimodule K(X)XB with the mor- phism coming from the nondegenerate homomorphism κA : A → M(K(X)) = L(X) describing the left action of A on X (see Proposition 2.27 of [3]). The action u of G on X gives an action μ of G on K(X) such that μs(Θx,y) = Θus(x),us(y), and then κA satisfies κA αs = μs κA. So the morphism [(A,α)(X,u)(B,β)] in C*act(G) fac- tors as [(K(X),μ)(X,u)(B,β)][κA]. Now κA is a morphism in (C0(G), rt) ↓ C*actnd(G) from (A,α, ) to (K(X), μ, κA ), and hence by Theorem 11 restricts to a mor- phism κA| from Fix(A,α, ) to Fix(K(X), μ, κA ). We want to define Fix so that it is a functor, so our definition must satisfy
(4) Fix([X,u]) = Fix([(K(X),μ)(X,u)(B,β)]) Fix([κA]).
Since we don’t want to change the meaning of Fix on morphisms in C*nd, our strategy is to define Fix([κA]) := [κA|], figure out how to Fix imprimitivity bimodules, and then use (4) to define Fix([X,u]).
So we suppose that (A,α, ) and (B, β, ψ) are objects in the semi-comma cat- egory C*act(G, (C0(T ), rt)), and that [X,u] is an equivariant (A,α) – (B, β) im- primitivity bimodule. We emphasise that, because of our choice of morphisms in C*act(G, (C0(T ), rt)), we do not make any assumption relating the actions of
and ψ on X. We let X := {(x) : x ∈ X} be the dual bimodule, and form the linking algebra
L(X) :=
L(u) :=
define an action L(u) of G on L(X) and a nondegenerate homomorphism L of C0(T ) into M(L(X)) which intertwines rt and L(u). Then (L(X), L(u), L) is an object in C*act(G, (C0(T ), rt)), and (reverting to Rieffel’s notation to simplify the formulas) we can form L(X)L(u) := Fix(L(X), L(u), L). It follows quite easily from the construction of Fix in [15, §2] that the diagonal corners in L(X)L(u) are Aα and Bβ, and we define Xu to be the upper right-hand corner, so that
L(X)L(u) =
) ;
with the actions and inner products coming from the operations in L(X)L(u), Xu
becomes an Aα –Bβ-imprimitivity bimodule (see [35, Proposition 3.1]). We now define Fix([X,u]) := [Xu], and use (4) to define Fix in general, as described above.
With this definition, Theorem 3.3 of [7] says:
Theorem 13. Suppose that T is a free and proper right G-space. Then the assignments
(A,α, ) → Fix(A,α, ) and [X,u] → Fix([X,u])
form a functor Fix from the semi-comma category C*act(G, (C0(T ), rt)) to C*.
Proving that Fix preserves the composition of morphisms is surprisingly compli- cated, and involves several non-trivial steps. For example, we needed to show that if
(A,α)(X,u)(B,β) and (B,β)Y(C,γ) are imprimitivity bimodules implementing isomor-
phisms in C*act(G, (C0(T ), rt)), then (X⊗B Y )u⊗v is isomorphic to Xu⊗Bβ Y v as Aα –Cγ imprimitivity bimodules.
It follows from [3, Theorem 3.7] that RCP is a functor from C*act(G, (C0(T ), rt)) to C* which takes a morphism [X,u] to the class of the Combes bimodule [Xu,rG]. We can now state the main naturality result, which is Theorem 3.5 of [7].
Theorem 14. Suppose that a locally compact group G acts freely and properly on a locally compact space T . Then the Morita equivalences Z(A,α, ) implement a natural isomorphism between the functors RCP and Fix from C*act(G, (C0(T ), rt)) to C*.
The proof of Theorem 14 relies on factoring morphisms; then Theorem 3.2 of [15] gives the result for the nondegenerate homomorphism, and standard linking algebra techniques give the other half.
We saw in Example 10 that Rieffel’s Morita equivalence can be used to gener- alise Mansfield’s imprimitivity theorem to crossed products by homogeneous spaces, and we want to deduce from Theorem 14 that this imprimitivity theorem gives a natural isomorphism. To get the imprimitivity theorem in Example 10, we applied Rieffel’s Theorem 9 to a crossed product C δ G. So the naturality result we seek relates the compositions of RCP and Fix with a crossed-product functor.
Suppose as in Example 10 that H is a closed subgroup of a locally compact group G. We know from Theorem 2.15 of [3] that there is a category C*coactn(G) whose objects are normal coactions (B, δ), and whose morphisms are isomorphism classes of suitably equivariant right-Hilbert bimodules. We also know from Theo- rem 3.13 of [3] that there is a functor CP : C*coactn(G) → C*act(H), and adding the canonical map jG makes CP into a functor with values in the comma cate- gory (C0(G), rt |H) ↓ C*act(H). We show in [7, Proposition 5.5] that there is a functor RCPG/H which sends (B, δ) to the crossed product B δ,r (G/H) by the homogeneous space G/H, and that this functor coincides with Fix CP. We saw in
Example 10 that Rieffel’s bimodules Z(Bδ G, δ|H , jG) implement a Morita equiv- alence between (Bδ G)δ|,r H and Bδ,r G/H. Write RCPH for the functor from
C*act(G) to C* sending (C, γ) → C γ|,r H. Then the general naturality result above gives the following theorem, which is Theorem 5.6 of [7].
Corollary 15. Let H be a closed subgroup of G. Then Rieffel’s Morita equiv-
alences Z(Gδ G, δ|H , jG) implement a natural isomorphism between the functors RCPH CP and RCPG/H from C*coactn(G) to C*.
Corollary 15 extends Theorem 4.3 of [3] to non-normal subgroups, and extends Theorem 6.2 of [15] to categories based on C* rather than ones based on C*nd.
6. Induction-in-stages and Fixing-in-stages
Rieffel’s theory of proper actions seems to be a powerful tool for studying systems in the comma or semi-comma category associated to a pair (T,G). Corol- lary 15 is, we think, an impressive first example. As another example, we discuss an approach to induction-in-stages which works through the same general machinery, and which we carried out in [8].
The original purpose of an imprimitivity theorem was to provide a way of recognising induced representations (as in, for example, [20]), and Rieffel’s theory of Morita equivalence for C∗-algebras was developed to put imprimitivity theorems in a C∗-algebraic context [36, 37]. One can reverse the process: a Morita equivalence X between a crossed product CαG and another C∗-algebra B gives an induction process X-Ind which takes a representation of B on H to a representation of C on X ⊗B H, and for which there is a ready-made imprimitivity theorem (see, for example, [6, Proposition 2.1]). The situation is slightly less satisfactory when one has a reduced crossed product, but one can still construct induced representations and prove an imprimitivity theorem.
Mansfield’s imprimitivity theorem, as extended to homogeneous spaces in [9],
gives an induction process IndGG/H from Bδ,r(G/H) to BδG which comes with an imprimitivity theorem. One then asks whether this induction process has the other properties which one would expect. For example, we ask whether we can induct-in-
stages: if we have subgroups H, K and L with H ⊂ K ⊂ L, is Ind G/H G/K(Ind
G/K G/L π)
unitarily equivalent to Ind G/H G/L π? If the subgroups are normal and amenable, then
the induction processes are those defined by Mansfield [22], and induction-in-stages was established in [14, Theorem 3.1]. For non-normal subgroups, not much seems to be known. There are clearly issues: for example, the subgroups H and K have to be normal in L for the three induction processes to be defined.
We tackled this problem in [8] using our semi-comma category. Suppose that (T,G) is as usual, N is a closed normal subgroup of G, and (A,α, ) is an object
in C*act(G, (C0(T ), rt)). Then N also acts freely and properly on T , so we can form the fixed-point algebra FixN (A,α|N , ). The quotient G/N has a natural action αG/N on Aα|N := FixN (A,α|N , ), and the map induces a homomorphism N : C0(T/N) → M(Aα|N ) such that (Aα|N , αG/N , N ) is an object in the semi- comma category C*act(G/N, (C0(T/N), rt)). We prove in [8] that FixN extends to a functor
Fix G/N N : C*act(G,C0(T ), rt) → C*act(G/N, (C0(T/N), rt)),
and that the functors FixG/N FixG/N N and FixG are naturally isomorphic (see [8,
Theorem 4.5]). The first difficulty in the proof is showing that the functor FixN has an equivariant version: because the functor Fix is defined using the factorisation of morphisms, we have to track carefully through the constructions in [7] to make sure that they all respect the actions of G/N .
Applying this result on “fixing-in-stages” with (T,G) = (L/H,K/H), gives the following version of induction-in-stages, which is Theorem 7.3 of [8].
Theorem 16. Suppose that δ is a normal coaction of G on B, and that H, K and L are closed subgroups of G such that H ⊂ K ⊂ L and both H and K are normal in L. Then for every representation π of B δ,r (G/L), the representation
Ind G/H G/K(Ind
G/H G/L π.
Obviously this is not the last word on the subject, and the normality hypotheses on subgroups are irritating. However, Mansfield’s induction process is notoriously hard to work with, and it seems remarkable that one can prove very much at all about an induction process which is substantially more general than his. We think that Rieffel’s theory of proper actions is proving to be a remarkably malleable and powerful tool.
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