C * - algebras and Mackey ' s theory of group representations

The subjects of C*-algebras and of unitary representations of locally compact groups are both approximately 50 years old. While it was known from the start that these subjects are related, it was noL originally appreciated just how close the relationship is, especially in the case of Mackey's theory of induced representations and of representations of group extensions. U G is a (locally compact) group, then the unitary representation theory of G is precisely that of its group C*-algebra C· (G). And if G contains a normal subgroup N, then C' (G) may be identified with a "twisted crossed product" C'(G, C'(N), 7") of C'(N) by GIN. The aim of the "Mackey machine" is to describe the representation theory of such a crossed product in terms of knowledge of C· (N) and of the action of G on it by conjugation. For many years results were confined to describing the irreducible representations or primitive ideals of the crossed product as a set, or at best as a topological space. To do the latter is already usually difficult. However I will try to focus on cases where one can describe the "fine structure" of the Mackey machine, that is, the way the crossed product sits over its spectrum, and possible twisting in this structure. I am honored to have heen asked to participate III this Special Session on "C*-Algebras: 1943-1993," especially since two of the originators of the subject of C*-algebras, Israel Gelfand and Irving Segal, are participating in this Session. The year 1993 marks the fiftieth anniversary of two landmark papers in functional analysis, [GeNl], which effectively originated the study of C*-algebras, and [GeR], which marked the beginning of much of the theory of unitary representations of locally compact groups. Accordingly, this seems to be an appropriate time for reflection on the history of both subjects, and especially of the relationship between them. While this paper contains a discussion of this history, it is rather personal in taste and far from complete, so I hope the reader will forgive me for the lapses and inaccuracies which will undoubtedly be present. 1991 Mathematics Subject Classification. 46L05; Secondary 22025, 22030, 01A60, OlA6S. This work was partially supported by NSF Grant # OMS-90-02642. This paper is in final form and no other version of it will be submitted for publication elsewhere. 151 © 1994 American Mathematical Society 0271-4132/94 11.00 + '.25 per page 152 JONATHAN ROSENBERG §1. Some Early History The subject of group representations on complex vector spaces dates back to work of G. Frobenius and I. Schur starting around 1897, but was originally viewed as an area of algebra and confined to the study of finite groups. The famous work of Peter and Weyl [PeW] showed that the theory could be extended, with relatively minor changes, to compact groups, where it could be used for harmonic analysis in the same way that Fourier series are used for harmonic analysis on the circle. But the study of unitary representations of non-compact, non-commutative locally compact groups came much later, starting with the famous study by Wigner [Wi] of representations of the "inhomogeneous Lorentz group" or Poincare group, the semi direct product SO(3, 1) K JR3,1. This paper marked the first appearance (for non-compact groups) of what would later come to be known as the "Mackey machine." In particular, three features of Wigner's work are worthy of mention: the study of the orbits of a group on the dual of a normal subgroup, construction of irreducible representations by unitary induction, and the need to study projective unitary representations, here viewed as "double-valued" representations, in other words, representations of a double covering group. Wigner did not develop any of these topics from a general point of view; in fact, each of the three was the subject of a more complete study later, the first two by Mackey ([M5] and [M3, M4]) and the third by Bargmann [Bar2]. Furthermore, had Wigner wanted to systematically study representations of semi-direct product groups, it would have been natural for him to have begun with simpler examples, such as the Heisenberg group or the 'ax + b' group of affine motions of the line, first studied by Gelfand and Naimark [GeN2] a few years later. Rather, it is fair to say that Wigner was more interested in possible physical applications of the Poincare group than in the theory of unitary representations for its own sake, and he developed his tools in an ad hoc manner as he needed them. The first substantial evidence that the unitary representations of locally compact groups were a subject worth developing systematically thus dates to the famous Gelfand-Raikov Theorem [GeR]' published in 1943. Gelfand and Raikov showed, using the theory of positive-definite functions, that an arbitrary locally compact group has enough irreducible unitary representations to separate points. However, they did not indicate methods for actually constructing these representations. The first major developments in this direction were the classifications of the irreducible unitary representations of the 'ax + b' group by Gelfand and Naimark [GeN2], of the Lorentz group by Gelfand-Naimark [GeN3] and Bargmann [Barl]' and of SL(2, JR) by Bargmann [Barl]. Since my intention is not to give a complete history of the development of the subject of group representations, I will not go into further details; the interested reader can consult the historical sections of [M6] and of Kirillov's book [Ki], as well as the broader histories in [M7] and in [H02]. Instead, I would like to focus for a moment on the early history of the relationC*-ALGEBRAS AND MACKEY'S THEORY 153 ship between unitary representation theory and the theory of C*-algebras. In the same journal (Mat. Sbornik) and in the same year (1943) that the GelfandRaikov paper appeared, Gelfand and Naimark published their famous characterization [GeNl) of what later came to be known (largely through the influence of I. Segal) as C·-algebras. While the details of their axiomatization were later simplified, what is important about their work was the accomplishment of giving an abstract algebraic characterization of those Banach *-algebras, or Banach algebras with involution (they called these "normed *-rings" at the time) which are isometrically *-isomorphic to a norm-closed *-algebra of operators on a complex Hilbert space. Among the class of Banach algebras, C* -algebras are important for two reasons: they are general enough to serve as models for all phenomena that can be described in terms of Hilbert space operators, yet "rigid" enough to be in many cases classifiable. The key to this rigidity may with hindsight be found in the original Gelfand-N aimark characterization, but was first made explicit in the famous papers of Gelfand-Naimark [GeN4) and Segal [52) describing the "GNS" construction of representations of C*-algebras from positive linear functionals, or states. In these two papers, new proofs were given of the Gelfand-Raikov Theorem, via the device of relating unitary representations of a locally compact group to representations of a suitable group algebra, first proposed in [51]. The key observation is that if one fixes a left Haar measure dx on a locally compact group G, then L1 (G) becomes a Banach algebra under convolution, which if G is finite is just the usual group ring CG. Furthermore, L1(G) has a'natural isometric (conjugate-linear) involution f 1--+ f*, where .6. the modular function (the Radon-Nikodym derivative of left Haar measure with respect to right Haar measure, normalized to be 1 at the identity element). It was undoubtedly clear to Gelfand and Naimark from the beginning that the subjects of C*-algebras and of unitary representations of locally compact groups a.re closely related. After all, several of the most interesting examples of von Neumann algebras (or in the original terminology, "rings of operators") constructed by Murray and von Neumann in the mid-30's came from unitary representations of groups. But full exploitation of C*-algebra techniques in describing unitary representations of groups was slow in coming. The proof of the Gelfand-Raikov Theorem in [GeN4J, basically the same as the one later printed in Naimark's influential book [NJ, depends on the fact, already pointed out in [51 J, that there is a natural bijection (preserving such features as irreducibility) between strongly continuous unitary representations of a locally compact group G and non-degenerate *-representations (representations sending the involution * of the algebra to the adjoint operation on Hilbert space operators) of the L1-algebra £1(G). Segal's proof in [52] is similar, but already begins to rely implicitly on the rigidity properties of C*-algebras, stated as separate theorems in [53]: any quotient of a C*-algebra by a closed two-sided ideal is 154 JONATHAN ROSENBERG again a C*-algebra, in fact in a unique way, and any (*-preserving) homomorphism of C*-algebras is automatically continuous with closed range. Thus Segal introduces what in modern terminology would be called C;(G), the reduced C*algebra of a locally compact group G: the norm closure of the image of LICG) in B(L2(G», the bounded operators on U(G), via the action of LI(G) on L2CG) by left convolution. The Gelfand-Raikov Theorem is then deduced from the general fact, as applied to C;( G), that any C*-algebra has a faithful family of irreducible *-representations. Many of the early papers on unitary representations of locally compact groups, such as the papers of Mautner, Godement, and Segal on decomposition theory and Plancherel formulas, relied in some way on the correspondence between group representations and algebra representations. But as far as I can tell, the first published statement of the equivalence of unitary representations with representations of a certain C*-algebra associated with the group seems to be found in the following statement of Kaplansky in [K2], buried in the proof of Theorem 7 in that paper. The footnotes are my own parenthetical comments. Let G be a locally compact group, A its Ll-algebra1 , B the result of re-norming A by assigning to every element the sup of its norms in all possible *-representations , and C the completion of B in this new norm. Then C is a C*-algebra, and it is known that there is a 1-1 correspondence between *-representations3 of C and (strongly continuous) unitary representations of G. In modern terminology, the algebra C described by Kaplansky is called the Cfull or universal) group C*-algebra, C*CG). The algebra C;CG) that had been introduced by Segal is a quotient of C· (G); it was shown much later that the two coincide if and only if G is amenable (see [P, Theorem 4.21]). The assignment G .,..... C· C G) is not quite a functor in the usual sense because of the technical complication that G does not sit inside C* C G) unless G is discrete, and thus in general a homomorphism G --> H does not induce a *-homomorphism C*CG) --> C*CH). However, one has functoriality under the class of group homomorphisms generated by quotient maps and embed dings of open subgroups. It is also convenient from the modern point of view (dating from [J]) to note that G naturally embeds in the multiplier algebra of C' C G), M (C' C G», which is the largest C*-algebra in which C*CG) can be embedded as a (closed two-sided) essential ideal. Thus the universal property of C* (G) shows that a homomorphism G --> H induces a *-homomorphism C*(G) --> M(C*(H». The correspondence between *-representations of C· (G) and unitary representations of G is now easily described: any non-degenerate *-representation 11" of C·CG) extends canonically to a representation ;r of M(C*(G» on the same 1 Kaplansky writes L J . 2Though not mentioned here, this sup is finite since all such representations are nOTmdecreasing. 3 Assumed non-degenerate. C'"-ALGEBRAS AND MACKEY'S THEORY 155 Hilbert space, and the associated unitary representation of G is then obtained by restricting 7r to G. In the other direction, given a unitary representation 7r of G, it defines a non-degenerate *-representation of U(G) by f 1-+ f f(x)7r(x) dx, which by the universal property of C*(G) must factor through C*(G). This correspondence is even better than the way Kaplansky described it; essentially all questions one can ask about the unitary representation theory of G are equivalent to questions about the structure of C* (G). Around the year 1950, the structure theory of C*-algebras began to take on an independent flavor of its own, related to, but distinct from, the study of von Neumann algebras and direct integral decompositions of representations on the one hand (a subject that occupied many operator algebraists at the time), and the study of more general classes of Banach algebras (typified by U-algebras or function algebras) on the other hand. However, the subject was a bit slow in getting off the ground; up until the time of publication of Dixmier's book [Dil, which brought the subject to the attention of a larger audience, the structure theory of C*-algebras was never studied by more than a handful of people at a time. The first major development in this study of the structure theory of C*algebras came in Kaplansky's paper [KIl, in which the notions of CCR and GCR C*-algebras (later called liminal and postliminal algebras in [Dil) were first introduced. The postliminal algebras were later shown in a deep paper of Glimm ([GIl]-see also [Di, §9]) to coincide with the type I C*-algebras, that is, with the C*-algebras all of whose factor representations are of Murray-von Neumann type I. However, at the time, the importance of Kaplansky's work was to point out that at least there are some type I C*-algebras with an interesting structure theory. At the same time, Kaplansky noticed that many C*-algebras can be described as algebras of continuous sections of "fields" or "bundles" of algebras, a point of view that was to be taken up again la:ter in greater detail by Fell [F2] and by Dixmier and Douady ([DiD], [Di, §lO]). And in [K2l, he noted that liminal algebras must play an important role in unitary representation theory, as could be seen from the recently annou.nced theorems of Harish-Chandra [He] implying that the C*-algebras of semisimple Lie groups are liminal.4 At about this same time, George Mackey began to systematize the unitary representation theory of general locally compact groups, with the aim of finding an algorithm for classifying the irreducible unitary representations of a group G having a closed normal subgroup N, assuming that one already has sufficient information about Nand G / N. Precedents for this work could already be seen in the papers of Frobenius and Schur, in an influential paper of Clifford [ell (working out the details of such an algorithm in the case of finite-dimensional 4 A simplified version of (a slightly more restrictive version of) Harish-Chandra's theorem appears in [Di. 15.5.6). We will discuss a refinement below in §3, in connection with Research Problem 2. 156 JONATHAN ROSENBERG representations of discrete groups), in the work of Stone and von Neumann on uniqueness of irreducible representations of the Heisenberg commutation relations [vNl. and in the famous work of Wigner which we have already discussed [Wi]. Mackey's main idea was to give a means, known as (unitary) induction, for constructing a unitary representation Ind~ 11" of a locally compact group G out of a unitary representation 11" of a closed subgroup H, and to give a criterion, known as the Imprimitivity Theorem, for recognizing when a representation is induced from H. The construction was largely given in [MIl. with further details (such as the "induction in stages" theorem Ind~ Ind~ 11" ~ Ind~ 11") in [M4, I]. Mackey's generalization in [M2] of the Stone-von Neumann Theorem basically amounted to a special case of the Imprimitivity Theorem with H = {I}. He noted incidentally that this result could be used to recover the classification of the irreducible unitary representations of the 'ax + b' group by Gelfand and Naimark [GeN2]. Mackey then proceeded in [M3] and [M4, I] to show how to decompose tensor products of induced representations, or restrictions of induced representations to closed subgroups. Finally, in [M4, I] he gave an analogue of the Frobenius Reciprocity Theorem, which in the case of finite groups basically says that induction of representations from H to G and restriction of representations from G to H are adjoint functors. (Some of this work was duplicated in papers of Mautner such as [Ma] , that were written at roughly the same time.) Mackey's methods were largely measure-theoretic; as a result he had to work hard to overcome technical difficulties that arise from dealing with equations that are only true almost everywhere. No C*-algebraic methods appear in Mackey's early papers (though there is plenty of use of direct integral decomposition of von Neumann algebras). Mackey's algorithm for classifying the irreducible unitary representations of a group G having a "regularly embedded" closed type I normal subgroup N, known in the trade as the "Mackey machine," appears in his paper [M5]. Again Mackey's methods were largely measure-theoretic, which forced him to restrict attention to representations of second-countable groups on separable Hilbert spaces, though this is not much of a restriction since almost all cases of interest satisfy these conditions. We will describe the Mackey machine in informal terms, since another version (due primarily to Rieffel and Green) will be given in the next section. Suppose 11" is an irreducible unitary representation of G and N is a closed type I normal subgroup. The restriction 1I"IN of 11" to N is then a unitary representation of N, though usually not irreducible. By the decomposition theory for representations of type I C* -algebras or groups, it splits up in an essentially unique way as a direct integral t fJ m1f (p)p df.l" (p) of irreducible representations of N, each occurring with a certain multiplicity, with respect to some measure f.l1f on !V, the space of equivalence classes of irreducible unitary representations of N. Because N is normal in G, G acts on unitary representations of N by the formula 9 . p( n) = p(g-l ng), and since clearly g. 1I"IN ~ 1I"IN for all 9 E G, it follows that J1" is G-quasi-invariant and the multiplicity function m" C"-ALGEBRAS AND MACKEY'S THEORY 157 is G-invariant. Furthermore, since 7r was assumed irreducible, it is easy to see that the measure Jl7r is ergodic and that the multiplicity function m". is constant almost everywhere. Mackey calls the class of Jl1l" a quasi-orbit. In cases where the action of G on N is nice enough (this is the "regular embedding" condition), Jl1f will be supported on a single G-orbit, say G· p, and we may assume that where m1f is now a constant and dg is a quasi-invariant measure on the homogeneous space G/Gp . This basically verifies that the hypothesis of Mackey's Imprimitivity Theorem applies to 7r, and thus 7r is induced from an irreducible unitary representation of G p whose restriction to N is a multiple of p. One can also go in the other direction: given p E IV, one can compute its stabilizer G p in G (for the action of G on N), and if (Y is an irreducible unitary representation of G p whose restriction to N is a multiple of p, then Indg (Y will be an irreducible p unitary representation of G "lying over" the G-orbit of p. In this way, 6 is partitioned into "fibers" over the various orbits of G on N, and this concludes the first part of the Mackey machine. The second part of the Mackey machine gives a mechanism for determining all irreducible unitary representations of G p whose restriction to N is a multiple of p, and in particular for showing that this set is non-empty for any pEN. This is where the theory of projective representations comes in. By definition of G p , if 9 E G p , then 9 . P ~ p, where "~" denotes unitary equivalence. In other words, if 9 E Gp , there is a unitary operator Ug on the Hilbert space 'lip where p is acting, such that p(g-lng) = Ug-1p(n)Ug for all n EN. It is also clear from the fact that p is a representation that if 9 EN, we may take Ug = p(g). Since p is irreducible, Schur's Lemma says that Ug is uniquely defined up to a scalar, and thus for any 9 and g' in G, Uggi must agree with UgUgl up to a scalar. Thus 9 ...... Ug is a projective unitary representation of G p on 'lip which extends p on N. In general, there is a cohomological obstruction, the Mackey obstruction, to our being able to choose the Ug's so that Uggi = Ug Ugl, in other words, to our being able to extend p to an actual unitary representation of Gp on 'lip. This obstruction is defined as a certain member of a cohomology group H2( G p/ N, 'If), the second cohomology of the "little group" G p/ N with coefficients in the circle group, where since G p is a second-countable locally compact group one uses a version of group cohomology for such topological groups, cohomology with Borel co chains. More precisely, Mackey shows that one can choose the Ug's to be constant on cosets of N and to vary "measurably", so that Ugg , = w(g, g')UgUgl for some Borel measurable function w : (Gp/N) x (Gp/N) --+ 'If. (Here 9 denotes the N -coset of g.) The function w satisfies the cocycle identity, and its cohomology class [w] is independent of the choice of the Ug's. One can extend p to an actual unitary representation of G p on 'lip if and only if this 158 JONATHAN ROSENBERG Mackey obstruction vanishes. When G is a semi direct product N )<l Hand N is abelian, so that 'lip is one-dimensional and p is a one-dimensional character, we actually have p(g-l ng) = p( n) for all n E Nand 9 E H p, so one may choose Ug to be identically 1 on H p == G pi N and the Mackey obstruction vanishes. But in general, any class in H2( G pi N, 1I') can occur as a Mackey obstruction for some