3 OPERATOR THEORY _ ,
14(1985), 249-261 © Copyright by INCREST, 1985
THE TYPE OF THE REGULAR REPRESENTATION OF CERTAIN TRANSITIVE GROUPOIDS
SHIGERU YAMAGAMI
1 INTRODUCTION
Let G be a (not necessarily connected) Lie group If GxG is provided with ’ a groupoid structure in a trivial manner (i.e., for (g, , go), (g1, øz) € GX G, (21, 82) 1 = = (g, 2,), and (g,, go) is composable with (gi, g3) if and only if g, = g, with the resulting composition equal to (g,, 23)), we can use Haar measure to form a convo- lution algebra C2(G x G) and its regular representation on L?(G x G) Then the gene- rated von Neumann algebra is a type I factor and acts on L(G G) as a standard representation Although these facts are all trivial, if we take a quotient of Gx G by a suitable subgroup of GXG, the situation becomes rather complicated and the analysis of its regular representation becomes an interesting problem
In this paper, we will carry out the type analysis of the von Neumann algebra associated with a groupoid of this type More precisely, let H be a closed subgroup of G and let H be a normal subgroup of H which contains the connected component of H We form a closed subgroup D of GxG:
() D={(0N,h;)e HxH; hị1h,c H
Then Ƒ' = đxŒ/Ð has the structure of groupoid induced from that of GxG, which is transitive in the sense that the canonical equivalence relation of the grou- poid is transitive (see [10], for example) We assume that H has a unitary character y, which is invariant under the action of H, ie y(hhA-) = x(A) for he H, he H We obtain a character y, of D defined by
(2) 7a(4, 6) = x(a~"b) for (a,b) € D Let y be a C®-function on GX G which satisfies
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for (g1,82)€ GxG, (a, a@)€ D, and has a compact support modulo D Here Ag ala) = detg,sAda; (G and § denote Lie algebras of G and H respectively) We denote the set of all these g's by YC and equip it with the following <-algebra structure: For ©, Ø;, Ø; € 9Í, ø; # Ø; € 9{ and ø* e 9í are defined by (4) (; # @s)(#¡, g;) = $ dg Ø;(#¡ #)@s(8, #›), GIH (5) 9*(81, 82) = @(, 8) (see [2], for example, for the meaning of & } Furthermore, the following inner GH
product makes % a (unimodular) Hilbert algebra:
(6) (|e) = } dg, dge | (gi, g2)/?
GxG/D
We denote by Y’’ (resp Wl’) the left (resp right) von Neumann algebra of this Hilbert algebra which acts on the Z2-completion L£2(21) of YW
Our first result is a concrete realization of the center of 2’ as a subalgebra of a convolution algebra over H/H (Theorem 1) Next, we work out the type analysis of 2’ under the assumption that the commutator subgroup of H is contained in H; QW’ is of type I or type If according as the coset space H/S is finite or infinite (Theo- rem 2) Here S is a normal subgroup of H defined by S = {a € H; y(aba-*b-4) = 1
for all b € H}
The shortest way to obtain the results is as follows: Take a nowhere-vanish- ing function s on GXG satisfying s(gyh,, Bolte) = x(hzhy)5(g1, 99), (ty, he) © D Then o((g,, g)D, (g, 82)D) = s(g,, g)s(g, 25)8(21, Z)~ defines a 2-cocycle on F and the left von Neumann algebra W’’ is spatially equivalent to the von Neumann algebra M(F,o) generated by the o-left regular representation of I Now define a 2-cocycle o, of H/H by o,(h,H, hgH) = o((1, h,)D, (1, /tg)D) Then by the inclusion H/H 3hH c> +> (1,4)De I, (H/H, o,) is similar to (7, ¢) and then M(H/H, o,) is stably-isomorphic to M(L, c) ((4]) In particular the center of M(H/H, o,) is isomorphic to the center of M(I, o) (a version of Theorem | in this paper) Furthermore, the type analysis of M(HỊH,ø)) 1s given in [7] (see also [1]), so the above Theorem 2 is obtained Although this may be sufficient in the abstract, a much more direct proof is presented in this paper, which has several advantages:
(i) concrete realization of the center of 9É”,
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The last point will be important when one applies (or extends} to compiex polarizations (in geometric quantization) I would like to discuss these applications in a future paper
2 DESCRIPTION OF THE CENTER OF UY”
Main result of this section is a concrete realization of the center of YW’ To do this, we need distributions (more precisely generalized sections of a line bundle) in order to express bounded linear operator on L?(%) in kernel form For this reason, we have restricted ourselves to Lie group, but the result itself may be valid for arbi-
trary locally compact groups To begin with, we define some line bundles over G/H:
We regard G as a principal H-bundle by the right translation, and form a line bundle B = GX, C over G/H, where the action of H on C is given by fi-z = = Ás,u(h)~}!2y(h)z,h e H,z eC We denote by B the conjugate bundle of B (B is constructed as B if we replace the action of H on C by hv z= Aa.w(h)~1?z(h)z)
Since z is H-invariant, for any a € HH, B3 g-z+> Ác u(2)%ga-z 6 B gives rise to a bundle morphism of B and hence HX H acts on the exterior tensor product bundle BEB (this is a line bundle over G/H x G/H with the fibre at (x, y) equal to B,@B,) In particular, restricting the action to the diagonal subgroup of Hx H, we get an action of H Then the quotient B, = BE|B/H is a line bundle over I’, and there is a 1-1 correspondence between elements in 9 and support-compact C™-sections of By oT
Lemma 1 For any L € W”, there is a generalized section le C-°(G/H x G/H; BRB) (= dual space of C°(G/HxXG/H; BRB), see [5] for the information of generalized section) such that
(7) I(xya, x;aXa~!, a~?) = U(x, Xe)
for ae H, and for FEN < C(G/HXG/H; BRIB);
(8) (LE\(x, 5 x2) = \ l(x,, x)F(v, x;)
x€G/H
(both sides should be regarded as elemenis in C"®(G|HxGIH; BbEIB))
Comments (i) In (7), the action of a € H on/ is through the bundle morphism of B described above
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Proof By Schwartz's kernel theorem, ZØ(/3U) is imbedded into C "(FP xT; B,XB,) On the other hand, C-™(r™xTI; B;X%B,) is identified with
H = (9 € C°(G/HXGHxG;)HXG;H; BE BXBXB):
(9)
` : —1 ~ - > ° ‘yy, ‘
@(xiø ; JìỚn; Xe › eGo) (Qj ;ới 1, ay an ay 1) = oy, > Fas Xo, 32) for ays Qy € HN, so we have a continuous linear map x of 2(1?@M)) ino C~®(G'H::G:H»‹G¡Hx
x G/H; BRIBE BEB), where A(L7(Q0) is equipped with the weak operator topology and the space of generalized sections is topologized by the weak* topology Let 6 be a generalized section of BDgB defined by
(10) (0, f> = f(x, x)
xE€G/H
‘for fe C2(G/H x G/H; Bf) B) and let 1 be a linear mapping of C-°(G/AxG/H; Bix}B) into C-°(G/H x G/H x G/H x G/H; BE) BE) BR)B) defined by
(1p), Y15 Xa, Vo) = POX, Xz) @S(W,, Yo) €
(11)
c5, ©B,,@B, @B,, = B, OB, @B.,@By,
Then : is a topological imbedding of C-°(G/HxG/H; BEB) into a closed sub-
space £ of C°-°(G/H x G/H X G/H x G/H; BRIBRIBEIB) Since xQ c #' n Z by
), (4), and since # \n 4 is a closed subspace, we have x(W) c Z'n.# Now set /=i~1ox(7) for a given £ € W'’ Then / satisfies (8) by construction and (7) follows from
q2) ôŒa, x;đ)(4a~1,a~}) = ô(ị,x;), ae H
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Proof In (8) and (14), let L = R;
(15) ly, X)f(%, Xe) = \ F(x,, x}(x x;)
J
xEG H „c@G H
for FEW < C(GjH»G;H; BXB) Take x) € GiH and an open neighborhood U of xy such that Ua, 1 Uag # © (a,, ap € H) implies ay'a, € H If we choose F so that the support of F (considered as a section of BB) is contained in (J (Ua Ua), then the right hand side of (15) vanishes when x, ¢\J Ua, and a€H a€H \ \ Í(x), x)F(x; x;)@Œ;) = + cỗ: x,€G'H r = \ Mx, x) \ F(x, Xo) P(X.) x€U x,€G'H for p € C&(U, B), because the support of x > F(x, Xe)p(%_) is contained in *,€G/H U In view of the density of pe F(x, 0G] in C.(U, B), we have, from x,€G/H these facts, that /(x,, x.) = 0 if x,¢JUa and x,e€U Since U can be chosen aGH
arbitrarily small, we obtain the desired result Z
LEMMA 3 Let X be a C®-manifold with a nowhere vanishing C® measure dx and k(x, x’) be a distribution on XXX with its support contained in {(x,x)€@ XXX; x eX} Suppose hat CẸŒ(X) 3 š t> (Kệ (x) = dx' k(x,x)š(x) vields a bounded
x
linear operator in L?(X) Then there exists a bounded measurable function f on X such that (KE)(x) = f(x)E(x)
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By applying this lemma to / in Lemma 2, we can find 1, © L©(G/H) for every a2¢€H which satisfies
{16) loxn=Z4(A)-U, for heH,
IC, Xa) fly) folX9) = (44, )€GIH x G/H q?) => \ I(x)q()(72(GX3)a~1) 2acH:H xeŠ:n for f,, fz c C?(G/H; B) ( y; means that summation is taken over a repre- acTfH
sentative of coset space H/H, which does not depend on a special choice of repre- sentative by (16)) Symbolically, (17) is written as e (18) Iosif) = YL Le foyaa-, ae H/H x,€G/IH for fe C2(G/H; B) Similarly, there exists 7, € L©(GiH) for each ae H, which satisfies
(19) - rau = Krag, HEH,
(20) \ foots) = Yo rox(fona)a—, aCHIH
x, EG/H
for fe C2(G/H; B), and the equation (15) is reduced to
(21) beHIH Y hoaFab, mb, Y= Yo Foy, ed), b€HIH b- D469)
LEMMA 4 J, is @ constant function on G/H, and as a function of aé H, it is 4a class function; 1, = Iyapnt > Sor 4, bE H
Proof Take x9 € G/H, a € H, and a neighborhood U of xy as in the proof of Lemma 2 If, in (21), we choose F such that supp Fc LJ (UXxU/)(1, a)0, ð), then
Trang 7REGULAR REPRESENTATION OF TRANSITIVE GROUPOIDS 255 for x, € U and for x, e Ue (c is an arbitrary element in H) (23) r.h.s = ri (XS)F{Xị, xsc” 1a), a~°e) I.h.s = f1 (X)FfQ@j4—1e, xz(€~1a, 1) = {23) = l-t(XDFG¡; X;c~?2)(1, a~1e) Thus comparing (22) and (23), (24) r~a (Xa) = 7 -10%) for x, 6 U, x» € Uc Since a, c are arbitrary, we have obtained the desired conclusion Z At this place, we claim that the converse of Lemma 4 holds To be precise, let {hiwex be a function on H which is a class function,
{25) lio = ly, G@ bE H,
and is H-covariant;
(26) lan = x(h)—11, ? h = H
Further suppose that
(27) F(x, 2) SDL Fug, x2)(a73, ID)
acHIH
for Fe 9Í, gives rise to a bounded linear operator ⁄ in L°(9 Then we have Lemma 5 Ù belongs ío 9Ứ°n 9U
Proof Let f, Fe WM We must show that L(ƒ+ Ƒ) = fs LF and L(F sf) = = [Fx f These equalities follow from a direct computation ZB
COMMENT Conditions (25) and (26) insure that (27) is a well-defined map of YW into C~%Œ, Đụ)
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(s) If be H commutes with a modula H, ie., a babe H, then z(a~1b~1ab) =: |
(s%) The cardinality of the conjugate class of a in HiH is finite
Proof Here we use the original definition of Qf; elements in 9L are C-functions on G+ G (see (3)) Then the operation of L on Q is expressed as
(28) (LF \(g, , 82) = > 1,F(g1a, s;)Ác,u(a)~13
e€HH
for g,, g2€G, Fe Take a section ¿ € C&(G/Hx G/H; B&B) whose support is contained in a sufficiently small neighborhood of ([1], [1]) and use the same letter @ to denote the corresponding function on GXG Then YO Ag.(a)-*9(g,a g20) acH'H defines an element in 2, and if we use this function as F in (28), we have (LF )(g, , 82)? = (29) = ¥ Ag n(bb')- AG nlaa')~ Ugly plgyba, 828) —(g\b'a', g:a") = a,b,a?,b* = 3 )*4a¿ u(4)~?|0(guba, 9z4)1ẺAa u(b)~} a,b
(the support of y should be assumed to be so small that y(g,ba, goa; and ~(g,b'a’, goa'y have overlapping supports if and only if a = a’ and b = 6’) Integrating (29) over Gx*G/D, we obtain (LF|LF) = dg, dgs Yi, *o( gb, B2)?Ag y(b)-? = b G/H :GIH = Zi? Ô- dadeloGi sò? = IBlŒ]#) GIH>G:H From this, we have (30) Yi? < |LIP
Now, in view of (25), boundedness of L implies the condition (#*) To prove (*), let b € H be such that a~15-1ab e H Then by (25), (26),
L => / —1 vas = fa b Tạ , = X(@~'b~'!ab)—1,
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To formulate the theorem in this section, we need some more definitions con- cerning the Hilbert algebra of H,/H Consider a function / on H with the following properties
(1) Ít) = y(h)-M(, heH,hecH,
(32) support of / is compact modulo H
We denote by # the totality of such functions F is furnished with s-algebra struc- ture: for /,, 4, 1€ 2, (33) (he l)a)= YJ, h(b),(b-*a), bcH/n (34) 1*(a) = Ha-}) For / € &, define a linear operator ®(J) on 9{ by (35) (@()fX#: 82) = ` l(a)f(gìa, #x)4c,u(a)~1⁄2 ¡ằ€ H/H
for F€ 9L Since the relevant summation is ñnie, ®(1) is bounded with respect to E3-norm and can be extended to a bounded hnear operator on Ƒ?(90 by continuity, which is also denoted by @(/) A direct computation shows that the correspondence + &(/) is a faithful «-representation of Z on L°(Q0, and its range is contained in QW’ Let S be the set of ae H which satisfies («), («#) in Lemma 6 and set @ = fle B; d(ab) == I(ba) for a, b € H and the support of / is contained in S} Since &(¢@) = = Ở(2)n 9Ú“ n 9U by Lemma 5 and Lemma 6, @ forms a x-subalgebra of @ and is commutative
THEOREM | The center of the left von Neumann algebra of % is generated by O(@)
Coro.wary The left von Neumann algebra of % is a factor if and only if S== H Before the proof of Theorem 1, we will give some preparatory discussions First we remark that the #-algebra #, described in (31)—(34), becomes a Hilber algebra by the inner product
(36) 0|)= ¥ ila), 1eB
aGH/H
We construct an isometry of L7(2) into L2Q0) which intertwines ® Take an ap- proximate 6-function ée¢ HW; (€ | Ế} = 1 and € is supported by a sufficiently smal neighborhood of (1, 1))De GxG/D, Define a linear map J of # into L7(20) by
@7) 1Œ #8) = 3; l(a)6(6¡3, g)4gc (4)! le Ø
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Then, by a computation as in the proof of Lemma 6, we have
(38) q@)|70)) — (1!) eB
Since [(/) = ®(@é, I intertwines ® Now we will show that @ can be extended to a norma! homomorphism of #” into I’" Set
Mf = {Le A(L2Q0) ; there is a function / on H satisfying (31) such that (39)
(LF)(g1,82)= Y, Ac,u(4)~*?l(a)F(g,a g;) for F e 90 8cH('H
Then Z becomes a von Neumann subalgebra of 9U“ and makes 72} invariant (1(2} đenotes the /2-closure of /(2)) Eurthermore, the restriction of.Z to the subspace 1{2) yields an isomorphism of - onto Me , and the induced algebra Mrs 1S transfered into a subalgebra of #’’ by the isometry / Thus we get a normal iso- morphism ¥ of &@ into #” Since ¥(O(B)) = Ø and Z is weakly dense in #”, we conclude that ¥ is an isomorphism of 4 onto 4” and the restriction of ¥-} to 8 coincides with ® In other words, we can extend # to 8” as a normal homomor- phism and 0(2”) =.#
Proof of Theorem 1 Since Wn W' <.#, we need to prove that @” - : = ®-19ƒ'n9U) By Lemma 5, # '° c 6-4’ AW’) For the reverse inclusion, we will show that ' c ®~!(9U' n 9U} Let K be a bounded linear operator in E?(2) Then K can be expressed in a matrix form:
(40) (KƒXa)= 3) k(,4)ƒ/(4)
a€H|H
where k (call the matrix function of K) is a function of Hx H satisfying
(41) k(ah, a'h') = (h-4h')K(a, a’),
here A, h’ € H For L € JƯ' n W’, we can find a function / on H which satisfies (25), (26) and represents L by the relation (28) Then a direct computation shows that the matrix functions of K6-1(L) and @-1(L)K are given by
(42) 3, M@k(a, , aa2),
aH, ỊH
(43) Š` l(a)k(a~1a, q;)
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Thus K€ @-!QU' ^ 9U} if and only if (42) = (43) for a,.a.¢H By Lemma 6, we can find a family {/,} of elements in @, with pairwise disjoint support, such that
(44) Ka) = %` la)
Now suppose that K commutes with @ Then K satisfies (42) = (43) with / replaced by /,, and therefore it holds for / = }) /, (Note that the summations in (42), (43)
are absolutely convergent.) In other words, K commutes with (QU 9 QW’) This
shows that #7 c ®-1!(9U“ˆ r 9Ú)” and the proof of Theorem 1 is completed Z
3 TYPE ANALYSIS OF W”
In this section, we assume that
(C) the commutator subgroup of H is contained in #7
Then the condition (**) is trivially satisfied and the set S (see before Theorem 1} is characterized as follows:
(45) S = {ae H;7(a-b-ab) = | for all b © H}
From this, we see that S is a normal subgroup of H To investigate the type of QU’’, we first give an explicit central decomposition of W’’ Set
(46) Ss, = {y;n is a unitary character of S whose restriction to H is equal to x} By fpointwise multiplication, S, is an (S/H)*-principal homogeneous space (The- irreducible decomposition of Indy; 57% shows that S, is non-empty.) For each y e Sy > we can construct a Hilbert algebra Mf, as in §1, here we use S, 4 in place of HT, x Take fe UW and set
(47) S(Br> 82) = YY 4g wla)-?n(@)-F(er4, ga)
acs/H
(Note that S/H > [a] > [(a, 1)] € Ds/D is an isomorphism, where Dy, is the ’D’ for- S.) Then f, € 2, and one sees that { f= { !hÌ;e§ ; f € UW forms a continuous field
z
{M,} ¢g¢ of Hilbert algebras, by a routine calculation Also a direct computation
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shows that
(48) (FIF) = \ dn (E,{F,), for Fe
S,
where ( | ), is the inner product in YW, and dy is a Haar measure through identifica- tion Sy =~ (S'H)* Note that dy is unique up to positive constant Furthermore, by applying Theorem | to %,, one sees that 2,’ is a factor and hence the above decom- position is central (see [3], p 191 Corollaire) Summarizing these considerations, we have: Proposition Through Wasa fre f= {fi neg? the Hilbert algebra MU is iden- ⁄# ® tified with the direct integral of Hilbert algebras, \ dyn &,, and the center of 9 a Sy
is identified with the algebra of diagonalizable operators L*(S,)
THEOREM 2 Under the condition (C), the left von Neumann algebra SÚ' is oƒ type Ml if H/S is infinite and it is of type Y otherwise
Proof By Proposition, it suffices to prove the theorem when &”’ is a factor, ie, S=H Then B” is a factor by Corollary to Theorem 1 Let t,, (resp 1„) be the canonical tracs of 9“ (rasp Z””) associated with the Hilbert algebra structure
By (38),
(49) ty(®()* + ®() = tự(* +*l), for le BY
From this, one sees that 9U” is of typs II If #'” is so As Ø” ís a ñnite factor, it is of type IL if dim L°(@) = |H/H| = oo Thus we have showed that 2” is of type IE if |H/H| = co Now suppose that |H/H| < oo In this case, we can regard L2QQD as a subspace of L™G/H x G/H) and the projection P to L*(Q) is given by
(50) (PF)(g , 82) = |HHỊ~ y Ag nla)~*F(g14, 824),
ac H{H
for Fe C»(G/HXG/H; B&B) Through this identification, we see that 9Ú? c
c B(L(G/H))@ 1 and Ty, coincides with the restriction to 2’ of the canonical trace of B(L*(G/H)) In particular, the image of t,, is discrete and hence W is of type I
This completes the proof of Theorem 2 Z
Trang 13REGULAR REPRESENTATION OF TRANSITIVE GROUPOIDS 261 awh — wa 10 REFERENCES Baccert, L.: KLeppner, A., Multiplier representations of abelian groups, J Functional Ana- édpsis, 14(1973), 299 324
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