complemented *-primitive ideals inl1-algebras of exponential lie groups and of motion groups

Math. Z. 204, 515 526 (1990) Mathematische

Zeitschrift �9 Springer-Verlag 1990

Complemented *-primitive Ideals in L 1-algebras of Exponential Lie Groups and of Motion Groups M.E.B. Bekka 1 and J. Ludwig 2 1 Mathematisches Institut der Technischen Universitiit Mfinchen, Arcisstrasse 21, D-8000 M~inchen 2, Federal Republic of Germany 2 Centre Universitaire de Luxembourg, Seminaire de Math6matique, 162a, Avenue de la Fa'iencerie, L-I 511 Luxembourg, Luxembourg

O. Introduction

Let G be a locally compact group with Haar measure dx and LI(G)=I~(G, dx) the group algebra of G. Let I be a closed left and right translation-invariant subspace (i.e. a closed two-sided ideal) of L 1 (G). Rosenthal [25, Lemma 1.4] gave, in the case of an abelian G, a necessary condition on I to be complemented in L l(G) (or, equivalently, to be the range of a continuous projection on L 1 (G)). The problem of the characterization of all complemented ideals in L 1 (G) for an abelian G is still unsolved (see however, Alspach, Matheson [1] and Alspach, Matheson and Rosenblatt [2] for some partial results).

On the other hand, the related problem of describing the ideals 1 in L ~(G) for which the annihilator I l is complemented in L~(G) has been completely solved by Gilbert [11, Theorem A] in the case of an abelian G. It turns out that these ideals are exactly the ideals in L 1 (G) which possess bounded approximate units (see Reiter [22, Sect. 17. Theorem 2] and Liu, van Rooij and Wang [15, Theorem 13]). This is not surprising, for the following has been shown in [5, Corollary 2.63 : If G is amenable, then a closed two-sided ideal I in L ~ (G) has bounded approximate units if and only if I • is complemented in L ~ (G).

If one tries to study these problems for non-abelian groups G, it is clear that one has first to look at the ,-primitive ideals in L~(G), i.e. the kernels of topologically irreducible .-representations of L ~ (G) in Hilbert spaces. In this paper we are concerned with exponential Lie groups and with connected motion groups. We characterize for both classes of groups G the .-primitive ideals I in Li(G) which are complemented in L~(G) as well as those for which I • is complemented in L ~ (G). Our results are as follows. First, let G be an exponential Lie group with Lie algebra 9. Then, by Kirillov theory, the set G of all (equivalence classes ot) irreducible .-representations of L 1 (G) is parametrized by the orbits of the coadjoint representation Ad* of G in the dual 9* of 9 (see [6]). We prove that a *-primitive ideal kerTr, T r ~ , is complemented in U (G) if and only if the corresponding coadjoint orbit is affine linear. Moreover, it turns out that these ideals are exactly the .-primitive ideals I for which I • is complemented in L ~ (G) (or, equivalently, which have bounded approximate

516 M.E.B. Bekka and J, Ludwig

units). This extends [4, 3.1 Theorem]. The situation in the case of a mot ion group is much simpler to study. By a mot ion group G we mean a connected locally compact group which is at the same time a semi-direct product G = K~<N of a compact subgroup K and an abelian normal subgroup N. The result here is that a ,-primitive ideal is complemented in L t (G) if and only if it is finite codimensional in Lt(G). Again, these are exactly the ,-primitive ideals I for which I • is complemented in L ~ (G).

1. Some General Results

We prove here some results which may be of independent interest and which will be needed in the sequel.

1.1. Let G be a locally compact group and let N be a closed normal subgroup of G. Then G acts on L 1 (N) b y f X ( n ) = f ( x - i n X) for f e L 1 (N), x e G and L ~ (G/N) acts on LI(G) by ( z . f ) ( x ) = z ( x ) f ( x ) for f eL l (G) , z eL~ iG/N)~L~(G) , xeG. It is shown in [13, 2.3 Theorem] that the following is a one-to-one correspondence between the set of all closed two-sided G-invariant ideals in LI(N) and the set of all closed two-sided L ~ (G/N)-invariant ideals in L I(G). For a closed two- sided G-invariant ideal J of L 1 (N) let e(J) be the closed linear span of CooiG)*J, where Coo(G) denotes the space of all continuous functions on G with compact support and J is viewed as a set of measures on G. Then e(J) is a closed two-sided L~(G/N)-invariant ideal of L I(G) and J is the closure of the set { ~ f l N ; f e e i J ) * Coo (G), x e G}, where f i n denotes the restriction to N of a function f on G and ~f is defined by x f (y)= f (x y). We are going to prove that complemented G-invariant ideals in L t (N) correspond, at least in the case where G is separable, to complemented L~(G/N)-invariant ideals in L~(G). We will apply this result in the following situation: Let Z be a unitary representation of N and let n=indN a X be the unitary representation of G induced by Z. Let J = ~ L 1 - k e r z ~, where ~ is defined by x ~ ( n ) = x ( x - l n x ) , neN.. Then by [13,

x e G

2.2 Lemma] , e(J) = L 1 - ker n.

Theorem 1.1. Let N be a closed normal subgroup of the separable locally compact group G. Let J be a closed two-sided G-invariant ideal in L 1 iN). Then J is complemented in L 1 (N) if an only if e(J) is complemented in L 1 (G).

Proof Since G is separable, there exists a Borel cross-section s: G/N--*G (see [19, Proposition 1]). This cross-section defines an isometric isomorphism S of Banach spaces between LI(G) and LI(G/N, L I (N)) in the following way, For f eLI (G) ,Coo(G) the mapping G - - * L t ( N ) , x ~ f I N is continuous. Define S f e L ~ (G/N, L ~ iN)) by

iS f ) (s = ~(x)f[N, ~ e G/N

Then S extends to an isometric isomorphism between L t (G) and L 1 (G/N, L 1 (N)). Let P: I_}(N)--,J be a continuous projection onto J For f eLI (G) ,Coo(G)

define f e L t (G/N, L 1 (N)) by

]'(~) = P (~(~fl N), 2c e G/N,

,-Primitive Ideals in L~-algebras 517

and Q feLl(G) by

Qf _.=S-1j 7.

Then

IIQf! l ,=l l f l l~ = ff IIP(~(x)fIN)lld~c~llell ~ L~flglld~c=IIPII IlfN1. G/N GIN

Thus Q extends to a continuous operator on U(G). If fee(J)*Coo(G), then x f lNeJ for all xeG and therefore ~ = S f and Q f = f Since e(J)*Coo(G) is dense in e(J), we have Q f = f for all fee(J). Since s~x)QflN=N-a(~(x~)[N for all 2eG/N and since S-I(~(x)~)[N=~(2) for almost all 2eG/N, it follows that s(~)Qf[N~J for almost all :~eG/N and hence Q fee(J). Thus Q is a continuous projection onto e(J).

Conversely, let Q:L I(G) ~ e(J) be a continuous projection onto e(J). Choose heCoo(G/N), h>O with f h(2)d2= 1. For feL~(N) define ~ D ( G ) by

G/N

f(x)=h(:c)f(s(Yc)-lx), :i=xN, xeG,

and PfeL 1 (N) by means of the L 1 (N)-valued integral

Pf= I (SQ~)(2)d2. GIN

Then

IIP f ll~ ~ S II(SQ~(~)Ilt dx= I;SQTII = IIQTII~ ~ IIQII IlYlII=IIQ[I llfll~, G/N

since II 7 IIi : II f II1- Thus P is a continuous linear operator on U (N). Moreover, i f f eJ then j~ee(J) and therefore

P f= ~ ( S ~ ( 2 ) d 2 = S h(2)fd2=f. GIN GIN

Now, P f can be arbitrary approximated in the norm of L ~ (N) by linear combina- tions of functions of the form xQ~IN. Since such functions belong to J, we have PfeJ. Thus P is a continuous projection onto J. This completes the proof of the theorem.

1.2. Let G be a locally compact group and N a closed normal subgroup of G. We now study relationships between complemented subspaces of L I(G/N) and certain complemented subspaces of LI(G). Our result will not be used in the sequel and is given for the sake of completeness.

There exists a canonical bounded linear mapping T from L 1 (G) onto L ~ (G/N) defined by

(Tf) (:~) = ~ f(xn)dn, feCoo(G), 2=xN,, xeG, N

518 M.E.B. Bekka and J. Ludwig

Let Y be a closed subspace of I2(G/N) and let X = T - t ( Y ) . An important example arises in the following way. Let n be a unitary representation of G and assume that n is trivial on N and hence factors to a representation ff of G/N. If Y= L 1 - ker r~, then T - 1 (y) = L 1 _ kern.

Theorem 1.2. Let N be a closed normal subgroup of G, T the canonical mapping L ~ (G) ~ L x (G/N), Y a closed subspace of L 1 (G/N) and X = T - 1 (y). Then X is complemented in L ~ (G) if and only if Y is complemented in L 1 (G/N).

Proof Let /~ be a Bruhat function for (G, N), that is, fl is a continuous positive function on G such that S f l ( xn)dn= 1 for every x e G and s u p p f l n K N is com-

N

pact for every compact K~_G (see [21, Chap. 8, 1.9]). Let P: LI(G)--*X be a continuous projection onto X. For f eL I (G/N) define f= f l . ( f on ) , where n: G ~ G/N is the canonical mapping. Then T ] ' = f and IN f [rl = mr f JJl. Define now Q f e L I ( G / N ) by Q f = TP~. It is then easily verified that Q is a continuous projection onto Y with It Q l[ </J P [I. Conversely, let Q: L 1 (G/N) ~ Y be a continuous projection onto Y. For f e L 1 (G) define P f E L ~ (G) by

( P f ) ( x ) = f ( x ) + f l ( x ) ( Q T f - T f ) ( n ( x ) ) , xeG.

It is readily shown that P is a continuous projection onto X with 1[ P l] < Jl Q rl + 2.

Corollary 1.2. ker T is complemented in L 1 (G).

1.3. For a locally compact group G we denote by Co(G) the space of all continuous functions on G that vanish at infinity and by M(G) the Banach algebra (under convolution) of all bounded Radon measures on G.

We will need the following version for two-sided ideals of the implication "(6)~(f l )" in [15, Theorem 4].

Theorem 1.3. Let G be a locally compact amenable group. Let I be a closed two-sided ideal of L I(G) with the property that I• ~ Co(G) is dense in I • with respect to the weak *-topology a(L ~ (G), L 1 (G)), where 1• { q)eL ~ (G); ( q 0 , f ) = 0 for all f d } . Suppose that I • is complemented in L~(G). Then there exists a central idempotent measure l~ e M ( G) such that I = I~ * L 1 ( G).

Proof It is not necessary to assume that I is a(M(G), Co(G)) - closed in 1-15, Theorem 4]. The proof of this theorem works also under the assumption that I -L c~ Co(G) is weak .-dense in I ~. Regarding I as a left ideal, we get an idempotent measure #eM(G) such that I=LI(G)*# , by the implication "(6)~(/r from [15, Theorem 4]. Now, I is also a right ideal. Thus the obvious right hand version of [15, Theorem 4] yields an idempotent fie M(G) such that I = ~ , L ~ (G). But then we have

g *l~=f i*(g*#)=( f i*g)*#=f i*g

for all g e l 1 (G). This shows that # = ~ and that/~ is a central measure.

2. Exponential Lie Groups

2.1. Let G be a connected, simply connected Lie group with Lie algebra g. Then G is called exponential if the exponential map exp: g --, G is a diffeomor-

.-Primitive Ideals in L ~-algebras 519

phism. Let g* be the dual space of 9 and Ad* the coadjoint representation of G in g*. Recall that there is a bijection between the set (~ of (equivalence classes of) irreducible ,-representations of L t (G) (or, equivalently, irreducible unitary representations of G) and the orbit space g*/Ad*.

For n e d denote by O~ the corresponding Ad*-orbit. Let feO~, and let G ( f ) = { x e G , Ad*(x) f= f} be the stabilizer o f f in G. Then G(f) is connected (see [-6, Chap. I, 3.3]) and g ( f ) = { X e g ; ( f , IX, g ] ) = 0 } is the Lie subalgebra corresponding to G(f). Hence G ( f ) = e x p g(f). Let

P={xEG;Tt(x) is amultiple ,~(x)I

of the identity I on H~}

be the projective kernel of n. Then clearly, P is a closed normal subgroup of G and )~. is a unitary character of P. We first give a description of P

Theorem 2.1. Let G be an exponential Lie group, 7 z ~ , O~ the corresponding Ad*-orbit and P the projective kernel of n. Let K be the kernel of n, and denote by A the intersection of all G(g), g~O~. Then P =A. In particular, P is connected. Moreover, P /K is compact. More precisely, P = K or P/K is isomorphic to IR/71.

Proof. It is clear that A is a closed normal subgroup of G and that A=exp a, where a is the intersection of all g(g), g~O~. In particular, A is connected.

Take feO~. There exists a real polarization I) of f such that n is equivalent to ind~n )~, where H = exp h and )~ is the unitary character of H defined by

g(exp X ) = e i(f 'x) , X ~ D.

We first show that P is contained in H. To this end, recall that the Hilbert space ~ of n = i n d ~ Z consists of the measurable functions r on G which are square-integrable modulo H and satisty

~(xh)= A (h) �89 ~(h) ~(x), he l l , almost all x~G,

where A(h)=AH(h)/AG(h) and An, Ar denote the modular functions of H and of G. Then G acts by left translation on 3r ~. Suppose that x(gH for some x~P. Then choose a continuous r with r and q~eCoo(G/H) such that q~(H)=0 and q)(xH)40. Then ~ ' = q ~ and for h~H

(n (x- 1) ~,) (h) = 2, (x) 3' (h) -- 0,

since x~P. On the other hand

(re (x- 1) ~,) (h) = {'(xh) = A (h) ~ ~(h) r 40, h~H.

This is a contradiction. Hence P ~ H . Moreover, taking any continuous ~o~r with ~(e)= i, one easily sees that, for x~P, 2,~(x)=A(x) ~' )~(x). Hence IA(x)] = 1 and therefore A(x)= l for all x~P. Thus ;.~(x)=x(x) for all x e P Since ~ is G-invariant, Z(Y- l xy) = Z(x) for all x~P, y~G. Hence, by differentiation, x~G(g)


for all gsAd*(G)f , x~P. Thus P c A . On the other hand, for all aeA, ~eJCf, y~G,

(zc(a) ~)(y)= ~(a- l y)= ~(yy- X a-1 y)= ~(y- l a- , y ) ~(y)

= ~(a- 1) ~ (y) = Z (a) ~ (y),

since A ~ G ( f ) c H and since A is normal in G. That means that A cP . Hence P = A. In particular, P is connected. Since K- - ker ~ is the kernel of the unitary character 2, of P, P/K is isomorphic to a connected subgroup of the torus IR/~ Thus, either P / K = {e} or P/K~-~/7Z.

Recall that a unitary representation ~ of G is square integrable modulo its projective kernel P (resp. modulo its kernel K) if the absolute value of each coefficient function (re(.)~, t/) of rc belongs to L2(G/P)(resp. U(G/K)).

Corollary 2.1. Let G be an exponential Lie group. Then 7~ G is square integrable modulo its projective kernel if and only if ~ is square integrable modulo its kernel.

2.2. We intend to prove the following theorem which is the central result of Sect. 2.

Theorem 2.2. Let G = e x p 9 be an exponential Lie group. Let ~E~, On the corresponding coadjoint orbit in g* and I = L 1 - k e r ~z. I f the annihilator l • of I is complemented in L ~ (G), then O, is affine linear.

The proof is quite involved and breaks into several lemmas.

Lemma 2.2.1. Let K be the kernel of re and G = G/K. Denote by "ff the representation of ~J corresponding to n and let "[=L 1 - k e r ~. I f I • is complemented in L~(G), then there exists an idempotent central measure f~ on & such that "[ ---#. L 1 (CJ). Moreover, ~ is concentrated on P/K, where P is the projective kernel ofn.

Proof Since I j- is the weak *-closed linear span of the set of all coefficient functions of n, it is clear that i ( I ~) = I • where i denotes the isometric embedding of L ~ (~) into L ~ (G). Hence, if Q is a continuous projection onto I • then i - l o Q o i

is a continuous projection onto 7 "• In view of Theorem 1.3 it suffices to show that T• Co((~) is weak *-dense in I'• But this follows from [14, Theorem 7.1]. Indeed, it is proved there that for some k~]N the k-th tensor product n | of n is square integrable modulo R Hence all coefficient functions of n tend to zero at infinity modulo P (see remark after Theorem 6.1 in [14, p. 74]). Since, by Theorem 2.1, P/K is compact, it follows that all coefficient functions of ff vanish at infinity and therefore T~n Co(~) is weak .-dense in T • Thus there exists an idempotent central #EM(~) such that I = # * L 1((~). It remains to show that /~ is concentrated on P/K. Let S be the support group of #, that is the smallest closed subgroup of C which supports ft. By [23, Lemma 2], S/S' is compact, where S' denotes the closure of the commutator subgroup of S. Denote by H the quotient ~/(P/K) and by p: (~-~H the canonical mapping. Since P/K is connected (see Theorem 2.1), H is a connected, simply connected Lie group. Let H o = H, H , =H' ,_ ~, n > 1, be the derived series of H. Since H is solvable there is some m e n with Hm= {e}. Consider now the subgroup S=p(S) of H which is closed, since P/K is compact. Let k be the largest integer with O<-k<<-m and such that g ~ H k. Then the image of S under the canonical mapping Ilk-* Hk/H'k is a compact subgroup of the vector group Hk/H'k. Therefore

.-Primitive Ideals in LX-algebras 521

ScH'k=Hk+l and this is possible only if k=m. This means that S={e ) , i.e. S ~ P/K and the proof is complete.

We now determine the measure ~ in Lemma 2.2.1. Let O~ be the coadjoint orbit corresponding to rc and let f e OR. We have seen in the proof of Theorem 2.1 that ~(x)= Z(x)I for every x E P, where X is the character of P defined by

z(exp X ) = e i<f'x> , X e a

and a, the intersection of all g(g), gEO,, is the Lie algebra of P. It is clear that X factors to a character, denoted again by X, of the quotient P/K.

Lemma 2.2.2. Let d2 be the normalized Haar measure on the compact group P/K. Let # be the measure from Lemma 2.2.1. Then #=6e-~d2 , where fie is the Dirae measure at e and X the character on P/K defined as above.

Proof It is clear that P = P / K is central in G=G/K. Hence for every pE(G) ^ there exists a character Zp of P such that for all xeP,,

; (x) = z~ (x) L

In particular, Z~=Z. Let v=Se-~ds We claim that ~ = v . Since ~ and v are concentrated on P,, we have

p(#)=/~(zo)l and p(v)=f(Xo)I

for all pE(G) ^, where /i (resp. ~) denotes the Fourier transform of ~ (resp. v). It is clear that f(Z)=0. Moreover, s i nce / ~ .L I (G)=L 1 - k e r k, ~(/~)=0 and hence O(Z)=0. Therefore p(l~)=p(v) for all p~(G)^ with X0=Z. Let now pE(d) ^ be such that Zo#Z. Then clearly, 0(Zp)=l and hence p(v)=I. On the other hand, since /~ is idempotent, fi(Zo)=0 or ~(Zp)=l . As 0(Z)=0, v*L~(G)cLl-ker~. Hence, it would follow from O(Zo)=0 that v*La(G) is contained in L t - k e r p and this would be a contradiction to p(v)=I. Thus,/~(Zo) = f(Zp) and therefore p(#)=p(v). Consequently, p (#)=p(v) for all pE(G) ^ and that means # = v.

We can now easily describe the .-primitive ideal I = L 1 - ker~. Recall that our assumptions are that G is an exponential Lie group and that I • is complemented in L ~ (G).

Lemma 2.2.3. Under the above assumptions the following holds: L a - k e r r c = L ~ - k e r ind~ Z, where P is the projective kernel of a and Z is the character of P defined by ~.

Proof K being the kernel of n, r7 the representation of G/K defined by z and d)~ the Haar measure on/~ = P/K, we have by Lemma 2.2.1 and Lemma 2.2.2

L 1 - ke r f f= (6e -~d2 ) ,L l (~ )

= { f eL1 (1~); (fie--)~d-~) , f = f }

= { f eL l (G); ~ * f = 0}.

Now, using the fact that X is G-invariant, it is readily verified that for f e L ~ (G), )~*f=0 if and only if j I P e L t - k e r x for almost every xeG. Hence, in the notation of Section 1.1, L ~ - k e r r ~ = e ( D - k e r z ) . Since e ( D - k e r z ) = L a-ker ind~ Z (cf. [ 13, 2.2 Lemma] ), L ~ - ker ~ = D - ker ind~ Z. But, if T den-


otes the canonical mapping from L 1G) onto LI(G) (see Section 1.2), then L 1 - ker n = T - 1 (L 1 _ ker ~) and

D - ker ind~ Z = T - 1 (L 1 _ ker ind~ Z).

This completes the proof of the Lemma.

We now show that Lemma 2.2.3 implies that the coadjoint orbit O~ is affine linear. This will complete the proof of Theorem 2.2. To this end we will need the notion of a .-regular group. Let G be an arbitrary locally compact group and let Prim C*(G) (resp. P r im. LI(G)) denote the space of all primitive ideals of the C*-algebra of G (resp. the space of all *-primitive ideals of L I(G)). Then G is said to be . -regular if the canonical mapping

Prim C* (G) --+ Pr im, L 1 (G),

P--*Pc~LI(G), PePrimC*(G), is a homeomorphism, when both spaces are equipped with the Jacobson topology (see [7], [8]).

In case G is . - regular (this is the case if, for instance, G is nilpotent) Lem- ma 2.2.3 implies that the C*-kernels of rc and of indv ~ Z coincide. Using [4, 2.2 Lemma] it is not difficult to see that then O~ has to be affine linear. But, in general, exponential Lie groups are not *-regular (see [8]) and this makes the proof more difficult.

Lemma 2.2.4. Let G be an exponential Lie group, rccG and On the corresponding coadjoint orbit. Let P be the projective kernel of n and Z the corresponding character of P. I f L 1 - k e r n = L 1 - ker inde a Z, then O, is affine linear.

Proof Set p = inde ~ X. Take fE O~ and let n = ~ r k e r r where a, the intersection of all g(g), g e O , , is the Lie algebra of P. Then n/n is central in g/n. Since n and p are representations of G/exp n, we may assume that n = {0} and hence that P = exp a is the center of G.

We first show that n is equivalent to a subrepresentation ofp. There exists, by [17, Theorem 4.1], an hermitian h e L 1 (G) such that n(h) is a projection of rank one. Then, since L 1 - ker ~z = L 1 - ker p and since h �9 h - h ~ L 1 - ker n, p (h) is also a projection and p(h)+O, Let r q be vectors of norm 1 from the range of n(h) and p(h), respectively, Then, for any geLl(G), h * g * h - ( n ( g ) 4 , ~)h~L 1 - k e r n and hence ( p ( g ) q , q ) = ( p ( h * g * h ) t l , t l )=(n(g)~ , 4). This implies that n is a subrepresentation of p. This means that n is square-integrable modulo the center P (see [24, 2.3 Proposition]). Hence, by [24, 4.4 Proposition], O, is open in the affine subset f + a • Thus, dim a • ~ and therefore

g ( f ) = a. Consider now the ideal m = [g, fl] + g(f) . Since f l ( f )= a is the center of g, m

is nilpotent. Let p: g*---,m* be the restriction map p(g)=gJm, gEg*. Set O1 = p(O~). Then, as is well known (see, for instance, [6, 4.2 Lemma] ), O~ = p - 1 (O 1).

We claim that O1 is dense in p ( f ) + a • (here, a • is taken in m*). Indeed, let g~p( f ) + a • with gr Let ~r be the corresponding irreducible representation of M = e x p m. Since m is nilpotent, M is *-regular. Hence, Pr im. L ~(M) and the orbit space m*Ad*(M) are homeomorphic. Thus, there exists a function h e L 1 (M) with a(h)4= 0 and with 6(h)=0 for all 6 e ~ r whose corresponding orbit is contained in O1. On the other hand, by [12, Proposit ion 2], ind~ 6 is quasi- equivalent to n for any such 6 e M and ind~ a is quasi-equivalent to indpaz .

,-Primitive Ideats in Ll-algebras 523

Hence, L 1 (G), h is contained in L ~ --ker rt but is not contained in L 1 - k e r ind~ X. This is a contradiction. Therefore, O1 = p( f ) + a • But then, since O1 is a G-orbit in the exponential G-module m*, [4, 2.2 Lemma] implies that O1 = p ( f ) + a 1. Hence

Or = p - 1 (O1) = f + a•

This completes the proof of the Lemma and at the same time the proof of Theorem 2.2.

2.3. We can now state our main result the difficult part of which is the already proved Theorem 2.2 Recall that a normed algebra A has a bounded right approximate unit if there is a bounded net (x,),~i in A such that lim II x x , - x Ik = 0

for all x e A .

Theorem 2.3. Let G be a (connected, simply connected) exponential Lie group with Lie algebra g. Let 7z be an irreducible unitary representation of G and denote by I the *-primitive ideal L l - ker ~ ~?f L I (G). Then the following conditions are equivalent:

(i) I is complemented in L I (G). (ii) I has a bounded right approximate unit.

(iii) The annihilator I l of I in L ~ (G) is complemented in L ~ (G). (iv) The coadjoint orbit On corresponding to ~ is affine linear.

Proof. The implication (i)~(ii) follows from 1-15, Theorem 2], as G is amenable. The implication (ii)~(iii) is contained in [15, Theorem 41 (see Remarks after Theorem 4 in [15]). That (iii) implies (iv) is the assertion of Theorem 2.2.

Suppose that O. is affine linear. Then Or = f + g(f)• for any f e Or. In particular, g(f)• is Ad*(G)-invariant. Hence g( f ) is an ideal in g. Let N = e x p g ( f ) be the corresponding normal subgroup of G and Z the G-invariant character of N defined by

z(exp X ) = e i<y'x>, X ag ( f ) .

Then, by [12, Proposition 2], rc is quasi-equivalent to ind~z. In particular L 1 - k e r 7: = L 1 - k e r ind~ Z. But L 1 --ker Z is complemented in L 1 (N), since it has a finite codimension. Hence, by Theorem 1.1, L 1 - k e r n is complemented in L 1 (G). Thus (iv)~(i) and this completes the proof.

Remarks. 1. The equivalence of (ii) and (iv) generalizes [4, 3.1 Theorem]. 2. The equivalence of (ii) and (iii) holds, more generally, for all amenable

groups G and all closed two-sided ideals I in L 1 (G) ([5, Corollary 2.6]). 3. In case G is a nilpotent Lie group there are several other characterizations

of the irreducible representations of G whose coadjoint orbits are affine linear (see, for instance, [9], [16], [18], 1-20]).

3. Motion Groups

We now turn to the case of motion groups. By a motion group G we mean a connected locally compact group G which is at the same time a semi-direct


product G = K v < N of a compact subgroup K and an abelian closed normal subgroup N.

Let zc be a unitary irreducible representation of the motion group G. Then, by Mackey's theory, there exist 2 o N and ~r~/~ such that

" G rc = lndK~ N o" | Z,

where K.~={kEK, 2k=.:t} is the stabilizer of 2 in K, 2 k is defined by 2k(n) =2(knk-1) , n~N, and (a | (kn)=2(n)a(k), h eN , keKa. Conversely, every representation of this form is an irreducible representation of G. Moreover, since ind~ 2 is a CCR representation (see [26, 9.3 Corollary]), z is contained in indN o2 (in particular, 7r is a CCR representation). It is easily shown that indN ~ 2 can be realized in U(K) by means of the action

(ind~ 2(kn) f ) (h)= 2h(n)f(k - l h),

f~LZ(K), n~N, k, hEK. It is apparent from this that for the group kernels the formula

ker ind~ 2 = (~ ker 2 k k e K

holds. With a method similar to that of Sect. 2 we can now prove the following

result.

Theorem 3.1. Let G = K~<N be a motion group. Let 7z be an irreducible unitary representation of G and denote by t the *-primitive ideal L 1-ker z~ of L I(G). Then the following conditions are equivalent:

(i) I is complemented in L t (G). (ii) I has a bounded right approximate unit.

(iii) The annihilator I l of I in L ~ (G) is complemented in L ~ (G). (iv) n is finite dimensional.

Proof. The implications (i)=~(ii) and (ii)~(iii) are shown as in the proof of Theorem 2.3. The implication ( iv)~(i) is clear since I has then finite codimensJon in I~(G). So, the only thing to prove is that (iii) implies (iv). Let 2 e N and a~/~a be such that

n = i n d ~ a | )~.

Denote by No the group kernel of ind ,2 . By the above, N0~ker n and No~ker2 . Thus, 2 and n factor to representations 2" of -~=N/No and ff of G = Kt,<bT,, respectively.

By [3, Theorem 3] (see also Remark (1) in [-3, p. 178]), the coefficient functions of ind~ 2 vanish at infinity. Hence, the coefficient functions of ~ vanish

• 3_ 1 at infinity, too. Therefore, I" ~ Co(G) is weak-* dense in I" , where T = L - k e r ~. Since/'z is complemented in L ~ ((~), it follows from Theorem 1.3 that

r=#,L'(G)

for a central idempotent/~EM((~).

.-Primitive Ideals in L a-algebras 525

N o w (~ is a C C R group. Hence, the primitive ideal space Prim C*(G) of C*((~) is a T 1 topological space. But if,, a compact extension of an abelian group, has polynomial growth. Thus, by [7, Satz 2], the space P r im, L I(G) of all , -primit ive ideals in L 1 ((~) is homeomorph ic to Prim C*((~) and hence a T~ space, too. Therefore, p( /~)=I for all pc(G) ^, p4=ff. Since ~ (# )=0 , it follows that {~} is the spectrum of the ideal ( 6~ -u )*C*(G) of C*(~). Thus {if} is open in (G) ^ ~ Prim C* (G). We claim that this implies that G is compact .

The connected abelian locally compact group N has the form N = ~ " x C for some compac t g roup C. Hence ~ = X | for some )~c~," and some ~SeC. Since C is discrete and K is connected, K fixes every element of C and acts on l~".

Suppose that n > 1. Then, since the orbit space I~"/K is 7"1 and connected, there is a sequence (Z,) in 1~" with lira L = Z and K.Z,4=K. Z for all ,. Set 2~ = Z , | Then 2,~(N) ^, l im2,=~" and K.2,4=K.~ for all 2. Hence, inducing being cont inuous [10, Theorem 4.2], we deduce from this that

f3 {C* -- ker ind-~ p; p c(N) ^ with K .p 4= K. ~} __ C* - ker fT.

Therefore, using [10, Proposi t ion 1.11, we get

r3 {c* - ker p; pc(G) ^ \{~}} _ C* - ker ~.

That means that {if} is not open in (~)~, a contradict ion. Hence, n = 0 , and /~ = C. Thus, G = Kt><N is compact . Since ff~(G)^, n is finite dimensional.

Corollary 3.2. Let n> 2 and G=SO(n)xIR" be the Euclidean group, where SO(n) acts by rotations on IR". Then U - k e r n, ne~, is complemented in LI(G) if and only if n~SO(n) ̂

Remark. One can show by the same method of proof that Theorem 3.1 is still valid for a separable connected locally compact group G which has a connected closed abelian normal subgroup N, such that G/N is compact .

References

1. Alspach, D., Matheson, A.: Projections onto translation-invariant subspaces of Lt(R). Trans. Am. Math. Soc. 277 (2), 815 823 (1983)

2. Alspach, D., Matheson, A., Rosenblatt, J.: Projections onto translation-invariant subspaces of L 1 (G), J. Funct. Anal. 59, 254~292 (1984)

3. Baggen, L., Taylor K.: Riemann-Lebesgue subsets of ~," and representations which vanish at infinity, J. Funct. Anal. 28, 168-181 (1978)

4. Bekka, M.B.: Primitive ideals with bounded approximate units in Ll-algebras of exponential Lie groups, J. Aust. Math. Soc. Ser. A 41,411-420 (1986)

5. Bekka, M.B.: Complemented subspaces of L ~ (G). Ideals of L 1 (G) and amenability. Preprint (1986)

6. Bernat, P., et al.: Repr6sentations des groupes de Lie r&olubles. Paris: Dunod 1972 7. Boidol, J., et al. : Rfiume primitiver Ideale in Gruppenalgebren. Math. Ann. 236, 1-13 (1978) 8. Boidol, J.: ,-regularity of exponential Lie groups, Invent. Math. 56, 231-238 (I980) 9. Felix, R.: When is a Kirillov orbit a linear variety? Proc. Am. Math. Soc. 86, 151-152

(1982) 10. Fell, J.M.G.: Weak containment and induced representations IL Trans. Am. Math. Soc.

110, 424-447 (1964)


11. Gilbert, J.E.: On projections of L~(G) onto translation-invariant subsapces. Proc. London Math. Soc. (3) 19, 69-88 (1969)

12. Gr61aud, G.: D6sint6gration des repr6sentations induites d'un groupe de Lie r6soluble exponentiel. C.R. Acad. Sci. Paris S6rie A, 277, 327 330 (1973)

13. Hauenschild, W., Ludwig, J.: The Injection and the Projection Theorem for spectral sets. Monatsh. Math. 92, 167-177 (1981)

14. Howe, R., Moore, C.C.: Asymptotic properties of unitary representations. J. Funct. Anal. 32, 72-96 (1979)

15. Liu, T.S., Rooij, A. van, Wang, J.K.: Projections and approximate identities for ideals in group algebras. Trans. Am. Math. Soc. 175, 469~482 (1973)

16. Ludwig, J.: Good ideals in the group algebra of a nilpotent Lie group. Math. Z. 161, 195-210 (1978)

17. Ludwig, J.: Irreducible representations of exponential solvable Lie groups and operators with smooth kernels. J. Reine Angew. Math. 339, L 26 (1983)

18. Ludwig, J.: On the Hilbert-Schmidt semi-norms of L 1 of a nilpotent Lie group. Math. Ann. 273, 383-395 (1986)

19. Mackey, G.W. : Induced representations of locally compact groups I. Ann. Math. 55, 101-139 (1952)

20. Moore, C.C., Wolf, J.: Square integrable representations of nilpotent Lie groups. Trans. Am. Math. Soc. 185, 445~t62 (1973)

21. Reiter, H.: Classical Harmonic Analysis and locally compact groups. Oxford Math. Mono- graphs 1968

22. Reiter, H.: Ll-algebras and Segal algebras. (Lect. Notes Math., vol. 231). Berlin Heidelberg New York: Springer 1971

23. Rider, D.: Central idempotent measures on SIN-groups. Duke Math. J. 38, 181-189 (1971) 24. Rosenberg, J.: Square integrable factor representations of locally compact groups. Trans.

Am. Math. Soc. 237, 1-33 (1978) 25. Rosenthal, H.P. : Projections onto translation-invariant subspaces ofLr(G). Mem. Am. Math.

Soc. No. 63 (1966) 26. Schochetman, I.E.: Integral operators in the theory of induced Banach representations.

Mem. Am. Math. Soc. no. 207 (1978)

Received May 25, 1989

complemented *-primitive ideals inl1-algebras of exponential lie groups and of motion groups

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