Multipliers on homogeneous Banach spaces on compact groups
~-~ARALD E. KROGSTAD
O. Introduction
The first section is m o t i v a t e d b y a result of K. de Leeuw [9, th. 4.2] and is con- cerned with Banach modules where the elements of the algebra act as weakly com- p a c t operators on the B a n a c h space. I n w 2 and w 3 t h e general properties of such modules are used to characterize the multipliers on homogeneous B a n a c h spaces on compact groups. The results are p a r t l y t a k e n from the author's dissertation for the licentiat degree at the Technical U n i v e r s i t y of N o r w a y which was w r i t t e n u n d e r t h e direction of Professor Olav Njgstad.
1. Banach modules with weakly compact action
We s t a r t b y recalling some basic definitions. I f A is a Banach algebra and V is a B a n a c h space, t h e n V is a left B a n a c h A-module if it is a left A-module in t h e algebraic sense, and ]lavl[ < [jail ][vii for all a C A, v E V. A right B a n a c h module is defined similarly. The closed linear subspace of V spanned b y A V is called t h e essential p a r t of V and is denoted b y Ve ([10 l, p. 454). I f Ve = V, t h e n V is called an essential A-module. I t is well known t h a t if A has a b o u n d e d approxi- m a t e identity, t h e n V , = A V e , i.e. for al v E Ve, t h e r e exists an a 6 - A , w E V8 such t h a t v = aw. (In w h a t follows, we always assume the b o u n d on a b o u n d e d appr. id. to be 1.) I f V is a left A-module, t h e n V* is a right A-module u n d e r the adjoint action: (att)(v) D #(av), a 6- A , v 6- V, # 6. V*. The essential p a r t of V*
is called t h e contragradient of V and is d e n o t e d b y g c ([10], p. 455).
I f V and W are B a n a e h A-modules, t h e n HomA(V, W) as usual denotes the space of linear, continuous operators T from V to W such t h a t T ( a v ) = a ( T v ) for all a 6. A , v 6. V.
204 ~ R A L D E . K ~ O G S T A D
W e s a y t h a t a ~ A is w e a k l y c o m p a c t if t h e o p e r a t o r v - ~ av is w e a k l y c o m p a c t . L e t ~: V--~ V** be t h e canonical injection.
LEM~[A 1.1. Let V be an essential A-module. T h e n u ( V ) ~ - ( V * ) ~ i f and only i f every a ~ A is weakly compact.
Proof. L e t a E A, v C V, # C V*. T h e n ,~(av)(#) = #(av) = z(v)(a#) = au(v)(#), t h a t is, u(av) C V *~. Since t h e linear span of A V is dense in V, ~ is a linear iso- m e r r y , a n d V *~ is closed, it follows t h a t z ( V ) c V *~. T h e l e m m a is n o w a n i m m e d i a t e c o n s e q u e n c e of t h e f a c t t h a t a linear o p e r a t o r on V is w e a k l y c o m p a c t if a n d o n l y if its second a d j o i n t s m a p s V** into x(V) ([2], p. 482).
Ln~MA 1.2. Let A have a bounded appr. id. {in} , and let module. T h e n there exists a natural isometric i s o m o r p h i s m
~: Vc___~ V ~~
such that
(e~)(~) = ~(~), r e V *~ ~ e V ~
V be a B a n a c h A -
Proof. One easily checks t h a t ~ c o m m u t e s w i t h t h e a c t i o n o f A which implies t h a t e ( V *~) - - V%
L e t f C V% T h e n t h e r e exists a n d a C A , g E V ~ such t h a t f ~ a g . I f n o w C V** is a c o n t i n u o u s e x t e n t i o n o f g to V**, t h e n
q(a~)(~) : a~(/~) ~-- ~(a/~) = g(att) : f(tt) for all ~ C V ~.
T h u s ~ is onto. Clearly l]o~bl[ ~ EIr a n d t o c o m p l e t e t h e proof, it r e m a i n s t o p r o v e t h e opposite i n e q u a l i t y :
L e t ~b E V *c. Since {i~} also acts as a n a p p r o x i m a t e i d e n t i t y for V *c, t h e r e is a n i n such t h a t Hi~q~-
~11
< s/2.L e t /t E V*, L/#ll = 1, so t h a t l~5(#)i > l/~ll - - e/2. T h e n ( ~ ) ( i ~ # ) = q~(i~/~) --~
q~(#) _ qS(/~) + q)(i~#), h e n c e
( ~ : R e a d " i s o m e t r i c i s o m o r p h i c " . )
PROPOS[TIO~ 1.1. Let A have a bounded appr. id., and let V be an essential B a n a c h A-module. T h e n the following are equivalent:
(i) V ~ (V*) c ~-- V%
(ii) A l l a E A are weakly compact.
(iii) A have a bounded appr. id. consisting of weakly compact elements.
NULTIPLIERS ON ttOMOGENEOTJS :BANACtt SPACES ON COIVIPACT GI~OU~S ~905
Proof. The equivalence of (i) and (ii) is obvious from the lemmas. Clearly (ii) implies (iii), and since the algebra norm majorizes the operator norm, and the composition of a bounded linear operator and a weakly compact operator is weakly compact, (iii) implies that every a E A is the limit in the uniform operator topology of weakly compact operators, and therefore is weakly compact. Thus (iii) implies (ii).
Let V @y W be the projective tensor product of the Banach A-modules V and W, and let M be the closed linear span of elements of the form av @ w - - v @ aw. The quotient space V @~ W / M is an A-module tensor product and is denoted b y V @A W (See [10], [11]).
Each element cf of V @A W has an expansion ~v = ~ - 1 vi @ wi where
~iml [IVi[lllWi[I < DO. The norm of qo is defined b y ]l~[I = i n f ~ L 1 ][vi[lHwill where the infimum is taken over all possible representations of ~0 (See [5], [10]). A bilinear operator gJ from V • W to a Banach space D is called A-balanced if it is con- tinuous and T ( a v , w ) = ga(v, aw) for all v q V, w E W, a E A . I f g*: V •
is A-balanced, there is a unique linear operator W: V @A W - + D such t h a t 1) V x W - - ~ T D
(~)A o / ~
V @A W / 2) LI~IL = I1~1[.
We refer to [10] for further properties of V @A W.
PROPOSITION 1.2. Let A be a B a n a c h algebra with a bounded appr. id. Let V and IV be essential A-modules and let the action of A on W be weakly compact.
Then
Hom~ (V, W) ~ (V | W~) * ~ (V | W*)*.
The isomorphism carries T E Horn A ( V, W) to lZT E ( V @ A Wc) * (resp.
(V @A W*)*) defined by
#T(V @A W*) = w*(Tv), v C V, w* C W e (resp. W*)
Proof. The conditions assert that W ~ W co, and since the natural isomorphism from W to W cc commutes with the action of A, we obviously have HomA (V, W) ~ HomA (V, Wcc). Thus
HomA (V, W) ~ HomA (V, W ~c)
Horn A (V, W c*) ([10], cor. 3.8)
~ (V @A We) *. ([10], cor. 3.21)
206 H A R A L D E . K R O G S T A D
t{omA (V, W) ~_ (V @A W*)* is p r o v e d similarly, a n d it is easily verified t h a t t h e resulting isomorphism has t h e f o r m stated. (Actually, if V a n d W are essential A-modules a n d A has a b o u n d e d appr. id., t h e n V @A W c ~-- V @~ W*:
Viewing A as b o t h a left a n d a r i g h t A - m o d u l e one has:
V @A W* ~_ (V | A) @A W* ~ V @A (A @A W*) ~ V @ A ( W * ) ~ _ V @ A W t ) PROt'OSITION 1.3. I f A, V, and W are as in prop. 1.2, and in addition the action of A on V is weakly compact, then:
Horn A (V, W) ~ H o m A (W *, V *) ~_ I-Iota A (~g*, V*).
Proof.
H o m A (W*, V*) ~ (W* @A V)* ~ (V @A W*)*
--~- Horn A (V, W) ~ (V @A W*) * ~ ( W* @A V*~) * HomA (W ~, Vc).
_Remark 1.1. P r o p o s i t i o n 1.1 generalizes a result of K . de L e e u w ([9], t h m . 4.2).
I f G is a locally c o m p a c t group a n d {Tg}~eG is a n isometric, s t r o n g l y continuous r e p r e s e n t a t i o n of G on a B a n a c h space V, t h e n V becomes an essential B a n a c h Ll(G)-module b y t h e composition
f o v = f Tgvf(g)dg, f 6 LI(G), v C V. (1.1) I n this case V c is e x a c t l y t h e closed linear subspace of V* on which t h e a d j o i n t r e p r e s e n t a t i o n is s t r o n g l y continuous, de L e e u w states t h a t if G is c o m p a c t a n d Abelian, t h e n V ~ V ~ if for each character (x, y), y 6 G, (x, y) o V has a finite dimension, or a t least is a reflexive subspace of V. N o w t h e first of these con- ditions implies t h a t t h e o p e r a t o r v - + (x, y) o v is compact, a n d t h e second t h a t it is w e a k l y c o m p a c t b y Corollary 3, p. 483 in [2]. W h e n G is compact, LI(G) has a b o u n d e d apr. id. consisting of t r i g o n o m e t r i c polynomials, a n d so condition (iii) in prop. 1.1 is clearly satisfied in de L e e u w ' s case.
_Remark 1.2. F o r a n e x a m p l e where V QL__ VcC, t a k e A = LI(R), V = C~
w i t h t h e usual convolution as composition:
T h e n V* = M ( R ) , V r --~ LI(R), V ~* = L ~ ( R ) , a n d V cc = C~(R), t h e space of bounded, u n i f o r m l y continuous functions.
2. Homogeneous Banach spaces
L e t V a n d T be as in R e m a r k 1.1, a n d let the group G be infinite, compact, a n d Abelian. I f we denote T a f b y fo, We have
M U L T I P L I E I ~ S O N H O M O G E N E O U S B A N A C H S P A C E S O N C O M P A C T G I ~ O U P S 207
(i) f ( - + 0 = (f~)b, f e V, a, b C G .
(ii) [lf~l[ =/[flL for all f ~ V, a C a. (2.1)
(iii) lira - fll ~ o ~" O.
W e also assume t h e existence of a F o u r i e r t r a n s f o r m on V which has t h e usual properties:
(i) f---~f(7) is a continuous linear functional for every y q G.
(ii) f
= 0 i f a n d o n l y i f f(7) = 0 for all 7 e G. (2.2) (iii) fa(7) = (-- a, y)f(y).W e denote a B a n a c h space which satisfies 2.1 a n d 2.2 a homogeneous Banach space (HBS). This n o t i o n has been used b y various a u t h o r s for certain subspaces of LI(G) where t h e F o u r i e r t r a n s f o r m has been i n h e r i t e d from LI(G), a n d T has been t h e regular representation, i.e. f a ( x ) = f ( x - a). The spaces C(G) a n d Lo(G), 1 < p < 0% are well k n o w n examples. B u t it is convenient n o t to restrict oneself to spaces of functions on G: LP(G), 1 < p < 0% becomes a H B S if we t a k e t h e " F o u r i e r t r a n s f o r m " to be t h e i d e n t i t y m a p p i n g a n d t h e action of G to be t h e m u l t i p l i c a t i o n w i t h a character: (Taf)(7) ~ (-- a, 7)f(7).
A H B S is a n essential Ll(G)-module i f t h e action is d e f i n e d as in eq. 1.I. Ob- / \
viously, g of(y) = ~(y)f(7) for all 7 C G , f E V, g E L~(G), a n d b y 2.2 (ii), (x, 7) o V is either {0} or a one d i m e n t i o n a l subspace of V. Thus t h e action of LI(G) on V is compact, a n d t h e results of w 1 apply. Define {ev}re ~ b y %(~) = ~ if (x, 7) o V r 0 i f (x, 7) o V = { 0 } . T h e n clearly {e~} spans V.
The a d j o i n t r e p r e s e n t a t i o n of G on V* is d e f i n e d b y #~(f) = #(fa), I ~ E V*, f C V. I t will be isometric b u t n o t necessarily s t r o n g l y continuous. W e define t h e
F o u r i e r t r a n s f o r m of a continuous linear f u n c t i o n a l b y
~(r) = ~(e), u e v*, r c ~.
The m a p p i n g /~-+/~ satisfies 2.2, a n d therefore, i f P inherits t h e F o u r i e r t r a n s f o r m from V*, it becomes a H B S .
PROPOSITION 2.1. Let V and W be H B S ' s which we also consider as modules. Then V @L~(r W is a H B S i f we define
(i) ( f @ g ) a = f Q g o , f e V , g e W , a c G .
/ \
(ii) f @ g(y) = f(7)~(7), f e V, g C W, y E G.
LI(G)
Proof. I t is well k n o w n a n d easily p r o v e d t h a t (i) defines a n isometric, strongly c o n t i n u o u s r e p r e s e n t a t i o n on V @n, W (See [I0], cor. 3.10). The F o u r i e r t r a n s f o r m d e f i n e d b y (ii) clearly satisfies eq. 2.2 (i) a n d (iii). L e t
208 H A ] ~ A L D E o : E : R O G S T A D o9
c v | w , v = ~ L o g,, Z [ILl[Hail < 00.
i = I i
I f we define t h e a c t i o n of LI(G) on V @L1 W b y eq. (1.1), we see t h a t if a n d o n l y if (x, 7) o ~ = 0 f o r all 7 E G . B u t
= X f , r ((x, 7) o g,)
i
= E L | ((x, 7)~ (x, 7)~ g,)
i
= ~ ((x, 7) o f,) | ((x, 7) ~ g,)
i
~ g
= ~f,(r)g,(r)e~ 0 e~,
i
which shows t h a t ~ ~ 0 if a n d only if
f d Y ) g , ( Y ) = F ( 7 ) = 0 for all y e G .
i
3. Multipliers o n H B S ' s
I f X a n d Y are H B S ' s or possibly t h e duals o f such spaces, we s a y t h a t
= {r is a (X, Y)-multiplier if for all f e X, ~ f is t h e F o u r i e r t r a n s f o r m of a n element in Y. The set of all (X, Y)-multipliers is d e n o t e d b y (X, Y). The connection between multipliers a n d " t r a n s l a t i o n i n v a r i a n t " operators is well known:
PROPOSITION 3.1. Let V and W be H B S ' s and let U: V - + W be a linear operator. Then the following are equivalent:
(i) There is a r C (V, W) such that U f ( 7 ) = r for all f C V, T C G.
(ii) U is continuous and (Uf)o = U(fa) for all f C V, a E G.
(iii) U is continuous and U(h o f ) ---- h o Uf for all f C V, h C Li(G).
W e o m i t t h e proof, b u t r e m a r k t h a t while (iii) ~=~ (i) ~ (ii) holds even i f we s u b s t i t u t e V a n d W b y V*, W* resp., (ii) does n o t necessarily i m p l y (i) in this ease. (See [12], p. 220 for a c o u n t e r example).
The n e x t proposition is also simple to prove:
PROPOSITIOS; 3.2. I f V and W are HBS's, then the following identities obtain:
(i) ( v , w ) = ( w * , v * ) = ( w ~, vo), (ii) (V, W * ) = (V, W~),
(iii) (V, W*) = (W, V*).
M U L T I P L I E R S O N t : I O I V I O G E N E O U S B A N A C / { S P A C E S O~q C O ~ s G R O U I ~ S 209
We assign to each r C (V, W) a corresponding multiplier operator U~ defined b y Ur = r We identify multipliers which generate the same operators, and we norm (V, W) b y setting 1[r = HU~H 9 This gives us b y prop. 3.1 an isometric isomorphism of (V, W) onto HomL~(c ) (V, W).
PROPOSITION" 3.3. Let V and W be H B S ' s . Then there exists an isometric iso- morphism
i: (V, W) - ~ (V @.(c) Wg*
such that
I x ,
i r 1 6 2 for all C e ( V , W ) , y e G .
Proof. The existence and the natural definition of i follows immediately from
A ,
prop. 1.2 and the remarks above. I t remains to prove that ir = r
/ x
l;[ie ~
i r : i r ff @ L 1 e• )
W c V
= er (Ucer) (Prop. 1.2)
V ^
= (U+e~) (~,)
V ^
= (r (~) = r
We remark t h a t b y prop. 2.1, V @L~ Wc is a H B S , and thus the multipliers between two arbitrary H B S ' s can always be identified with the dual space of a H B S .
COROLLARY 3.1. I f V is a H B S , then (V, V) ~ (V @L~(G ~Vc) *.
As in the case of the LP-spaces, 1 < p < 0% (See [4], [11]), we can identify V @Ll(c) V ~ with a Banach space of continuous functions:
Let •: V @LI(~)Vc---~C(G) be the lifting of the Ll(G)-balaneed operator T: V • Vc-+ C(G), T ( f , #)(t) = # ( f , ) . We first prove that ~ is injective:
Let ~ = Z ' f ' | ~' e ker W, Z ' Ilf, ti/)**i[ < oo.
( ~ ) ( t ) = ~ tti((fi)_,), and, as the series converges uniformly, we calculate its Fourier transform:
o = ~ r
= Z ~,((fl-,l^(Y)
i
(3.1)
210 ~ A R A L D E . ~ R O G S T N D
i
= ~ #,(7)fi(7).
i
B y prop. 2.1, ~ ~ 0.
L e t A r d e n o t e t h e r a n g e of T . T o m a k e A v a B a n a c h space, it n o w suffices
~o define ]ITalIA V ~ 1]~]1. Observe also t h a t t h e i n d u c e d r e p r e s e n t a t i o n a n d F o u r i e r t r a n s f o r m on A ~. coincide w i t h t h e canonical.
Remark 3.i. G non Abelian. L e t G be a n o n - A b e l i a n c o m p a c t g r o u p w i t h dual o b j e c t X, t h a t is, t h e set of e q u i v a l e n c e classes of c o n t i n u o u s i r r e d u c i b l e r e p r e - s e n t a t i o n s of G. F o r each ~ E X, we choose a f i x e d r e p r e s e n t a t i o n U ~ on t h e t I i l b e r t space H a, a n d t h e set ]-[oez H o m ( H ~ is d e n o t e d ~(X) (See [6], ch. 7).
I t is n o w c o n v e n i e n t to d e f i n e a H B S V as a B a n a c h space on w h i c h G acts i s o m e t r i c a l l y a n d s t r o n g l y c o n t i n u o u s in a w a y r e s e m b l i n g b o t h t h e left a n d t h e r i g h t r e g u l a r r e p r e s e n t a t i o n s . Moreover, we a s s u m e t o h a v e a m a p p i n g f r o m V t o
~ ( X ) , f - + f , similar t o (2.2):
(i) f = 0 i f a n d o n l y i f f ( a ) - 0 for all r
/ ~ , _ _ ^
(ii) Laf(a)-~ U:f(a), for a e G , a E X ,
/ % . ~ - -
(iii) R,f(a) ~-f(~)U:, for a E G, a e X.
F a m i l i e a r spaces like LP(G), 1 < 29 < oo, C(G), a n d also (~P(Z), 1 < p < oo, d e f i n e d in [6] p. 77, h a v e all t h e s e p r o p e r t i e s . T h e results in w 2 a n d 3 can n o w be generalized w i t h a p p r o p r i a t e modifications. T h e t e n s o r p r o d u c t V @ L l W equals V @7 W / K w h e r e K is t h e closed subspaee o f V @r W s p a n n e d b y e l e m e n t s o f t h e f o r m (fG R~fh(x)dx) @ g -- f @ ( f c L,gh(x)dx), f C V, g C W, h E I_P(G).
V QL, W b e c o m e s a H B S if we define:
(i) L~(f Q g) -~ (L~f) Q g, (ii) R , ( f ~ g) = f ~ (Rag),
/ \
(iii) f ~ g(a) - f(a)g(a).
P r o p o s i t i o n 3.1 has one left a n d one r i g h t h a n d e d version, a n d in p r o p o s i t i o n 3.2, t h e class o f left multipliers f r o m V t o W, d e f i n e d in t h e obvious way, is equal to t h e class o f r i g h t m u l t i p l i e r s f r o m W* t o V*.
Remark 3.2. A p p l i c a t i o n s of prop. 3.3. Rieffel [10] has s h o w n t h a t L~(G) ~L,(V) V ~ V GL,(G) L~(G) ~- V
i f V is an essential Ll(G)-module and LI(G) is c o n s i d e r e d as a n essential LI(G) - m o d u l e o v e r itself. This gives us t h e i d e n t i f i c a t i o n s :
(L~(G), V) ~_ (LI(G) Qc~(G) V~) * ~ ( V~) *, (V, C(G)) ~ (V QL~(G) L~(G)) * ~ V*.
N i U L T I P L I E R S O N t I O 2 r ] 3 A N A C I - I S P A C E S O N C O I V I P A C T G R O U P S 211
Then (LI(G),
L~(G)) ~ C(G)* ~ M(G), (LI(G), LP(G)) ~ Lq(G)* ~ LP(G), (LI(G), C(G)) ~ LI(G) * ~ L~(G)
etc.I f we look at the table p. 410--411 in [6], proposition 3.3 provides a tensor product .characterization of all pairs (X, Y) in the table where X is a
H B S
(Recall t h a t (V, W*)--~ (V, We)). T h a t is, all rows except the ones for~+, M(G), L~176
B u t if X and Y both are dual spaces, we m a y use prop. 3.2 (i).Multipliers between Lmspaces have been characterized by means of tensor products by I~ieffel [11]. I f 1 < p, q < o9, one identifies the tensor products with Banach spaces of integrable (or even continuous) functions by examining the lifting of the operator ~P:
LP(G) • Lq(G) --+
LI(G), ~ ( f , g) = f * g. One can prove t h a t ~P is injective by a similar calculation to eq. 3.1.R. Larsen has considered the multipliers on the spaces
Ap(G)
= {f; Ilf[IL~(G) +llfl[L~(~)
< ~}.I f 1 _< p < ~ , these are easily seen to be
HBS's,
in particular, if G is compact,(Ae(G), Ap(G))
can be identified with the dual space of a Banach space of continuous functions as proved by Larsen [8], pp. 207.Remark
3.3. F u r t h e r results. Here we mention some other simple results which might be of interest:a) In prop. 3.3, i({r U~ is compact}) = (V @L1 We) c. The proof is essentially the same as the proof of thm. 3.1 (ii) in [1].
b) The representation of multipliers as Fourier transforms of continuous linear funetionals on certain Banach spaces is essentially unique: I f V, M, X, Y are
HBS's
and X* ~ (V, W) ~ Y*, t h e n there exists a continuous iso-/ N
morphism j from X onto Y such t h a t
jx(y)
= y(~). This can be used to identify the tensor products if the multiplier classes are known.e) I f V is a
H B S
such t h a t (x,y) o V # { 0 } for all y C G , t h e nA(G) = { f i f e
LI(G)} ~A v C__ C(G).
d) (Ar, A v ) ~ ( V , V).
Remark
3.4. G non-compact. I f G is non-compact, much of the above t h e o r y breaks down. First, to assume the existence of a mapping like(2.2)
is now a more serious restietion t h a n it was in the compact ease. Secondly, as the example in remark 1.2 shows, it is easy to find eases where the action ofLI(G)
is not weakly compact. In this particular example, the conclusion of proposition 1.3 is nevertheless true:HomL1 (C ~ C ~ ~ M (We omit the elementary argument) HomLI(L 1,L 1 ) ~ M (See e.g. [8], p. 3)
H o m L, (M, M) ~ (M @L~ CO) * -~ M.
212 ~AR&LD E. KROGSTAD
However, if we compute Horn v (L ~, L~), we get
HOME1 (L ~, L ~) ~ (L ~ @L~ L1) * --~ (C~) *, that is,
H0mL~ (L 1, L 1) ~ - HomL~ (L ~, L~).
References
1. BACHELIS, G. F. a n d GILBERT, J. E.: Banach spaces of compact multipliers a n d their dual spaces, Math. Z. 125 (1972), 285--297.
2. DVX~OBD, N. a n d SCHWARTZ, J. T.: Linear operators, vol. 1, Interscience Publishers, Inc.
(1958).
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4. FIGX-TALA~ANCA, A.: T r a n s l a t i o n i n v a r i a n t operators in L/, Duke Math. J. 32 (1965), 495--502.
5. GROTHENDIECK, A.: Produits tensoriels topologiques et espaces nuclgaires, Mere. A m . Math. Soc., no. 16 (1955).
6. tiEWlTT, E. a n d Ross, K. A.: Abstract harmonic analysis, vol. 2, Springer (1970).
7. KROGSTAD, ~ . E.: Homogeneous Banach spaces and summability methods for Fourier series.
The Technical LTniversity of Norway (1973). (Norwegian, unpublished.) 8. LA~SEN, R.: A n introduction to the theory of multipliers, Springer (1971).
9. LEEUW, K. DE: Linear spaces with a compact group of operators, Ill. J. Math. 2 (1958), 367--377.
10. I~IEFFEL, M. A.: I n d u c e d Banach representations of Banach algebras a n d locally compact groups, J. Func. Anal. 1 (1967), 443--491.
11. --)>-- Multipliers a n d tensor products of LP-spaees of locally compact groups, Stud. Math.
33 (1969), 71--82.
12. RVDIN, V~r.: I n v a r i a n t measures on L ~ Stud. Math. 44 (1972), 219--227.
Received November 15, 1974 t t a r a l d E. Krogstad
