TENSOR PRODUCTS OF COMMUTATIVE | ANACH ALGEBRAS

Let A1, A2 be commutative semisimple Banach algebras and A1⊗∂A2 be their projective tensor product. We prove that, if A1⊗∂A2 is a group algebra (measure algebra) of a locally compact abelian group, then so are A1 and A2. As a consequence, we prove that, if G is a locally compact abelian group and A is a comutative semi-simple Banach algebra, then the Banach algebra L1(G,A) of A-valued Bochner integrable functions on G is a group algebra if and only if A is a group algebra. Furthermore, if A has the Radon-Nikodym property, then the Banach algebra M(G,A) of A-valued regular Borel measures of bounded variation on G is a measure algebra only if A is a measure algebra.

U.B. T#ARI, M. DUTTA AND SHOBHA MADAN locally compact abelian group H, then it is well known (Theorem 3.2 ot [3]) that LI(G,A) is isometrically isomorphic to LI(G x H).Thus LI(G,A) is a group algebra if A is a group algebra.We shall prove that the converse is also true.
There is another way of looking at this problem.It is known that LI(G,A) is iso- metrically isomorphic to LI(G) @2 A (see 6.5 of [4]).Thus, if A 1 and A 2 are group algebras, then so is A 1 @ A 2. Conversely, we shall show that, if A I and A 2 are two commutative Banach algebras and A 1 @2 A2 is a group algebra, then so are A 1 and A 2.
It seems proper to remark that we are concluding properties for A 1 and A2, assuming corresponding properties for A 1 @ A 2. This is in contrast to the appro.ch of Gelbaum [5] and [6].Our result for LI(G,A) readily follows from this.The main tool in our investigation is a theorem of Rieffel [7] characterizing group algebras.In this paper, Rieffel also characterized measure algebras.Accordingly, we investigate whether the fact that A 1 @2 A2 is a measure algebra implies that A 1 and A 2 are measure algebras.We shall show that this is indeed the case.As a consequence, we shall show that, if A is a commutative Banach algebra having the Radon Nikodym property and M(G,A) is the Banach algebra of A-valued regular Borel measures of bounded variation on G, then M(G,A) is a measure algebra only if A is a measure algebra.

PREL IMINARIES.
Let E and F be Banach spaces.The projective tensor product of E and F (see [8]) is denoted by E $ F. Every element t E @2 F can be expressed as t E e @ f with each e E and f F such that E II eil] fill < oo.here, as well as elsewhere, -fer o the Porms in the spaces containing the elements, t g o t and t f o t define bounded linear maps from E @3 F to E and E @3 F to F, respectively.These maps will be frequently used in the sequel.
L 1 Let (S,F,%.)be a measure space and X be a Banach space (S,X) denotes the Banach space of X-valued functions integrable with respect to %.We shall often use the fact that LI(s) O X is isometrically isomorphic to LI(s,x).
Gelbaum [5] and Tomiyama [9] have shown that, if A and B are commutative Banach algebras, then A @3 B forms a commutative Banach algebra whose maximal ideal space is homeomorphic to the cartesian product of the maximal ideal spaces of A and B. The maximal ideal space of a commutative Banach algebra A will be denoted by A(A).An element of A(A) will be regarded as a multiplicative linear functional (m.l.f.) of A. All the Banach algebras in our discussion will be taken to be corn- mutative and semisimple.It is proved in [6] that A @3 B has an identity if and only if both A and B have identities.It is also known [6] that A @3 B is Tauber- ian if A and B are Tauberian.The following lemma, though simple, does not seem to have appeared in print.
LEMMA 2.1 If A @ B is Tauberian, then so are A and B.
PROOF.Let us show that B is Tauberian.It can be shown in the same way that A is Tauberian.Let b B and e 0. Take A(A) and a A such that (a) i.
Let t a @ b. Choose s e A @3 B such that has compact support K and IIs-tll < .
Let K I { e A(B): (@,) K}.Then K I is compact.Let (S,F.)be a measurable space and X be a Banach space.Let be an X-valued set function on Y..The total variation V() of is defined for any E S as the supremum being taken for all possible choices of E.'s.
1 An X-valued measure on (S,7.) is a countably additive set function from 7. into X.
is said to be of bounded variation if V() is finite.The space M(S,Y.,X) of X-valued measures of bounded variation on S forms a Banach space under the norm Let , be a positive measure on (S,7.) and LI(S,X) be the Banach space of X- L 1 valued functions on S, integrable with respect to .If F (S,X), then we can define the mapping F: 7. X by F(E) f F d%. Then F is an X-valued measure E of bounded variation on S. Let (S,7.,X).We say that has the derivative F with respect to % if equals F for F LI(s,x).We say that X has the Radon- Nikodym property (X has RNP) if every X-valued measure of bounded variation on an arbitrary measurable space (S,7.) has a derivative with respect to V().If X is separable and the dual of a Banach space or is reflexive, then X has RNP (see [i0] and [ii]).An example of a separable Banach space which does not have RNP is LI[0,1] (see [12]).
Let G be a locally compact abellan group and let A be a commutative Banach algebra.M(G,A) denotes the Banach space of A-valued regular Borel measures of bounded variation on G. Suppose the range of every M(G,A) is separable.This is true if A has RNP or if G is second countable.Under these conditions, we can define the convolution of measures and belonging to M(G,A).This makes M(G,A) L I a co-utatlve Banach algebra (see [13]) The algebra (G,A) is an ideal in M(G,A) (see [14]).There is a natural isometric isomorphism from M(G) 83 A into M(G,A) (Theorem 4.2 of [15]).This is a Banach algebra isomorphism and, if A has RNP, following conditions are satisfied: (2) P is a lattice.IV and ^respectively denote supremum and inflmum.Re (a) a I where a a I + i a 2, a i E Rm].We note that if (i) (3) hold, then Rm forms a real ab- stract L-space in the sense of Kakutani [16], and hence R is a boundedly complete m lattice (see page 35 of [7]).Therefore, lal is well defined.
In [7], a L'-induclng m.l.f, is defined to be a m.l.f, which satisfies the following condition in addition to (i) (5).
However, White [17] has shown that a m.l.f, satisfying (i)-( 5) automatically satisfies (6), and hence our definition is equivalent to that of [7].We now state Rieffel's characterization of a group algebra.
THEOREM R I. Let A be a commutative semlsimple Banach algebra.A is a group algebra if and only if (a) every m.l.f, of A is L'-induclng, and (b) A is Tauberian.
Let A be a commutative semisimple Banach algebra and let D be the collection of L'-inducing m.l.f.'s of A. Consider the *-topology on D. A continuous function p on D is said to be a D-Eberleln function if there exists a constant k > 0 such that for any choice of points ml"'''mn of D and scalars l'''''an; we have n n IE 1 i P(mi) < k lllE i mill ,"

A
The following theorem of Rieffel characterizes a measure algebra.
THEOREM R 2. Let A be a commutative Banach algebra and let D be the set of L'inducing m.l.f.'s of A. Then A is a measure algebra if and only if (1) D is a separating family of linear functionals of A, (ii) D is locally compact in the *-topology, and (lii) every D-Eberlein function is the restriction to D of the Gelfand trans- form of some element of A.
The 'if' part is nothing but Theorem B of [7].The 'only if' part follows from the following and the familiar properties of Fourier-Stieltjes transforms.
PROPOSITION 2.1.The L'-inducing m.l.f.'s of M(G) are precisely those that are given by F, the dual of G.
PROOF.Let S be the structure semigroup of M(G) (see 4.3 of [18]).M(G) can be identified (3.2 of [18]) with a weak*-dense subalgebra of M(S).Under tlis identification, the m.l.f.'s of M(G) are given by , the collection of semicharact- ers of S. Let f e .Then, using the arguments of Proposition 2.5 of [7] (see also Proposition 2.8 of [7]), we can prove that f represents an h'-inducing m.l.f.

IN RESULT.
Our main result is the following theorem.All oter results are derived as a consequence of this.
PROOF.Suppose ,] is L'-inducing.We shall show that satisfies (i) (5) for # to be L'-inducing.Since I ii]ii _ iI%'II -< i, if follows thai iii.il'il i.Let P it A: (t) iltil} and P {r A I" (r) ri[ .Choose A 2 = a fixed s such that %(s) iIs;.i.Let t P and r o t.Then ,(r) 'Ii.Therefore, r e PC.. On the other hand, if r P then .l(r@ s) %(r) v,(s) (r) ril ir ilsll .r@ s ', and so r @ s P rl,r2 A and r i -r2, then r I @ s >_ r 2 @ s.
Now, let ,r 2 P. len it is easy to see that r o ((r I @ s) v (r 2 @ s)) and r I A ro_ o ((r @ s) A (r @ s)).Fr ex,mple, if- r o ((r I @ s) v (r 2 @ s)), then, since (r @ s) v (r 2 @ s) r! @ s [f follows that r r].Similarly, r r 2. On the other hand, if r' r and r' r2, tle and not on s.Therefore, P is a lattice.We can also see that (r I v r2) s (r I @ s) v (r 2 @ s) and (r I ^r2) @ s (r I O s) (r 2 @ s).For example, it is obvious that (r I v r2) @ s -> (r I O s) v (r 2 @ s) and fur thermo re, ll(r I v r2) @ s-(r I s) v (r 2 @ s)l I] [(r I v r2) @ s-(r I @ s) v (r 2 @ s)] Next, if t R and r < R, then # o t R and r 0 s R;,.Moreover, all the above relations are true for r I v r 2 and r I ^r2 for rl,r2 e R. Now, let rl,r 2 R and r I A r 2 0. Then r I s r 2 @ s R and (r I s) A (r 2 @ s) 0. Therefore lit I 8 s + r 2 @ all =llr I @ s r 2 si, and hence llr I + r 2 =llr I r21 I. Hence sat- isfies (3).
Suppose now that r A I. Then r @ s A and r @ s t I + it2, with tlt2 R.
Then r # o (r @ s) o t I + i o t 2. Also, if r r I + ir 2 r 3 + ir 4 for r i e R, then r @ s r I @ s + i r 2 @ s r 3 @ s + i r 4 @ s.Therefore, We have also --(Re r@ s Re (r @ s).Thus satisfies (4).We now show that satin- shown that fies (5).Let r A I. First, we show that Irl ' o Ir 8 s and Ir 8 s Ir 8s.
We have o r O sl Re (eir) o r O sl .o (Re (e i0 r@s)) o [ir @ s Re (e i rOs)], for every @ [0,2].Therefore, q o Ir s -> Irl.On the other hand, rl _> Re (e i i0 (el0 r).Hence Irl O s >-Re (e r) O s Re (r @ s)).Therefore, Therefore, Irl O s Ir O s I. Now [I ;rl II llsll =If ir; II.This proves that satisfies (5).Hence is L'-inducing.
We can show similarly that is L'-inducing.
U.B. TEWARI M. DUTTA AND SIOBBA MADAN Conversely, suppose and are L'-inducing.We shall show that Q is L'-indu- cing.It is obvious that IIII I. Since is L'-inducing m.l.f, of AI, by Propo- sition 2.3 of [7], there exists a locally compact Hausdorff space X and a positive regular Borel measure on X such that A I is isometrically limear isomorphic and, under the order induced by , order isomorphic to LI(x,).The dual of A I is then represented by L(X,) and, under this representation, is represented by the corn- sCant function II#II i on X.Now, O A 2 LI(x,) , A 2 LI(x,,A2 ereaf- ter, we shall not distinguish between elemnts of A and L I(x,,) and, for L 1 F (X,,A2), statements llke "F A" will be used without elanatlon.For F A, we observe that F P n if and only if o F P. This is so, because IIFII >-Ii o o F) I.

du(x)
This shows that F E P if and only if F(x) P a.e. (U).Let FI,F 2 E Pn" Using the continuity and other properties of the lattice operations, it is easy to prove that the function F 1 v F 2 defined a.e.(U) by (F 1 v F2)(x),-Fl(X v F 2(x), belongs to LI(x,,A2) and consequently defines an element of PQ.This proves that P is a lattice.Other details involved in showing that N is L'-inducing are also now easy to verify and hence we omit them.This completes the proof of our Theorem.
Having proved our main theorem, we now proceed to give its consequences.
THEOREM 3.2.Let A 1 and A 2 be commutative semislmple Banach algebras.Then A 1 8 A 2 is a group algebra if and only if A 1 and A 2 are group algebras.
PROOF.As mentioned in the introduction, it is well known that, if A 1 and A 2 are group algebras, then so is A 1 8 A 2. The converse follows from Lemma 2.1, Theorem R 1 and Theorem 3.1.
The following is an immediate consequence of Theorem 3.2.
THEOREM 3.3.Let G be a locally compact abelian group and let A be a commu- tative semlslmple Banach algebra.ThnLI(G,A) is a group algebra iff A is a group algebra.
PROOF.The result follows from Theorem 3.2 and the fact that the Banach al- gebras LI(G,A) and LI(G) A are isometrically isomorphic.
THEOREM 3.4.Let A I and A 2 be commutative semisimple Banach algebras and A A I A 2. If A is a measure algebra, then A I and A 2 are measure algebras.
PROOF.Let D, D1, D 2 be the set of L'-inducing m.l.f.'s of A, AI, and A 2 re- spectively.Theorem 3.1 implies that D D 1 x D 2. Since D satisfies condition (i) of Theorem R2, it easily follows that D I and D 2 also satisfy this condition.
Since D is locally compact in the *-topology, D I and D 2 are also locally compact in the *-topology.It remains to show that A I and A 2 satisfy condition (iii) of Theorem R 2.
We shall do this for A2, the case of A I being similar.Since A is a measure algebra, it has an identity.It follows that A I and A 2 have identities.
Let e be the identity of A I. Let p be a D2-Eberleln function.Define the function P on D by P(,) p().Obviously, P is continuous.Moreover, n n lie i P($i'i )I lie i P(i )I < kll i i II A However, for any a A2, n n <a, 1 i i > <e @ a, 1 7 .i(i,i)> n Therefore, I I i i II < I 1 i($1 i)llA ,.This shows that P is a D-Eberlein funct- i A ion and therefore there exists t A such that () P() for every t A(A).
Choose A(A I) and let b o t.Then () p() for every A(A2).This shows that A 2 satisfies condition (iii) of Theorem R 2 and the proof of our theorem is complete.
THEOREM 3.5.Let G be a locally compact abelian group and A be a commutative semisimple Banach algebra having RNP.Then M(G,A) is a measure algebra only if A is a measure algebra.
PROOF.The theorem follows from Theorem 3.4 and the fact that the algebras M(G) A and M(G,A) are isometrically isomorphic under the hypothesis of our theorem.
then it is onto (Theorem 4.4 of [15]) Let A be a commutative and semlslmple Banach algebra and m 6 A(A).Let {a e A: re(a) IIm[[ [I[}.Then P is a cone in A and therefore introduces P m m an order in A. Let R {a-b: a,b e P }. m is said to be L'-inducing if the m m

m ( 3 ) 2 ( 5 )
If a,b e R and a ^b 0 then (4) If a A, then there exists unique elements, al, a 2 Rm, such that a a I + i a Let lal V{Re (e i8 a): 8 [0,2]}.Then flail II lal II The norms