I nternat. J. Math. Math. Si.
Vol. 5 No. 3 (1982) 503-512
503
TENSOR PRODUCTS OF COMMUTATIVE |ANACH ALGEBRAS
U.|.
TEWARI, M. DUTTA mid SiNi04A MADAN
Department of Mathematics
Indian Institute of Technology
Kanpur-208016, U.P.
INDIA
ABSTRACT.
Let
AI,
be commutative semlslmple Banach algebras and A
1
their projective tensor product.
We prove that, if A
10
02
A be
2
A is a group algebra
2
(measure algebra) of a locally compact abelian group, then so are A and A
2. As a
1
consequence, we prove that, if G is a locally compact abelian group and A is a
comutatlve semi-simple Banach algebra, then the Banach algebra
LI(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 easure algebra.
KEY WORDS AND PHRASES. Comive semisimple Banach algebra, pjective tensor
pdu, group algebra, measure algebra, locally .mlz. abelian group, RadonNikodym property.
AMS (MOS) SUBJECT CLASSIFICATION CODES (1970).
i.
Primary 43AI0, 43A20, 46J35.
INTRODUCTION.
Let A be a commutative Banach algebra.
We shall say that A is a group algebra
(measure algebra) if A Is isometrically isomorphic to
compact abelian group G.
LI(G)
(M(G)) for some locally
Let G be a locally compact abellan group and A be a com-
mutative semi-simple Banach algebra.
The space
LI(G,A)
of A-valued Bochner inte-
grable functions on G becomes a commutative Banach algebra (see [I], [2] and [3]).
A natural question arises:
when is
LI(G,A)
a group algebra?
If A
LI(H)
for some
U.B. T#ARI, M. DUTTA AND SHOBHA MADAN
504
locally compact abelian group H, then it is well known (Theorem 3.2 ot [3])
LI(G,A)
that
LI(G
is isometrically isomorphic to
algebra if A is a group algebra.
x H).
Thus
LI(G,A)
is a group
We shall prove that the converse is also true.
It is known that
There is another way of looking at this problem.
LI(G,A)
is iso-
LI(G) @2 A (see 6.5 of [4]). Thus, if A1 and A2 are group
A
Conversely, we shall show that, if A and A 2 are
algebras, then so is A
I
1 @ 2.
so are A
then
A
and
is
a
two commutative Banach algebras
group algebra,
1
1 @2 A2
metrically isomorphic to
and A
It seems proper to remark that we are concluding properties for A
2.
assuming corresponding properties for A
of Gelbaum
[5] and [6].
1
Our result for
@ A2.
LI(G,A)
readily follows from this.
and A
2
@2 A2
1
The
Accordingly,
is a measure algebra implies that A
As a
We shall show that this is indeed the case.
are measure algebras.
A2,
[7] characterizing group
In this paper, Rieffel also characterized measure algebras.
we investigate whether the fact that A
and
This is in contrast to the appro.ch
main tool in our investigation is a theorem of Rieffel
algebras.
1
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.
2.
Let E and F be Banach spaces.
[8]) is denoted by E
$
F.
E
II tII=
E
Every element t
E and f
e @ f
with each e
i’
i
i
i=l
norm of t is given by
t
The projective tensor product of E and F (see
inf
i
F
@2
F can be expressed as
E II eil] fill
such that
< oo.
The
i=l
{xi II eill llfi]l
t
iF e.1
@
f’}i
where the infimum is taken over all possible expressions of t.
Let f
E*
Ei f,’e
f o
g
o
tl
llgli
an d t e E
E ieil Ifiil
1Ito
@2
F with t
E1 g(fi)ei
:f. and g o t
converge because
ilg
F*
<
.
E
e. @ f..
We define
i-1
The series defining f o t and g o t
It: is obvious that
11 f
o
tll illl IIll and
The norms here, as well as elsewhere, -fer o the
Porms
1
TENSOR PRODUCTS OF COMMUTATIVE BANACH ALGEBRAS
in the spaces containing the elements,
linear maps from E
F to E and E
@3
@3
g o t and t
t
505
f o t define bounded
F to F, respectively.
These maps will be
frequently used in the sequel.
1
L (S,X) denotes the
Let (S,F,%.) be a measure space and X be a Banach space
Banach space of X-valued functions integrable with respect to %.
LI(s)
use the fact that
X is isometrically isomorphic to
O
We shall often
LI(s,x).
[5] and Tomiyama [9] have shown that, if A and B are commutative
Gelbaum
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
The maximal ideal space of a commutative Banach algebra A will be denoted
and B.
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-
It is proved in [6] that A
mutative and semisimple.
only if both A and B have identities.
ian if A and B are Tauberian.
B has an identity if and
@3
It is also known [6] that A
B is Tauber-
@3
The following lemma, though simple, does not seem
to have appeared in print.
LEMMA 2.1
PROOF.
Let K
{
e
B and e
Let b
Choose s e A
a @ b.
I
B is Tauberian, then so are A and B.
Let us show that B is Tauberian.
A is Tauberian.
Let t
If A @
A(B):
supported in K
I
(@,)
0.
It can be shown in the same way that
A(A) and a
Take
@3
B such that
K}.
Then K
llqbo
t
I
A such that (a)
IIs-tll
has compact support K and
is compact.
Let x
o s.
Then
i.
<
.
is
and
lib-x[[
qb o
s[I
_<
lit- st[
< e
This proves that B is Tauberian.
Let (S,F.) be a measurable space and X be a Banach space.
set function on
Y..
The total variation V() of
Let
be an X-valued
is defined for any E
fol lows.
n
V()(E)
Sup
{E [[ (Ei)[l:
1
E.’s
disjoint
E
the supremum being taken for all possible choices of
i
E for i < i < n}
E.’s.
1
S as
U B. TEWARI, M. DUTTA AND SHOBHA MADAN
506
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
valued functions on S, integrable with respect to
define the mapping
7.
F:
X by
of bounded variation on S.
with respect to % if
F
LI(s,x).
for F
be the Banach space of X-
Then
E
(S,7.,X). We say
Let
equals
f F d%.
F(E)
.
LI(S,X)
(S,7.) has
is an X-valued measure
F
has the derivative F
that
We say that X has the Radon-
Nikodym property (X has RNP) if every X-valued measure
an arbitrary measurable space
1
L (S,X), then we can
If F
of bounded variation on
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]
An example of a separable Banach space which does
and [ii]).
LI[0,1]
not have RNP is
(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
is true if A has RNP or if G is second countable.
define the convolution of measures
a co-utatlve Banach algebra (see
(see [14]).
M(G,A)
belonging to M(G,A).
and
[13])
This
Under these conditions, we can
The algebra
I
L
This makes M(G,A)
(G,A) is an ideal in M(G,A)
There is a natural isometric isomorphism from M(G)
(Theorem 4.2 of [15]).
is separable.
83
A into M(G,A)
This is a Banach algebra isomorphism and, if A has
RNP,
then it is onto (Theorem 4.4 of [15])
Let A be a commutative and semlslmple Banach algebra and m
P
m
{a e A:
re(a)
an order in A.
IIm[[ [I[}.
Let R
{a-b:
m
Then P
m
a,b e P m }.
following conditions are satisfied:
(2)
P
(3)
If a,b e R
m
is a lattice.
and a
^
b
0
then
6
A(A).
Let
is a cone in A and therefore introduces
m is said to be
L’-inducing if the
507
TENSOR PRODUCTS OF COHHUTATIVE BANACH ALGEBRAS
(4)
(5)
IV
a
and
a
^
A, then there exists unique elements, al,
If a
a
a
Let
lal
+
I
i a
V{Re (e i8 a):
E
Rm,
2
such that
2
[0,2]}.
8
Then
respectively denote supremum and inflmum.
a
I + i 2, a i
a
Rm].
lattice (see page 35 of [7]).
Re (a)
[16], and hence Rm
lal
Therefore,
a
(3) hold, then
We note that if (i)
stract L-space in the sense of Kakutani
flail II lal II
where
I
Rm forms
a real ab-
is a boundedly complete
is well defined.
In [7], a L’-induclng m.l.f, is defined to be a m.l.f, which satisfies the
(5).
following condition in addition to (i)
(6)
For a,b
A,
la.bl
<
Ibl.
lal
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.
p on D is said to be a D-Eberleln
that for any choice of points
A continuous function
function if there exists a constant k > 0 such
ml"’’’mn
of D and scalars
n
IE1 i P(mi)
l’’’’’an;
we have
n
< k
lllE i mill A
,"
The following theorem of Rieffel characterizes a measure algebra.
THEOREM R2.
Let A be a commutative Banach algebra and let D be the set of
inducing m.l.f.’s of A.
L’-
Then A is a measure algebra if and only if
(1)
D is a separating family of linear functionals of A,
(ii)
D is
(lii)
every D-Eberlein function is the restriction to D of the Gelfand trans-
locally compact in the *-topology, and
form of some element of A.
U.B. TEWARI, M. DUTTA AND SHOBHA MADAN
508
The ’only if’ part follows
The ’if’ part is nothing but Theorem B of [7].
from the following and the familiar properties of Fourier-Stieltjes transforms.
The L’-inducing m.l.f.’s of M(G) are precisely those that
PROPOSITION 2.1.
F,
are given by
the dual of G.
Let S be the structure semigroup of M(G) (see 4.3 of [18]).
PROOF.
.
[18]) with
be identified (3.2 of
,
subalgebra of M(S).
a weak*-dense
identification, the m.l.f.’s of M(G) are given by
Let f e
ers of S.
M(G) can
Under tlis
the collection of semicharact-
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.
if(s)
if and only if
i for all s
F
the canonical image of
in S.
By 4.3.3 of [181, f e
S.
E
i} is
;f
This proves our proposition.
IN RESULT.
3.
Our main result is the following theorem.
All oter results are derived as a
consequence of this.
THEOREM 3.1.
A=
A1
AI,A2
Let
is L’-indu ing if and only if
PROOF.
Suppose
il’il
A
a fixed s
(v
,(r)
P
r
o
A:
D(t)
t)
-
’Ii.
%(r) v,(s)
Thus we have shown that, if t
A
and r
i
r2,
((r
r
o
I
@ s) v (r
((r
I
2
@ s) v
i.
Similarly, r
r’
r
s and
r’
O s
{r
P
and r
Therefore, r
e A and t
t
I
e
ir
ril
2
2’
A
PC..
ro_
A
r
r
2.
2
satisfies (i)
(5)
I"
ri[
(r)
.
Choose
Then
o t.
ilsll .r @
then
P
s ’, and so r @ s
o t
o
and
2
if
@ s.
o
((r
@ s) A
@ s) v (r
On the other hand, if
O s
Then
On the other hand, if
v r
len it is easy to see that r
I
i:(A2).
if follows thai
P
Let t
(r)
2
and
(r 2 @ s)), then, since (r
r].
I
iI%’II -< i,
iltil}
@ s >_ r
I
@ s)) and r
that r
O s
t
I
then r
P.
Now, let r ,r
i 2
o
=
iIs;.
it
then .l(r @ s)
rl,r 2
ii]ii
.
L’-inducing.
We shall show that
(t)
such that %(s)
2
_
are
I
Since
it
Let P
i.
and
L’-inducing.
,] is
for # to be L’-inducing.
iii.
’(A I) and
(+/-) be given by (4,) for
Let ’%
@
be commutative semisimple Banach algebras and
Therefore, r’
r
(r
2
r’
@ s)).
Fr ex,mple, if-
@ s)
r!
@
r
and
r’
Note tllat r
s
[f
r2,
v
follows
tle
d
509
TENSOR PRODUCTS OF COMMUTATIVE BANACH ALGEBRAS
r
r
A
I
depend only on r
2
can also see that (r
(r
@ s).
2
r2)
v
I
and r
I
(r
s
s) v (r
@
I
For example, it is obvious that (r
P
Therefore,
and not on s.
2
@ s) and (r
2
v
I
r2)
^ r 2)
I
-> (r
@ s
We
is a lattice.
O s) v (r
I
(r
@ s
O s)
I
@ s) and
2
fur thermo re,
ll(r I
v
r2)
@ s- (r
[(r
I]
[r
R
Next, if t
relations are true for r
and r
lit I
r
A
I
8 s
+
0.
2
r
all =llr I
@
2
Then r
@ s- (r
o ((r
2
and r
2
r
s
I
@ s
r
I
r
^
si,
2
@ s) v
I
for
2
(r 2 @ s)]
(r 2 @ s))]
R;,.
r 0 s
and hence
s)
I
@ s)
2
0.
r21 I.
=llr I
llr I + r 2
R
Now, let rl,r 2
(r
A
0.
Moreover, all the above
R.
e
rl,r 2
and (r
R
@ s
2
@ s) v
I
R and
then # o t
v r
I
r2)
v r
I
R,
and r <
v
I
s)l
s) v (r 2 @
I
Therefore
Hence
sat-
isfies (3).
Suppose now that r
A
# o (r @ s)
o t
o t
@
@ s
Then r
r
r
i
I
R,
e
then r @ s
r
o s
shown that
o s and r
3
--(Re r@
fies (5).
r
+ i
I
s + i r
2
@ s
2
r
A
Also, if r
2.
r
Irl
r O
sl
Re
(eir)
o
.
o
[0,2].
for every @
rl
Irl
_>
Re (e
O s >-
i
;r
Also, since
r).
O
Irl
sl,
q
Therefore,
o
s >- Re (e
so
Irl O
that Iri
->
Ir
we get
O s _>
irl
O
sl,
;r
O s
,
o
[ir
Ir
Hence
ir
sl
r O
s
i0
Irl O s Ir
[I ;rl II llsll =If ir; II.
O s
r
2
+
3
We have also
4.
We now show that
o
(Re (e
and
8 s
i
4
for
Therefore,
r
Ir
R.
tlt 2
ir
i0
Ir
8 s
satin-
Ir
8s.
r@s))
rOs)],
On the other hand,
Re
(el0
(r @ s)).
Therefore,
O s
@ s
I.
,
o
Ir
O
sl]
0.
Now
This proves that
We can show similarly that
with
it2,
ir
2
o
Irl.
r) O s
[Irl
Therefore,
->
+
@ s.
4
Re (e
@ s
I
+
I
and r
3
’
We have
o
r
i r
r
t
satisfies (4).
Thus
First, we show that
I.
+
@ s
3
Hence, r I
@ s.
4
Re (r @ s).
s
Let r
I
A and r @ s
Then r @ s
I.
is
satisfies (5).
L’-inducing.
Hence
is
L’-inducing.
are L’-inducing.
and
Conversely, suppose
IIII
It is obvious that
cing.
M. DUTTA AND SIOBBA MADAN
TEWARI
U.B.
510
I.
We shall show that Q is L’-indu-
is L’-inducing m.l.f, of
Since
on X such that A
regular Borel measure
,
under the order induced by
represented by
L(X,)
sCant function
II#II
is isometrically limear isomorphic and,
I
order isomorphic to
LI(x,).
and, under this representation,
i on X.
O
Now,
LI(x,)
A
2
The dual of A I is then
, I(x,,)LI(x,,A2
is represented by the corn-
A
(X,,A2),
L
F
A, we observe that F
o F)
o
"F
statements llke
F
A"
and, for
will be used without
P if and only if
o
n
P.
F
ereaf-
2
ter, we shall not distinguish between elemnts of A and L
IIFII >-Ii
by Propo-
[7], there exists a locally compact Hausdorff space X and a positive
sition 2.3 of
1
AI,
For
elanatlon.
This is so, because
I.
du(x)
P if and only if F(x) P a.e.
This shows that F E
(U).
Let
FI,F2
E
Pn"
Using the continuity and other properties of the lattice operations, it is easy to
prove that the function F
belongs to
P
LI(x,,A2)
is a lattice.
v F
1
2
defined a.e. (U) by (F
1
v
and consequently defines an element
F2)(x),- Fl(X v F 2(x),
of PQ. This proves that
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.
A
1
8
A
2
Let A
1
and A be commutative semislmple Banach algebras.
2
is a group algebra if and only if A
PROOF.
1
1
and A
2
are group algebras.
As mentioned in the introduction, it is well known that, if A1 and A2
are group algebras, then so is A
Theorem R
1
Then
8 A2.
The converse follows from Lemma 2.1,
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.
algebra.
ThnLI(G,A)
is a group algebra iff A is a group
511
TENSOR PRODUCTS OF COMMUTATIVE BANACH ALGEBRAS
The result follows from Theorem 3.2 and the fact that the Banach al-
PROOF.
gebras
LI(G,A)
and
A
Let D,
PROOF.
and A
I
2
be commutative semisimple Banach algebras and
If A is a measure algebra, then A
A2.
I
A are isometrically isomorphic.
Let A
THEOREM 3.4.
A
LI(G)
D1,
D
2
are measure algebras.
be the set of L’-inducing m.l.f.’s of A,
2
D
Theorem 3.1 implies that D
spectively.
and A
I
x
1
D2.
AI,
and A
2
re-
Since D satisfies condition
it easily follows that D
also satisfy this condition.
I and D 2
Since D is locally compact in the *-topology, D and D are also locally compact
2
I
It remains to show that A and A satisfy condition (iii) of
in the *-topology.
I
2
(i) of Theorem
Theorem R
R2,
We shall do this for
2.
Since A is a
I being similar.
It follows that A and A have identities.
2
I
the case of A
A2,
measure algebra, it has an identity.
Let e be the identity of A
P on D by P(,)
lie
Let p be a
n
lie i P(i )I
i P($i’i )I
However, for any a
D2-Eberleln
function.
Obviously, P is continuous.
p().
n
I.
<
Define the function
Moreover,
kll i i II
A
A2,
n
<a,
1
i i>
<e
n
@ a,
17. i(i,i)>
n
Therefore,
Ii
i i IIA
<
I1 i($1 i)llA,.
ion and therefore there exists t
A(A I) and let b
Choose
shows that A
2
This shows that P is a D-Eberlein funct-
A such that
o t.
Then
()
()
P() for every t
p() for every
satisfies condition (iii) of Theorem R
2
A(A).
A(A2).
This
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.
M(G)
theorem.
The theorem follows from Theorem 3.4 and the fact that the algebras
A and M(G,A) are isometrically isomorphic under the hypothesis of our
U.B. TEWARI, M. DUTTA AND SHOBHA MADAN
512
REFERENCES
Group algebras of vector-valued functions, Bull. Amer. Math.
Soc. 62 (1956), 383 (abstract 493).
i.
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