WEAK AMENABILITY OF TRIANGULAR BANACH ALGEBRAS 1. Introduction In general, one can ...

18 Pages·2001·0.24 MB·English

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AMERICANMATHEMATICALSOCIETY
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S0002-9947(01)02957-9
ArticleelectronicallypublishedonDecember4,2001
WEAK AMENABILITY OF TRIANGULAR BANACH ALGEBRAS
B. E. FORREST AND L. W. MARCOUX
Abstract. Let Aand B b(cid:20)e unital Ba(cid:21)nach algebras, and let MbeaBanach
A M
A;B-module. Then T = becomes a triangular Banach algebra
0 B
(cid:20) (cid:21)
a m
whenequippedwiththeBanachspacenormk k=kakA+kmkM+
0 b
kbkB. A Banach algebra T is said to be n-weakly amenable if all derivations
from T into its nth dual space T(n) are inner. In this paper we investigate
Arensregularityandn-weakamenabilityofatriangularBanachalgebraT in
relationtothatofthealgebrasA,BandtheiractiononthemoduleM.
1. Introduction
In general, one can obtain a good deal of information about the structure of a
BanachalgebraAbystudyingthevariousHochschildcohomologygroupsHn(A;X)
ofAwithcoe(cid:14)cientsinaBanachA-bimoduleX. Forexample,thestudyofthe(cid:12)rst
cohomologygroupH1(A;X)isessentiallythestudyofinner(vs. outer)derivations
fromAinto X, while ifX is(cid:12)nite-dimensional,the study ofH2(A;X) isrelatedto
that of strong splittings of extensions of A by X [1]. For this reason, it is perhaps
discouraging to know that explicit calculations of cohomology groups tends to be
rather di(cid:14)cult, and results often focus on whether a givengroupHn(A;X) is or is
not trivial.
In [7], motivated by work of Gilfeather and Smith [8] in their study of joins of
operator algebras, we began an investigation of derivations of triangular Banach
algebras; these are algebras of the form
(cid:20) (cid:21)
A M
T = ;
0 B
whereAandBarethemselvesBanachalgebrasandMisaBanachA;B-module. In
thatpaperatechniquewasdevelopedwhich,inmanycases,allowsonetoexplicitly
compute H1(T;T) for speci(cid:12)c choices of A, B, and M.
In this paper, we show that the (cid:12)rst cohomology group for these algebras with
coe(cid:14)cients in their nth dual spaces T(n) is also often tractable. In particular, we
willdevelopcriteriafordecidingwhenH1(T;T(n))=0. Thisnotion,referredtoas
n-weak amenability and de(cid:12)ned by Dales, Ghahramani and Gr(cid:28)nb(cid:26)k in [5], splits
along two distinct lines in the setting of triangular Banachalgebras. Indeed, when
n is odd, the ((cid:12)rst) cohomology group of T depends only upon that of A and B
[see Theorem 3.7 below]. When n is even, the techniques developed in [7] can be
ReceivedbytheeditorsOctober9,1998and,inrevisedform,July20,1999.
2000 Mathematics Subject Classi(cid:12)cation. Primary46H25,16E40.
ResearchsupportedinpartbyNSERC(Canada).
(cid:13)c2001 American Mathematical Society
1435
1436 B. E. FORREST AND L. W. MARCOUX
extended to this setting, where it allows us to compute H1(T;T(2n)) for many
choices of A, B and M [see Theorem 3.12].
The body ofthis paper is divided into three parts. In Section2,we examine the
natural actions of T on its dual spaces T(n), n (cid:21) 1, and characterize when T is
Arensregular. InSection3,weobtainourtwomainresultsconcerningH1(T;T(n)),
the (cid:12)rstdealingwiththe casewhere nis odd,andthe seconddealingwiththe case
where n is even. Finally, in Section 4, we apply these results to three (classes)
of examples, including tensor products of a given C(cid:3)-algebra A by T , and (cid:12)nite
2
dimensional nest algebras.
2. Preliminaries
2.1. Let A and B be Banach algebras, and suppose X is a Banach A;B-module;
thatis,X isaBanachspace,aleftA-moduleandarightB-module,andtheactions
of A and B are continuous in that
ka(cid:14)x(cid:14)bk(cid:20)kakA kxkX kbkB:
If A has a unit eA and B has a unit eB, then X is said to be unital provided that
eA(cid:14)x=x(cid:14)eB =x for all x2X.
We cande(cid:12)ne a rightactionof A onthe dualspaceX(cid:3) ofX anda left actionof
B on X(cid:3) via
((cid:30)(cid:14)a)(x) = (cid:30)(a(cid:14)x);
(b(cid:14)(cid:30))(x) = (cid:30)(x(cid:14)b);
for all a2A; b2B; x2X, and (cid:30)2X(cid:3).
Similarly, the second dual X(2) of X becomes a Banach A;B-module under the
actions
(a(cid:14)(cid:8))((cid:30)) = (cid:8)((cid:30)(cid:14)a);
((cid:8)(cid:14)b)((cid:30)) = (cid:8)(b(cid:14)(cid:30));
for all a2A; b2B; (cid:30)2X(cid:3), and (cid:8)2X(2).
Of course, we may continue this process to higher order dual spaces of X, with
theconclusionbeingthatX(2m) isaBanachA;B-moduleandX(2m(cid:0)1) isaBanach
B;A-module for all m (cid:21) 1. When X is unital, so is each X(m), m (cid:21) 1. When
A=B, we simply have that each X(m) is a Banach A-bimodule. A particular but
important example of this is when X = A = B, and the module actions are given
by a(cid:14)x=ax and x(cid:14)a=xa for all a;x2A. The above analysis then shows that
A(m) is a Banach A-bimodule for all m(cid:21)1.
Suppose X is a Banach A-bimodule. A derivation (cid:14) : A ! X is a linear map
which satis(cid:12)es (cid:14)(ab) = (cid:14)(a)(cid:14)b+a(cid:14)(cid:14)(b) for all a;b 2 A. In this paper, we shall
only consider continuous derivations. The derivation (cid:14) is said to be inner if there
exists x 2 X such that (cid:14)(a) = (cid:14) (a) := a(cid:14)x(cid:0)x(cid:14)a for all a 2 A. Denoting
x
the linear space of bounded derivations from A into X by Z1(A;X) and the linear
subspaceof (necessarilybounded) inner derivations byN1(A;X), we may consider
the quotient space H1(A;X) = Z1(A;X)=N1(A;X), called the (cid:12)rst Hochschild
cohomology group of A with coe(cid:14)cients in X.
The Banach algebra A is said to be amenable if H1(A;X(cid:3)) = 0 for all Banach
A-bimodules X. This notion was (cid:12)rst de(cid:12)ned by Johnson in [10], who showed
that the groupalgebra L1(G) is amenable precisely when G is an amenable group.
In particular, the equivalence of amenability and nuclearity for C(cid:3)-algebras [9]
WEAK AMENABILITY OF TRIANGULAR BANACH ALGEBRAS 1437
indicates that amenability canbe viewedas a generalized\(cid:12)niteness"condition on
a Banach algebra.
Bade, Curtis and Dales [1] later de(cid:12)ned the notion of weak amenability for a
commutative Banach algebra A. They called A weakly amenable if every bounded
derivation from A into a symmetric Banach A-module is inner and hence zero.
They also showed that this is equivalent to H1(A;A(cid:3)) = 0. This latter condition,
thatH1(A;A(cid:3))=0, has been adoptedas the de(cid:12)nition ofweak amenability for an
arbitrary Banach algebra. That this is strictly weaker than amenability is clear,
since all C(cid:3)-algebras are weakly amenable [9].
More recently, Dales, Ghahramani and Gr(cid:28)nb(cid:26)k [5] have de(cid:12)ned the notions of
n-weak amenability and permanent weak amenability for a Banach algebra A as
follows: let n be a positive integer. Then A is said to be n-weakly amenable if
H1(A;A(n))=0,andAispermanently weakly amenable ifitisn-weaklyamenable
for all n(cid:21)1. Amongst other things, they were able to show that everyC(cid:3)-algebra
is permanently weakly amenable ([5], Theorem 2.1).
Inthisnoteweshallconsiderthen-weakamenabilityofaclassofBanachalgebras
we (cid:12)rst studied in [7], and which we call triangular Banach algebras. They are
de(cid:12)ned in the following way: let A and B be Banach algebras and suppose that
M is an essential Banach A;B-module; that is, A(cid:14)M = M (cid:14)B = M. (This
is guaranteed when M is unital.) We de(cid:12)ne the corresponding triangular Banach
algebra
(cid:20) (cid:21)
A M
T = ;
B
with the sum and product being given by the usual 2(cid:2)2 matrix operations and
obvious internal module actions. The norm on T is
(cid:20) (cid:21)
a m
k k:=kakA+kmkM+kbkB:
b
An important class of examples arises when H is a complex Hilbert space, A
and B are operator subalgebras of B(H), and M is an essential A;B-submodule of
B(H). T is still not quite an operator subalgebra of B(H(cid:8)H), however, since the
given norm and the operator norm do not coincide.
2.2. Arens regularity for triangular Banach algebras. Given a triangular
Banach algebra T as above, we can make A(2) and B(2) into Banach algebras by
giving them the (cid:12)rst Arens product as follows:
Let fa g;fa g be nets in A with (cid:0) =w(cid:3)-lim a and (cid:0) =w(cid:3)-lim a : Then we
i j 1 i i 2 j j
de(cid:12)ne
(cid:0) (cid:3)(cid:0) =w(cid:3)-limlima a :
1 2 i j
i j
Similarly, we can de(cid:12)ne
(cid:9) (cid:3)(cid:9) =w(cid:3)-limlimb b
1 2 i j
i j
whenever (cid:9) = w(cid:3)-lim b and (cid:9) = w(cid:3)-lim b : Moreover, we can extend the
1 i i 2 j j
actions of A on M and of B on M to actions of A(2) and B(2) on M(2) via
(cid:0)(cid:3)(cid:5)=w(cid:3)-limlima m
i j
i j
1438 B. E. FORREST AND L. W. MARCOUX
and
(cid:5)(cid:3)(cid:9)=w(cid:3)-limlimm b ;
j k
j k
where (cid:0) = w(cid:3)-lim a , (cid:5) = w(cid:3)-lim m , and (cid:9) = w(cid:3)-lim b : This allows us to
i i j j k k
show th(cid:20)at the (cid:12)rst(cid:21)A(cid:20)rens multip(cid:21)lication de(cid:12)ned on T(2) behaves in a natural way.
(cid:0) (cid:5) (cid:0) (cid:5)
Let 1 1 ; 2 2 2T(2): Assume also that
(cid:9) (cid:9)
1 2
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:0)1 (cid:5)1 =w(cid:3)-lim ai mi
(cid:9)1 i bi
and
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:0)2 (cid:5)2 =w(cid:3)-lim aj mj :
(cid:9)2 j bj
Thenit iseasytosee that(cid:0) =w(cid:3)-lim a ;(cid:0) =w(cid:3)-lim a ;(cid:9) =w(cid:3)-lim b ;(cid:9) =
1 i i 2 j j 1 i i 2
w(cid:3)-lim b ;(cid:5) =w(cid:3)-lim m and (cid:5) =w(cid:3)-lim m Moreover,
j j 1 i i 2 j j:
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
(cid:0)1 (cid:5)1 (cid:3) (cid:0)2 (cid:5)2 =w(cid:3)-limlim ai mi aj mj
(cid:9)1 (cid:20) (cid:9)2 i j (cid:21) bi bj
=w(cid:3)-limlim aiaj aimj +mibj
(cid:20) i j bibj (cid:21)
w(cid:3)-lim lim a a w(cid:3)-lim lim (a m +m b )
= i j i j i j i j i j
w(cid:3)-lim lim b b
(cid:20) i(cid:21) j i j
(cid:0) (cid:3)(cid:0) (cid:0) (cid:3)(cid:5) +(cid:5) (cid:3)(cid:9)
= 1 2 1 2 1 2 :
(cid:9) (cid:3)(cid:9)
1 2
Thus the (cid:12)rst Arens product of T behaves just like matrix multiplication, with
coordinate-level operations behaving like the (cid:12)rst Arens product of the building
blocks.
There is a second way of de(cid:12)ning a product on the second dual of a Banach
algebra, namely by interchanging the order of the limits taken in the (cid:12)rst Arens
product. The result of doing this is called the second Arens product.
We now set
(cid:0) (cid:5)(cid:0) =w(cid:3)-limlima a ;
1 2 i j
j i
and similarly,
(cid:9) (cid:5)(cid:9) = w(cid:3)-limlimb b ;
1 2 i j
j i
(cid:0) (cid:5)(cid:5) = w(cid:3)-limlima m ;
1 i j
j i
(cid:5)(cid:5)(cid:9) = w(cid:3)-limlimm b :
1 j i
i j
WEAK AMENABILITY OF TRIANGULAR BANACH ALGEBRAS 1439
The previous analysis of the (cid:12)rst Arens product can now be extended to the
second Arens product to see that
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(cid:0)1 (cid:5)1 (cid:5) (cid:0)2 (cid:5)2 =w(cid:3)-limlim ai mi aj mj
(cid:9)1 (cid:20) (cid:9)2 j i (cid:21) bi bj
=w(cid:3)-limlim aiaj aimj +mibj
(cid:20) j i bibj (cid:21)
w(cid:3)-lim lim a a w(cid:3)-lim lim (a m +m b )
= j i i j j i i j i j
w(cid:3)-lim lim b b
(cid:20) (cid:21) j i i j
(cid:0) (cid:5)(cid:0) (cid:0) (cid:5)(cid:5) +(cid:5) (cid:5)(cid:9)
= 1 2 1 2 1 2 :
(cid:9) (cid:5)(cid:9)
1 2
In general, a Banach algebra D is said to be Arens regular provided that the
(cid:12)rstandsecondArensproductsonD(2) agree. GivenD;F 2D(2),itisknownthat
D(cid:3)F =D(cid:5)F whenever either D or F lies in the image of D under the canonical
embedding of D into D(2) ([13], Theorem 1.4.2) A fortiori, when this is the case,
the Arens product also coincides with the natural dual module action of D on
D(2) de(cid:12)ned at the beginning of Section 2.1. To see this, suppose for instance that
D; F 2D(2) withDbeingtheimageofsomed2Dunderthecanonicalembedding.
Suppose F =w(cid:3)-lim f and consider (cid:26)2D(cid:3). Then (D(cid:3)F)=w(cid:3)-lim df and so
j j j j
(D(cid:3)F)((cid:26))=lim(cid:26)(df ):
j
j
On the other hand, we have
(D(cid:14)F)((cid:26)) = F((cid:26)(cid:14)d)
= lim((cid:26)(cid:14)d)(f )
j
j
= lim(cid:26)(df )
j
j
= (D(cid:3)F)((cid:26)):
The case where F arises from the image of some f 2 D under the canonical em-
bedding and D 2D(2) is arbitrary is handled in a similar fashion.
Returning to the setting of triangular Banach algebras, the same observations
carry over to the case of the two A(2);B(2)-module actions (cid:5) and (cid:3) which we have
de(cid:12)ned on M(2). More precisely, (cid:0)(cid:5)(cid:5) = (cid:0)(cid:3)(cid:5) if either (cid:0) is the image of some
a2A under the canonical embedding of A into A(2), or if (cid:5) is the image of some
m 2 M under the canonical embedding of M into its second dual. (A similar
statement holds for the right action of B(2) upon M(2).) Again, when at least one
termistheimageofanelementinA(orM,orB),thesetwoactions(cid:3)and(cid:5)agree
with the dual module action (cid:14) de(cid:12)ned above. We shall return to this later.
Let us say that the Banach algebras A and B act regularly on M if for every
(cid:0)2A(2), (cid:5)2M(2) and (cid:9)2B(2) we have
(cid:0)(cid:3)(cid:5)=(cid:0)(cid:5)(cid:5) and (cid:5)(cid:3)(cid:9)=(cid:5)(cid:5)(cid:9):
The following proposition is now immediate:
(cid:20) (cid:21)
A M
2.3. Proposition. A triangular Banach algebra T = is Arens regular
B
if and only if both A and B are Arens regular and A and B act regularly on M.
1440 B. E. FORREST AND L. W. MARCOUX
2.4. Now that we have shownthat T(2) canbe made into a Banachalgebrausing
either Arens product, we may apply the same argumentto show that we canmake
T(4); T(6);andmoregenerallyT(2n) forn(cid:21)1intoaBanachalgebrabyrepeatedly
using the (cid:12)rst (or second) Arens product. Two interesting implications of this
analysis are the following:
(i) It provides us with an action of T upon T(2n) for each n (cid:21) 1 via restriction
ofthe ArensproducttothecanonicalembeddingofT intoT(2n). Aswehaveseen,
this action agrees with the dual module action de(cid:12)ned in Section 2.1.
(ii) The action of T upon T(2n) referred to in the previous paragraph in turn
de(cid:12)nes\actions"ofA(2n) andB(2n) onM taking values in M(2n) viathe formulae
A(cid:14)m=A(cid:5)m~ and m(cid:14)B =m~ (cid:5)B;
where A 2 A(2n), B 2 B(2n), and m~ denotes the (canonical) embedding of an
elementm2Minto M(2n). We shalluse this inourstudy ofderivationsofT into
T(2n), n(cid:21)1.
2.5. Iterated duals - the actions of T upon T(m). Since, as a Banach space,
T is isomorphic to the ‘1-direct sum of A, M and B, it is clear that T(2m(cid:0)1) ’
A(2m(cid:0)1) (cid:8)1M(2m(cid:0)1) (cid:8)1B(2m(cid:0)1), while T(2m) ’ A(2m) (cid:8)1M(2m) (cid:8)1B(2m) for
each m (cid:21) 1. Let us now consider the action of T upon T(m), m (cid:21) 1. To reduce
notational clutter, we shall denote the module actions of A and B upon M simply
by juxtaposition. (cid:20) (cid:21) (cid:20) (cid:21)
a m (cid:11) (cid:22)
The case m = 1. Suppose t = 2 T and (cid:28) = 2 T(cid:3).
b (cid:12)
(Althoughwewrite(cid:28) asanupper triangularmatrix,thisis simplyforconvenience.
The action o(cid:20)f T(cid:3) up(cid:21)on T is given by (cid:28)(t)=(cid:11)(a)+(cid:22)(m)+(cid:12)(b).)
x y
Let w = 2T. From above, the action of w on (cid:28) is given by
z
(w(cid:14)(cid:28))(t) = (cid:28)(tw)
(cid:20) (cid:21)
ax ay+mz
= (cid:28)
bz
= (cid:11)(ax)+(cid:22)(ay+mz)+(cid:12)(bz)
= (x(cid:14)(cid:11))(a)+(y(cid:14)(cid:22))(a)+(z(cid:14)(cid:22))(m)+(z(cid:14)(cid:12))(b)
(cid:20) (cid:21) (cid:18) (cid:19)
x(cid:14)(cid:11)+y(cid:14)(cid:22) z(cid:14)(cid:22) a m
= :
z(cid:14)(cid:12) b
(cid:20) (cid:21)
x(cid:14)(cid:11)+y(cid:14)(cid:22) z(cid:14)(cid:22)
Thus (w(cid:14)(cid:28))= :
z(cid:14)(cid:12) (cid:20) (cid:21)
(cid:11)(cid:14)x (cid:22)(cid:14)x
Similarly, one can check that ((cid:28) (cid:14)w)= :
(cid:22)(cid:14)y+(cid:12)(cid:14)z
(cid:20) (cid:21)
A(2) M(2)
The case m=2. We canidentify T(2) with . As we have seen,
B(2)
the action of T upon T(2) coincides with the restriction of the ((cid:12)rst or second)
Arens product on T(2) to the image of T in T(2) under the canonical embedding,
and hence
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
x y (cid:0) (cid:5) x(cid:14)(cid:0) x(cid:14)(cid:5)+y(cid:14)(cid:8)
(cid:14) =
z (cid:8) z(cid:14)(cid:8)
WEAK AMENABILITY OF TRIANGULAR BANACH ALGEBRAS 1441
and
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(cid:0) (cid:5) x y (cid:0)(cid:14)x (cid:0)(cid:14)y+(cid:5)(cid:14)z
(cid:14) = :
(cid:8) z (cid:8)(cid:14)z
Again,we observethattheseequationscoincide withthe standardmatrix\mul-
tiplications" of w and T, with multiplication at the coordinate level corresponding
to the appropriate module action.
The case m(cid:21)3. It is not hard to see that the action of T on T(3) agrees with
the dual action of T(2) on T(3) when we restr(cid:20)ict to th(cid:21)e image o(cid:20)f T in T((cid:21)2) under
(cid:18) (cid:17) (cid:0) (cid:5)
the canonicalembedding. Indeed, note that if 2T(3), 2T(2),
(cid:20) (cid:21) ! (cid:9)
a m
and 2T, then
b
(cid:20) (cid:21) (cid:20) (cid:21)(cid:18) (cid:19) (cid:20) (cid:21)(cid:18)(cid:20) (cid:21) (cid:20) (cid:21)(cid:19)
(cid:18) (cid:17) a m (cid:0) (cid:5) (cid:18) (cid:17) a m (cid:0) (cid:5)
(cid:14) = (cid:14)
! b (cid:9) ! b (cid:9)
(cid:20) (cid:21)(cid:18)(cid:20) (cid:21) (cid:20) (cid:21)(cid:19)
(cid:18) (cid:17) a m (cid:0) (cid:5)
= (cid:5) ;
! b (cid:9)
where(cid:14)denotestheappropriatedualactionand(cid:5)denotestheArensproduct(recall
thatbothArensproductsagreesinceoneofthetermsistheimageofanelementof
T underthecanonicalembedding). Moregenerally,itfollowsthatthecomputation
of the action of T upon T(cid:3) carries over mutatis mutandis to the action of T upon
T(2m(cid:0)1), m (cid:21) 2. Similarly, the action of T upon T(2m), m (cid:21) 2, also satis(cid:12)es
the corresponding formulae for the action of T upon T(2), and hence \looks like"
standard matrix multiplication, as we saw in Section 2.4.
3. Derivations and n-weak amenability
3.1. We now wish to(cid:20)consider t(cid:21)he question of n-weak amenability of triangular
A M
Banach algebras T = . From this point on, we shall assume that the
B
corner algebras A and B are unital and that M is a unital Banach A;B-module.
Although this is not crucial for weak amenability (i.e. when n = 1), it guarantees
thatM(n) isanessentialmodulewhenn(cid:21)2. Asweshallsee,therearetwodistinct
phenomena which occur, depending upon whether n is even or odd. We begin by
considering the case where n is odd, and in fact, (cid:12)rst considering the case where
n=1.
Weakamenability. Theproofofthefollowinglemmainvolvesnothingmorethan
routine calculations which are left to the reader.
3.2. Lemma. Let D : T ! T(cid:3) be a continuous derivation. Then there exist
continuous derivations (cid:14) : A ! A(cid:3), (cid:14) : B ! B(cid:3), and an element (cid:13) 2 M(cid:3) such
1 4 (cid:14)
that (cid:20) (cid:21) (cid:20) (cid:21)
x 0 (cid:14) (x) (cid:13) (cid:14)x
(i) D = 1 (cid:14) for all x2A,
(cid:20) 0 (cid:21) (cid:20) 0 (cid:21)
0 0 0 (cid:0)z(cid:14)(cid:13)
(ii) D = (cid:14) for all z 2B,
(cid:20) z (cid:21) (cid:20) (cid:14)4(z) (cid:21)
0 y (cid:0)y(cid:14)(cid:13) 0
(iii) D = (cid:14) for all y 2M.
0 (cid:13) (cid:14)y
(cid:14)
1442 B. E. FORREST AND L. W. MARCOUX
3.3. Lemma. Suppose (cid:14)A :A!A(cid:3) is a continuous derivation. Then
D(cid:14)A : (cid:20) T (cid:21) ! (cid:20) T(cid:3); (cid:21)
a m 7! (cid:14)A(a) 0 ;
b 0
is also a continuous derivation. Furthermore, (cid:14)A is inner if and only if D(cid:14)A is
inner.
Similarly, suppose (cid:14)B :B !B(cid:3) is a continuous derivation. Then
D(cid:14)B : (cid:20) T (cid:21) ! (cid:20) T(cid:3); (cid:21)
a m 0 0 ;
7!
b (cid:14)B(b)
is a continuous derivation, and it is inner precisely when (cid:14)B is inner.
Proof. That D and D are derivations are again elementary calculations which
(cid:14)A (cid:14)B
we leave to the reader.
Suppose (cid:14)A is inner. Then there exists (cid:11) 2 A(cid:3) such that (cid:14)A(a) = (cid:14)(cid:11)(a) :=
a(cid:14)(cid:11)(cid:0)(cid:11)(cid:14)a. (cid:20) (cid:21)
(cid:11) 0
Consider (cid:28) = 2T(cid:3). Then
0
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
a m a m (cid:11) 0 (cid:11) 0 a m
D = (cid:14) (cid:0) (cid:14)
(cid:28) b b 0 0 b
(cid:20) (cid:21)
a(cid:14)(cid:11)(cid:0)(cid:11)(cid:14)a 0
=
0
(cid:20) (cid:21)
(cid:14) (a) 0
= (cid:11)
0
(cid:20) (cid:21)
a m
= D ;
(cid:14)A b
and so D(cid:14)A is inner. (cid:20) (cid:21)
(cid:11) (cid:22)
Conversely, suppose D is inner. Then there exists (cid:28) = 2 T(cid:3) such
(cid:14)A (cid:12)
that
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
a m a m (cid:11) (cid:22) (cid:11) (cid:22) a m
D = (cid:14) (cid:0) (cid:14)
(cid:14)A b b (cid:12) (cid:12) b
(cid:20) (cid:21)
a(cid:14)(cid:11)+m(cid:14)(cid:22)(cid:0)(cid:11)(cid:14)a b(cid:14)(cid:22)(cid:0)(cid:22)(cid:14)a
= :
b(cid:14)(cid:12)(cid:0)(cid:22)(cid:14)m(cid:0)(cid:12)(cid:14)b
But
(cid:20) (cid:21) (cid:20) (cid:21)
a m (cid:14)A(a) 0
D = :
(cid:14)A b 0
It follows that (cid:14)A(a) = a(cid:14)(cid:11)(cid:0)(cid:11)(cid:14)a+m(cid:14)(cid:22) for all a 2 A; m 2 M. We may
assume without loss of generality that m=0. Then for any a2A, we have
(cid:14)A(a)=a(cid:14)(cid:11)(cid:0)(cid:11)(cid:14)a;
and so (cid:14)A =(cid:14)(cid:11) is inner, as claimed. The proofs for (cid:14)B and D(cid:14)B are similar.
WEAK AMENABILITY OF TRIANGULAR BANACH ALGEBRAS 1443
Wementionedintheaboveproofthatthecalculationthatshowsthatderivations
from the corner algebra A into its dual space lift to derivations from T into T(cid:3)
is elementary. It is worth pointing out that the conclusion is false if we consider
lifting derivations from A into A to derivations from T into T. See [7], Example
3.6 for a counterexample.
3.4. Theorem. Let A(cid:20)and B be (cid:21)unitalBanach algebras and M bea unitalBanach
A M
A;B-module. Let T = be the corresponding triangular Banach algebra.
B
Then
H1(T;T(cid:3))’H1(A;A(cid:3))(cid:8)H1(B;B(cid:3)):
Proof. Let (cid:14) : T ! T(cid:3) be a (continuous) derivation. From Lemma 3.2 above, we
obtain derivations (cid:14) : A ! A(cid:3) and (cid:14) : B ! B(cid:3), as well as an element (cid:13) 2 M(cid:3)
1 4 (cid:14)
such that
(cid:20) (cid:21) (cid:20) (cid:21)
x y (cid:14) (x)(cid:0)y(cid:14)(cid:13) (cid:13) (cid:14)x(cid:0)z(cid:14)(cid:13)
(cid:14) = 1 (cid:14) (cid:14) (cid:14) :
z (cid:13) (cid:14)y+(cid:14) (z)
(cid:14) 4
Consider the linear map
(cid:20): Z1(T;T(cid:3)) ! H1(A;A(cid:3))(cid:8)H1(B;B(cid:3));
(cid:14) 7! ((cid:14) +N1(A;A(cid:3));(cid:14) +N1(B;B(cid:3))):
1 4
Clearly (cid:20) is linear.
Weclaimthat(cid:20)issurjective. Indeed,given(cid:14)A 2Z1(A;A(cid:3))and(cid:14)B 2Z1(B;B(cid:3)),
Lemma 3.3 shows that the lifted maps D and D both lie in Z1(T;T(cid:3)), and
(cid:14)A (cid:14)B
thus D =D +D 2Z1(T;T(cid:3)). But then
(cid:14)A (cid:14)B
(cid:20)(D) = (cid:20)(D )+(cid:20)(D )
(cid:14)A (cid:14)B
= ((cid:14)A+N1(A;A(cid:3));0)+(0;(cid:14)B+N1(B;B(cid:3)))
= ((cid:14)A+N1(A;A(cid:3));(cid:14)B+N1(B;B(cid:3))):
Thus (cid:20) is onto, as claimed.
Next, suppose (cid:14) 2 ker (cid:20). Then (cid:14) 2 N1(A;A(cid:3)) and (cid:14) 2 N1(B;B(cid:3)). By
1 4
Lemma 3.3, D ; D 2 N1(T;T(cid:3)), and hence D = D + D 2 N1(T;T(cid:3)).
But (cid:14) (cid:0) D = (cid:14)(cid:14)12 (cid:14)4 3 2 N1(T;T(cid:3)), and so (cid:14) is an (cid:14)i1nner d(cid:14)e4rivation. That
0 (cid:13)
4 (cid:14) 5
0
(cid:14) 2 N1(T;T(cid:3)) implies (cid:14) ;(cid:14) are also inner is again Lemma 3.3. Thus ker (cid:20) =
1 4
N1(T;T(cid:3)).
Elementary linear algebra theory now implies that
H1(A;A(cid:3))(cid:8)H1(B;B(cid:3))=ran(cid:20)’Z1(T;T(cid:3))=ker (cid:20)=H1(T;T(cid:3)):
3.5. Corollary. LetA(cid:20)andB be(cid:21)unitalBanach algebras andMbeaunitalBanach
A M
A;B-module. Let T = be the corresponding triangular Banach algebra.
B
Then T is weakly amenable if and only if both A and B are weakly amenable.
1444 B. E. FORREST AND L. W. MARCOUX
3.6. Recall that the action of T on T(3) is a restriction of the action of T(2) on
T(3). FromthiswecanagaindeterminethebehaviourofaderivationD :T !T(3).
As before,
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(cid:18) (cid:17) a m (cid:18)(cid:14)a (cid:17)(cid:14)a
(cid:14) =
! b (cid:17)(cid:14)m+!(cid:14)b
and
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
a m (cid:18) (cid:17) a(cid:14)(cid:18)+m(cid:14)(cid:17) b(cid:14)(cid:17)
(cid:14) = :
b ! b(cid:14)!
By repeating exactly the same calculations as for derivations (cid:14) from T into T(cid:3),
we (cid:12)nd that there exist derivations (cid:14)1 : A ! A(3) and (cid:14)4 : B ! B((cid:18)3)(cid:20)as well a(cid:21)s(cid:19)an
element(cid:13) 2M(3)(arisingasthe(1;2)-coordinateoftheimageof(cid:14) eA 0 ))
(cid:14) 0
such that
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
a m (cid:14) (a) 0 a m
(cid:14)( )= 1 +(cid:14)2 3( ):
b (cid:14) (b) 0 (cid:13) b
4 4 (cid:14) 5
0
This shows that H1(T;T(3))’H1(A;A(3))(cid:8)H1(B;B(3)), and that, in particu-
lar,T is3-weaklyamenableifandonly ifbothAandB are. Infact, this argument
extends to any dual of the form T(2n(cid:0)1) with n(cid:21)1.
We therefore have:
3.7. Theorem. Let A(cid:20)and B be (cid:21)unitalBanach algebras and M bea unitalBanach
A M
A;B-module. Let T = be the corresponding triangular Banach algebra.
B
Suppose n is a positive integer. Then
H1(T;T(2n(cid:0)1))’H1(A;A(2n(cid:0)1))(cid:8)H1(B;B(2n(cid:0)1)):
It follows that T is (2n(cid:0)1)- weakly amenable if and only if both A and B are.
3.8. (2n)-weak amenability. The study of derivations from T into T(2n) di(cid:11)ers
radically from that of derivations into T(2n(cid:0)1), n (cid:21) 1. While the (2n(cid:0)1)-weak
amenabilityofT dependsonlyuponthe(2n(cid:0)1)-weakamenabilityofAandB,the
situationfor(2n)-weakamenabilitymorecloselyresemblesthesituationexploredin
[7]forderivationsfromT intoitself. Indeed,giventhatthe actionofT uponT(2n)
mimics that of matrix multiplication, many of the calculations from that paper
formally carry over intact to prove an analogue of [7], Thm. 2.8. For the sake
of completeness, we shall include a detailed outline of the proof here, elaborating
on the di(cid:11)erences which occur as a result of having changed the codomain of the
derivations.
3.9. Proposition. Let n be a positive integer and suppose (cid:14) : T ! T(2n) is a
continuous derivation. Then there exist (cid:13) 2 M(2n), and continuous derivations
(cid:14)
(cid:14) : A ! A(2n), (cid:14) : B ! B(2n) and a continuous map (cid:26) : M ! M(2n) which
1 4
satisfy:
(cid:20) (cid:21) (cid:20) (cid:21)
a 0 (cid:14) (a) a(cid:14)(cid:13)
(i) (cid:14) = 1 (cid:14) ;
(cid:20) 0 (cid:21) (cid:20) 0 (cid:21)
0 0 0 (cid:0)(cid:13) (cid:14)b
(ii) (cid:14) = (cid:14) ;
b (cid:14) (b)
4