σ-Contractible and σ-biprojective Banach algebras

Abstract

The notion of σ-amenability for Banach algebras and its related notions were introduced and extensively studied by M.S. Moslehian and A.N. Motlagh in [10]. We develop these notions parallel to the amenability of Banach algebras introduced by B.E. Johnson in [5]. Briefly, we investigate σ-contractibility and σ-biprojectivity of Banach algebras, which are extensions of usual notions of contractibility and biprojectivity, respectively, where σ is a bounded endomorphism of the corresponding Banach algebra. We also give the notion σ-diagonal. Then we verify relations between σ-contractibility, σ-biprojectivity and the existence of a σ-diagonal for a Banach algebra, when σ has dense range or is an idempotent. Moreover, we obtain some hereditary properties of these concepts.

Full text

Quaestiones Mathematicae 33(2010), 1–11.
c
 2010 NISC Pty Ltd, www.nisc.co.za
σ-CONTRACTIBLE AND σ-BIPROJECTIVE BANACH
ALGEBRAS
Hashem Najafi
Faculty of Mathematical Science, Persian Gulf University, Boushehr, 75168, Iran.
E-Mail najafi@pgu.ac.ir
Taher Yazdanpanah
Faculty of Mathematical Science, Persian Gulf University, Boushehr, 75168, Iran.
E-Mail yazdanpanah@pgu.ac.ir
Abstract. The notion of σ-amenability for Banach algebras and its related notions
were introduced and extensively studied by M.S. Moslehian and A.N. Motlagh in [10].
We develop these notions parallel to the amenability of Banach algebras introduced
by B.E. Johnson in [5]. Briefly, we investigate σ-contractibility and σ-biprojectivity
of Banach algebras, which are extensions of usual notions of contractibility and
biprojectivity, respectively, where σ is a bounded endomorphism of the correspondin g
Banach algebra. We also give the notion σ-diagonal. Then we verify relations
between σ-contractibility, σ-biprojectivity and the existence of a σ-diagonal for a
Banach algebra, when σ has dense range or is an idempotent. Moreover, we obt ain
some hereditary properties of these concepts.
Mathematics Subject Classification (2000): 16E40, 46H25.
Key words: Banach algebras, σ-contractibility, σ-biprojectivity, σ-diagonal.
1. Introduction. The concept of amenability for Banach algebras was
introduced by B.E. Johnson in 1972 [5]. In fact, he defined the amenability
of a Banach algebra A through vanishing of first cohomology g roups of A with
coefficients in X
∗
for e ach Banach A-bimpdule X, where X
∗
denotes the first dual
space of X, which is a Banach A-bimodule in the usual way. He showed that the
concept of amenability is of great importance in the theory of B anach algebras.
Other notions of amenability such as contractibility (or super amenability) and
weak amena bility were introduced by other authors (c.f. [1]). All of above notions
of amenability are generalized in some ways during recent decades. For example,
Ghahramani and Loy defined approximate versions of these notions ([2]) and
extensively studied them on different classes of Banach algebras through several
papers. Another extension of the mentioned notions of amenability, that we are
interested, were done by us ing an endomorphism of a Banach algebra (see, for
example, [6], [9], [10]). There are some other concepts related to the notions of
amenability such as biflatness and biprojectivity, introduced by Helemskii in [3]
(see also [4]).
In this paper, we develop definitions of contractibility, biprojectivity and
diagonal to σ-contractibility, σ-biprojectivity and σ-diagonal, respectively, where
1

2 H. Najafi and T. Yazdanpanah
σ is a dense r ange or an idempotent bounded endomorphism of the corres ponding
Banach algebra. Then we investigate relations between σ-contractibility, σ-
biprojectivity and the existence of a σ-diagonal for a Banach algebra. We also verify
relations between usual notions and σ-notions of contractibility, biprojectivity and
diagonal. Some hereditary properties of these concepts are given in the last part
of this paper.
2. Preliminaries. Let A be a Banach alg ebra and σ be a bounded endomor-
phism of A, i.e., a bounded Banach algebra homomorphism from A into A. A σ-
derivation from A into a Banach A-bimodule X is a bounded linear map D : A → X
satisfying
D(ab) = σ(a).D(b ) + D(a).σ(b) (a, b ∈ A).
For e ach x ∈ X, the mapping ad
x
: A → X defined by ad
x
(a) = σ(a).x − x.σ (a),
for all a ∈ A, is a σ-derivation called a n inner σ-derivation.
A number of examples of σ-derivations are listed below (see [7], [8] and [10]):
(i) Every usual derivation D : A → X is id
A
-derivation, where id
A
: A → A is
the identity operator.
(ii) Let ϕ : A → A be a bounded homomorphism. Then ϕ is a
ϕ
2
-derivation.
(iii) Let φ be a character on A. Then a φ-derivation is just a po int derivation at
φ.
Remark 2.1. Throughout this paper we shall assume that A is a Banach algebra
and σ is a bounded endomorphism of A unless o therwise specifie d. Also, we shall
omit the letter σ, when σ = id
A
.
We denote by Z
1
σ
(A, X) the space of all σ-der ivations from A into X, and by
B
1
σ
(A, X) the space of all inner σ-derivations from A into X. It is clear that
B
1
σ
(A, X) is a subspace of Z
1
σ
(A, X).
Definition 2.1. Let X be a Banach A-bimodule. The first σ-cohomology group of
A with coefficients in X, denoted by H
1
σ
(A, X), is the quotient space
Z
1
σ
(A, X)/B
1
σ
(A, X).
The following definition ex tends the concept of contractibility ([10]):
Definition 2.2. A is called σ-contractible if H
1
σ
(A, X) = 0 for each Banach A-
bimodule X.
Note that id
A
-contractibility is nothing than the usual notion of contractibility.
Definition 2.3. Let A be a Banach alge bra and let X and Y be Bana ch A-
bimodules. The linear map T : X → Y is called a σ- A- bimodule homo morphism
if
T (ax) = σ(a).T (x), T (xa) = T (x).σ(a) (a ∈ A, x ∈ X).

σ-Contractible and σ-biprojective Banach algebras 3
Let A be a Banach algebra and A
ˆ
⊗A be the projective tens or product of A and A.
The product map on A extends to a map ∆
A
: A
ˆ
⊗A → A determined by
∆
A
(a ⊗ b) = ab (a, b ∈ A).
The projective tensor product A
ˆ
⊗A becomes a Banach A-bimodule with the fol-
lowing usual module actions:
a.(b ⊗ c) = ab ⊗ c, (b ⊗ c).a = b ⊗ ca (a, b, c ∈ A).
Obviously, by above actions, ∆
A
becomes an A- bimodule homomorphism.
3. Characterization of σ -contractibility when σ has dense range. First
we bring some properties of σ-contractible Banach algebras which will be used in
the sequel.
Proposition 3.1. If A is σ-contractible, then there exists an element e ∈ A which
is a unit for σ(A) (i.e., eσ(a) = σ(a)e = σ(a)).
Proof. Consider X = A as a Banach A-bimodule with the trivial right action,
that is:
a.x = ax, x.a = 0 (a ∈ A, x ∈ X).
Then D : A → X defined by D(a) = σ(a) is a σ-derivation, because
D(ab) = σ(ab) = σ(a).σ(b ) = σ(a).D(b) = σ(a).D(b) + D(a).σ(b) (a, b ∈ A).
Since A is σ-contractible, D is inner and so there is u ∈ X = A such that D = ad
u
.
Hence
σ(a) = D(a) = ad
u
(a) = σ(a).u (a ∈ A),
which shows that u is a r ight unit for σ (A). Similarly, one can find a left unit (say
u
0
) for σ(A). Now e = u + u
0
− uu
0
is a unit for σ(A). 2
Corollary 3.2. If A is σ-contractible and σ(A) is dense in A, then A is unital.
The following proposition generalizes a classical res ult in [5].
Proposition 3.3. Let A be unital (with unit e), σ(A) be dense in A and X be a
Banach A-bimodule. Then H
1
σ
(A, X)
∼
=
H
1
σ
(A, e.X.e).
Proof. Since σ(e) is a unit for σ(A), and σ(A) is dense in A, we see that σ(e) = e.
Set X
1
= e.X.e , X
2
= (1 − e).X, X
3
= X.(1 − e) and X
4
= (1 − e).X.(1 − e), which
certainly are Banach A-bimodules. For e ach x ∈ X we have
x = exe + (x − ex) + (x − xe) + (ex + xe − exe − x)
= exe + (x − ex) + (x − xe) − (1 − e)x(1 − e),
which shows that X = X
1
⊕ X
2
⊕ X
3
⊕ X
4
. Let D : A → X be a σ-derivation.
Consider the de riva tions D
i
= p
i
◦ D : A → X
i
for 1 ≤ i ≤ 4, where p
i
: X → X
i

4 H. Najafi and T. Yazdanpanah
is the canonical projection. Then D = D
1
+ D
2
+ D
3
+ D
4
and D
i
∈ Z
1
σ
(A, X
i
)
for each 1 ≤ i ≤ 4. Now it can be easily seen that D
2
= ad
−D
2
(e)
, D
3
= ad
D
3
(e)
and D
4
= 0. Therefore, D is inner if a nd only if D
1
is inner. The proof is now
complete. 2
The following definition extends the definition of the diagonal in the classical
sense:
Definition 3.4. An element m ∈ A
ˆ
⊗A is called a σ-diag onal for A if
σ(a).m = m.σ(a), σ(a).∆
A
(m) = σ(a) (a ∈ A).
It follows immediately from the Definition 3.4 that if A has a σ-diagonal, then A
has an element e = ∆
A
(m) which is a unit for σ(A).
Theorem 3.5. Let σ(A) be dense in A. Then A is σ-contractible if and only if it
has a σ-diagonal.
Proof. Let m ∈ A
ˆ
⊗A be a σ-diagonal for A and let D ∈ Z
1
σ
(A, X), for a Banach A-
bimodule X. Then ∆
A
(m) is a unit for σ(A). Since σ(A) is dense in A, A is unital
with unit e = ∆
A
(m). By Propo sition 3.3 we have H
1
σ
(A, X) = H
1
σ
(A, e.X.e).
So, without loss of genera lity, we can suppose that X is a unital A-bimodule (i.e.,
e.x = x.e = x for all x ∈ X). Define P : A
ˆ
⊗A → X by
P (a ⊗ b) = D(a).σ(b) (a, b ∈ A),
and extend by linearity and co ntinuity. As sume that m =
P
∞
n=1
a
n
⊗ b
n
, where
(a
n
) and (b
n
) are bounded seq ue nc es in A such that
P
∞
n=1
ka
n
kkb
n
k < ∞. Since
σ(A) is dense in A, we have a.m = m.a, and so P (a.m) = P (m.a) for all a ∈ A.
Thus
∞
X
n=1
D(aa
n
).σ(b
n
) =
∞
X
n=1
D(a
n
).σ(b
n
)σ(a) (a ∈ A). (1)
Set x =
P
∞
n=1
σ(a
n
).D (b
n
), which belongs to X. Then for each a ∈ A we have
D(a) = D(ae) = D(a
P
∞
n=1
a
n
b
n
) =
P
∞
n=1
D(aa
n
b
n
)
=
P
∞
n=1
σ(aa
n
).D (b
n
) +
P
∞
n=1
D(aa
n
).σ(b
n
)
= σ(a).x +
P
∞
n=1
D(a
n
).σ(b
n
)σ(a)
= σ(a).x +
P
∞
n=1
D(a
n
b
n
).σ(a) −
P
∞
n=1
σ(a
n
).D (b
n
).σ(a)
= σ(a).x + D(e).σ(a) − x.σ(a)
= σ(a).x − x.σ(a).
(2)
Note that in the last e quality we used the fact that D(e) = 0; since X is unital.
Therefore, D = ad
x
and thus A is σ-contractible. Conversely, suppose that A is

σ-Contractible and σ-biprojective Banach algebras 5
σ-contractible. Then by Corollary 3.2 it pos sesses a unit (say e). Consider the
derivation ad
e⊗e
: A → A
ˆ
⊗A. For a ∈ A we have
∆
A
(σ(a).(e ⊗ e) − (e ⊗ e).σ(a)) = ∆
A
(σ(a) ⊗ e − e ⊗ σ(a)) = σ(a) − σ(a) = 0.
Therefore, ∆
A
vanishes on the range of ad
e⊗e
, and thus ad
e⊗e
: A → ker ∆
A
is a
derivation. Hence it is inner. Let n ∈ ker ∆
A
be such that ad
e⊗e
= ad
n
. Now it
can be easily checked that m = e ⊗ e − n is a σ-diagonal for A. 2
We define the σ-biprojectivity of a Banach algebra such that it could be regarded
as a generalization of the usual notion of biprojectivity:
Definition 3.6. A is called σ-biprojective if there exists a bounded σ-A-bimodule
homomorphism ρ : A → A
ˆ
⊗A such that ∆
A
◦ ρ = σ.
When σ = id
A
, this definition coincide with the definition of usual biprojectivity.
Theorem 3.7. Let σ(A) be dense in A. Then A is σ-contractible if and only if it
is unital and σ-biprojective.
Proof. Assume that A is σ-contractible. The n by Corollary 3.2 it is unital and
by Theorem 3.5 there is a σ-diagona l (say m) for A. Define ρ : A → A
ˆ
⊗A by
ρ(a) = m.σ(a). Since
m.σ(ab) = m.σ(a)σ(b) = σ(a).m.σ(b) (a, b ∈ A),
we see that ρ is a σ-A-bimodule homo morphism. Since σ is bounded, ρ is bounded
too. We also have ∆
A
◦ ρ(a) = ∆
A
(m.σ(a)) = ∆
A
(m).σ(a) = σ(a). Thus A is
σ-biprojective. Conversely, let A be unital (with unit e) and σ-biprojective, and
let ρ : A → A
ˆ
⊗A be as in Definition 3.6. Then it can be routinely checked that
m = ρ(e) is a σ-diagonal for A. The result now follows from Theorem 3.5. 2
4. Characterization of σ-contractibility when σ is idempotent. In this
section, we characterize σ-contractible Banach algebr as when σ is an idempotent
endomorphism (i.e., σ
2
= σ ◦ σ = σ).
Proposition 4.1. Assume that A has an element e which is a unit for σ(A).
Assume also that X is a Banach A-bimodule. If σ is a bounded idempotent
endomorphism, then H
1
σ
(A, X)
∼
=
H
1
σ
(A, σ(e).X .σ(e)).
Proof. Set X
1
= σ(e).X.σ(e), X
2
= (1 − σ(e)).X, X
3
= X.(1 − σ(e)) and
X
4
= (1 − σ(e)).X.(1 − σ(e)). Since σ is idempotent, it is easily seen that each X
i
is a Banach A-bimodule via σ. Take x ∈ X. We have
x = σ(e)xσ(e) + (x − σ(e)x) + (x − xσ(e)) − (1 − σ(e))x(1 − σ(e)),
which shows that X = X
1
⊕ X
2
⊕ X
3
⊕ X
4
. The rest of the proof is similar to the
proof of Prepos ition 3.3 (here D
2
= ad
−D
2
(σ(e))
and D
3
= ad
D
3
(σ(e))
). 2

6 H. Najafi and T. Yazdanpanah
Theorem 4.2. Let σ be an idempotent endomorphism of A. Then A is σ-
contractible if and only if it has a σ-diagonal.
Proof. Let m ∈ A
ˆ
⊗A be a σ- diagonal for A and let D ∈ Z
1
σ
(A, X), for a Banach
A-bimodule X. Then ∆
A
(m) is a unit for σ(A) and thus, by Proposition 4 .1, we
have H
1
σ
(A, X)
∼
=
H
1
σ
(A, σ(∆
A
(m)).X .σ(∆
A
(m))). Hence we can suppose that X
is a σ(∆
A
(m))-unital A-bimodule (i.e., σ(∆
A
(m)).x = x.σ(∆
A
(m)) = x for all
x ∈ X). Now for all a ∈ A we have
D(σ(a)) = D(∆
A
(m).σ(a))
= D (∆
A
(m)).σ(a) + σ(∆
A
(m)).D(σ(a))
= D (∆
A
(m)).σ(a) + D(σ(a)),
and thus D(∆
A
(m)).σ(a) = 0. The rest of the pro of is similar to the proo f of
Theorem 3.5 (use D(∆
A
(m)).σ(a) = 0 in the last equality of (2)). Conversely,
suppose that A is σ-contractible. Then by Proposition 3.1 there exists an element
e ∈ A which is a unit for σ(A). Since σ is idempotent, σ(e) is a unit for σ(A) too.
Now consider the derivation ad
σ(e)⊗σ(e)
: A → A
ˆ
⊗A. For each a ∈ A we have
∆
A
(σ(a).(σ(e) ⊗ σ(e)) − (σ(e) ⊗ σ(e)).σ(a))
= ∆
A
(σ(a) ⊗ σ(e) − σ(e) ⊗ σ(a)) = σ(a) − σ(a) = 0.
Therefore, ∆
A
vanishes on the range of ad
σ(e)⊗σ(e)
and so ad
σ(e)⊗σ(e)
⊆ ker ∆
A
.
Thus ad
σ(e)⊗σ(e)
: A → ker ∆
A
is inner. Let n ∈ ker ∆
A
be such that ad
σ(e)⊗σ(e)
=
ad
n
. Then m := σ(e) ⊗ σ(e) − n is a σ-diagonal for A. 2
Theorem 4.3. Let σ be an idempotent endomorphism of A. Then A is σ-contract-
ible if and only if it is σ-biprojective and σ(A) is unital.
Proof. The proof is similar to the proof of Theorem 3.7 (just use Theorem 4.2
instead of Theorem 3.5, and use an element of A which is a unit for σ(A) in place
of e). 2
5. Contractibility and σ-contractibility. In this section, we verify relations
between σ-contractibility and contractibility, σ-biprojectivity and biprojectivity,
and the existence of a σ-diagonal and diagonal.
Proposition 5.1. Let ϕ be a bounded endomorphism of A.
(i) If A has a diagonal, then it has a ϕ-diagonal.
(ii) If A has a σ-diagonal, then it has a (σ ◦ ϕ)-diagonal. Specially, A has a
σ
n
-diagonal for each natural number n.

σ-Contractible and σ-biprojective Banach algebras 7
(iii) If A has a σ-diagonal and σ(A) is dense in A, then it has a ϕ-diagonal. In
particular, A has a diagonal.
Proof. (i) This is obvious from the Definition 3.4.
(ii) Let m be a σ-diagonal for A. Then σ(a).m = m.σ(a) and σ(a).∆
A
(m) =
σ(a) for all a ∈ A. So
σ(ϕ(a)).m = m.σ(ϕ(a)) and σ(ϕ(a)).∆
A
(m) = σ(ϕ(a)) (a ∈ A).
Therefore, m is a (σ ◦ ϕ)-diagonal for A.
(iii) Let m be a σ-diagonal for A. Since σ(A) is dense in A, we have a.m = m.a
and a.∆
A
(m) = a for all a ∈ A. Thus
ϕ(a).m = m.ϕ(a), ϕ(a).∆
A
(m) = ϕ(a) (a ∈ A).
Hence, m is a ϕ-diagonal for A. 2
Proposition 5.2. Let A be σ-biporjevtive. Then it is (σ ◦ϕ)-biprojective for each
bounded endomorphism ϕ of A. Specially, A is σ
n
-biprojective for each natural
number n.
Proof. Let ρ : A → A
ˆ
⊗A be the map as in Definition 3.6. Define ρ
0
:= ρ ◦ ϕ.
Then
ρ
0
(ab) = ρ(ϕ(a)ϕ(b)) = ρ(ϕ(a)).σ(ϕ(b)) = σ(ϕ(a)).ρ(ϕ(b)) (a, b ∈ A),
and
∆
A
◦ ρ
0
(a) = ∆
A
◦ ρ(ϕ(a)) = σ(ϕ(a)) (a ∈ A).
So A is (σ ◦ ϕ)-biprojective. 2
Corollary 5.3. If A is biporjective, then it is ϕ-biporjevtive for every bounded
endomorphism ϕ of A.
Proposition 5.4. Let ϕ be a bounded endomorphism of A. If A is σ-contractible,
then it is (ϕ ◦ σ)-contractible too.
Proof. Let X be a Banach A-bimodule and let D : A → X be a (ϕ◦ σ)-deriva tion.
Then X is an A-bimodule via ϕ, that is, X is an A-bimodule with the following
module ac tions:
a ∗ x = ϕ(a).x, x ∗ a = x.ϕ(a) (a ∈ A, x ∈ X).
We have
D(ab) = (ϕ◦σ(a)).D(b)+D(a).(ϕ◦ σ(b)) = σ(a) ∗D(b) +D(a) ∗σ(b) (a, b ∈ A).

8 H. Najafi and T. Yazdanpanah
Thus D is a σ-de riva tion. Since A is σ-contractible, there is an element x ∈ X
such that D = ad
x
. In fact,
D(a) = σ(a) ∗ x − x ∗ σ(a) = (ϕ ◦ σ(a)).x − x.(ϕ ◦ σ(a)) (a ∈ A, x ∈ X).
2
Corollary 5.5. If A is contractible, then it is ϕ-contractible for each bounded
endomorphism ϕ of A.
Proposition 5.6. If A is σ-contractible and σ(A) is dense in A, then it is ϕ-
contractible for each bounded endomorphism ϕ of A. In particular, A is contractible.
Proof. By Theorem 3.5, A has a σ-diagonal. Since σ(A) is dense in A, by
Proposition 5.1, A has a ϕ-diagonal for each bounded endomo rphism ϕ of A. Now
again by Theorem 3.5, A is ϕ-contractible. 2
Example 5.1. Let G be a locally c ompact group, A = L
1
(G), the group algebra
of G, and σ be a bo unded dense range endomorphism of L
1
(G). It is known that
L
1
(G) is co ntractible if and only if it has a diagonal and if and only if G is finite
[11]. Therefore, we have:
(i) L
1
(G) has a σ-diagonal if and only if G is finite; by Proposition 5.1.
(ii) L
1
(G) is σ-co ntractible if and only if G is finite; by Coro llary 5.5 and
Proposition 5.6.
Example 5.2. Let G be a locally compa ct ab e lian group and σ be the canonical
involution on L
1
(G). It is well known that L
1
(G) is biprojective if and only if G
is compact [1]. Therefore, L
1
(G) is σ-biprojective if and only if G is compact; by
Proposition 5.2 and Corollary 5.3.
Example 5.3. Let G be a locally compac t group, A = M (G), the measure algebra
of G, and σ be a bounded dens e range endomorphism o f M(G). It is known that
M(G) is contractible if and only if G is finite [1]. Using Theorems 3.5 and 3.7, and
Proposition 5.6, we see that the following statements are equivale nt:
(i) M(G) is σ-biprojective.
(ii) M (G) has a σ-diagonal.
(iii) M(G) is σ-contractible.
(iv) G is finite.

σ-Contractible and σ-biprojective Banach algebras 9
6. Hereditary properties of σ-contractibility. Let A and B be Bana ch
algebras. Then A
ˆ
⊗B, the projective tensor product of A and B, is a Banach
algebra with the product defined by
(a ⊗ b).(a
0
⊗ b
0
) = (aa
0
) ⊗ (bb
0
) (a, a
0
∈ A, b, b
0
∈ B).
Let σ and τ be bounded endomorphisms of A and B, respe c tively. Then it is e asy
to see that σ ⊗ τ : A
ˆ
⊗B → A
ˆ
⊗B defined by
σ ⊗ τ (a⊗b) = σ(a) ⊗ τ(b) (a ∈ A, b ∈ B),
is a b ounded endomorphism of A
ˆ
⊗B.
Lemma 6.1. Let A and B be Banach algebras and let σ and τ be bounded
endomorphisms of A and B, respectively. If A has a σ-diagonal and B has a
τ-diagonal, then A
ˆ
⊗B has a (σ ⊗ τ)-diagonal.
Proof. Let m
1
=
P
∞
n=1
a
n
⊗ a
0
n
be a σ-diagonal for A and m
2
=
P
∞
m=1
b
m
⊗ b
0
m
be a τ -diagonal for B. Define
m =
∞
X
n,m=1
(a
n
⊗ b
m
) ⊗ (a
0
n
⊗ b
0
m
),
which c ertainly belongs to (A
ˆ
⊗B)
ˆ
⊗(A
ˆ
⊗B). Then for each a ∈ A and b ∈ B we
have
σ ⊗ τ (a⊗b).∆
A
ˆ
⊗B
(m) = (σ(a) ⊗ τ(b))
P
∞
n,m=1
(a
n
⊗ b
m
).(a
0
n
⊗ b
0
m
)
=
P
∞
n,m=1
σ(a)a
n
a
0
n
⊗ τ(b)b
m
b
0
m
=
P
∞
n=1
(σ(a)a
n
a
0
n
⊗
P
∞
m=1
τ(b)b
m
b
0
m
)
=
P
∞
n=1
(σ(a)a
n
a
0
n
) ⊗ τ(b).∆
B
(m
2
)
= σ(a).∆
A
(m
1
) ⊗ τ (b).∆
B
(m
2
)
= σ ⊗ τ (a⊗b).
Now define the map Φ : (A
ˆ
⊗A) × (B
ˆ
⊗B) → (A
ˆ
⊗B)
ˆ
⊗(A
ˆ
⊗B) by
Φ(
∞
X
i=1
a
i
⊗ a
0
i
,
∞
X
j=1
b
j
⊗ b
0
j
) =
∞
X
i,j=1
(a
i
⊗ b
j
) ⊗ (a
0
i
⊗ b
0
j
),
where
P
∞
i=1
a
i
⊗ a
0
i
∈ A
ˆ
⊗A and
P
∞
j=1
b
j
⊗ b
0
j
∈ B
ˆ
⊗B. Since σ(a).m
1
= m
1
.σ(a)
and τ(b).m
2
= m
2
.τ(b), for all a ∈ A and b ∈ B, we have Φ(σ(a).m
1
, τ(b).m
2
) =
Φ(m
1
.σ(a), m
2
.τ(b)) and so
σ ⊗ τ(a⊗b).m =
P
∞
m,n=1
(σ(a)a
n
⊗ τ(b)b
m
) ⊗ (a
0
n
⊗ b
0
m
)
=
P
∞
m,n=1
(a
n
⊗ b
m
) ⊗ (a
0
n
σ(a) ⊗ b
0
m
τ(b))
= m.σ ⊗ τ (a⊗b).

10 H. Najafi and T. Yazdanpanah
Therefore, m is a (σ ⊗ τ)-diagonal for A
ˆ
⊗B. 2
Theorem 6.2. Let A and B be Banach algebras and let σ and τ be bounded
endomorphisms of A and B, respectively. Let also σ and τ have dense ranges
(respectively, σ and τ are idempotents). If A is σ-contractible and B is τ-contractible,
then A
ˆ
⊗B is (σ ⊗ τ )-contractible.
Proof. This is immediate by Theorem 3.5 (respectively, 4.2) and Lemma 6.1. 2
It is clear that if two endomorphisms σ and τ are idempotent then so is σ ⊗ τ.
Proposition 6.3. Let I be a closed ideal of A such that σ(I) ⊆ I. If A is σ-
contractible, then
A
I
is ˆσ-contractible where ˆσ is the endomorphism of
A
I
induced
by σ (i.e., ˆσ(a + I) = σ(a) + I for a ∈ A).
Proof. Let X be a B anach
A
I
-bimodule and D :
A
I
→ X be a ˆσ-derivation.
Then X becomes an A-bimodule via the canonical homomorphism p : A →
A
I
, and
D ◦ p : A → X bec omes a σ-derivation. By the σ-contractibility of A there exists
x ∈ X with D ◦ p(a) = σ(a).x − x.σ(a) for all a ∈ A. Therefore,
D(a + I) = D ◦ p(a) = σ(a).x − x.σ(a)
= (σ (a) + I).x − x.(σ(a) + I) = ˆσ(a + I).x − x.ˆσ(a + I),
for each a ∈ A. Thus
A
I
is ˆσ-contractible. 2
Proposition 6.4. Let A and B be Banach algebras and let σ and τ be bounded
endomorphisms of A and B, respectively. Let also ϕ : A → B be a bounded
homomorphism with dense range such that ϕ◦σ = τ ◦ϕ. Then the σ-contractibility
of A implies that B is τ-contractible.
Proof. L et X be a Banach B-bimodule and D : B → X be a τ-derivation. Then
X becomes an A-bimodule via ϕ, a nd D ◦ ϕ : A → X is a σ-derivation. By the
σ-contractibility of A there exists x ∈ X with D ◦ ϕ(a) = σ(a).x − x.σ(a) for all
a ∈ A. Therefore,
D ◦ ϕ(a) = σ(a).x − x.σ(a) = ϕ(σ(a)).x − x.ϕ(σ(a)) = τ(ϕ(a)).x − x.τ(ϕ(a)),
for all a ∈ A. Thus D(b) = τ(b).x − x.τ(b) for all b ∈ B, and hence B is τ-
contractible. 2
Acknowledgement. The authors would like to thank the Persian Gulf University
Research Council for their financial support.

σ-Contractible and σ-biprojective Banach algebras 11
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Recei ved 28 February, 2010.