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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.