Amenability and Contractibility Modulo an Ideal of Banach Algebras

Abstract

We investigate the concept of amenability modulo an ideal of Banach algebra, showing that amenability modulo an ideal can be characterized by the existence of virtual and approximate diagonal modulo an ideal. We also study the concept of contractible modulo an ideal of Banach algebra. As a consequence, we prove a version of Selivanov's theorem for a large class of semigroups, including E-inversive E-semigroup and eventually inverse semigroups.

1. Introduction

Introduced and initiated in 1972 by Johnson [1] for groups, the concept of amenability of Banach algebras has since become an important group-theoretic and semigroup-theoretic tool, and since then many results have been obtained on the structure of this concept. This notion was coined by Johnson [1] in recognition of “a locally compact (discrete) group 𝐺 is amenable (in the usual sense) if and only if 𝐿¹(𝐺) (resp., 𝑙¹(𝐺)) is amenable.’’ For semigroups, in especial cases some necessary and sufficient conditions for amenability of semigroup algebra were introduced (see, e.g., [2–5] for more details).

Recently in [6], the first author and Amini introduced and studied the concept of amenability modulo an ideal. They showed that inducing the amenability of 𝑙¹(𝑆) modulo ideals by certain classes of group congruences on 𝑆 is equivalent to the amenability of 𝑆, restoring Johnson's theorem for a large class of semigroups.

In this paper, we study basic properties of amenability modulo an ideal such as virtual diagonal modulo an ideal, approximate diagonal modulo an ideal, and contractible modulo an ideal. Among other results, we show that the semigroup algebra 𝑙¹(𝑆) is contractible modulo an ideal if and only if 𝑆/𝜎 is finite, restoring Selivanov's theorem for a large class of semigroups [7].

The paper is organized as follows. In Section 2 we introduce the notions of virtual and approximate diagonal modulo an ideal. As an application, we characterize amenability modulo an ideal of Banach algebra. In Section 3 we consider the concept of contractible modulo an ideal of Banach algebra. In Section 4, constructing various examples we investigate the contractibility modulo an ideal of semigroup algebra.

2. Virtual and Approximate Diagonal Modulo an Ideal

Let 𝐴 be a Banach algebra and let 𝑋 be a Banach 𝐴-bimodule. By a derivation of 𝐴 to 𝑋 we mean a bounded linear map 𝐷∶𝐴→𝑋 such that 𝐷(𝑎𝑏)=𝑎⋅𝐷(𝑏)+𝐷(𝑎)⋅𝑏(𝑎,𝑏∈𝐴). Clearly the mapping 𝑎𝑑_𝑥∶𝐴→𝑋 defined by 𝑎𝑑_𝑥(𝑎)=𝑎⋅𝑥−𝑥⋅𝑎 is a derivation that is called inner (𝑥∈𝑋). A Banach algebra 𝐴 is called amenable if, for any 𝐴-bimodule 𝑋 and any derivation 𝐷∶𝐴→𝑋^∗, there exists 𝜙∈𝑋^∗ such that 𝐷=𝑎𝑑_𝜙.

In the following we recall the concept of amenability modulo an ideal and some results of [6] which we need in the future.

Definition 1. Let 𝐼 be a closed ideal of 𝐴. A Banach algebra 𝐴 is amenable modulo 𝐼 if for every Banach 𝐴-bimodule 𝐸 such that 𝐼⋅𝐸=𝐸⋅𝐼=0 and every derivation 𝐷 from 𝐴 into 𝐸^∗ there is 𝜙∈𝐸^∗ such that 𝐷=𝑎𝑑_𝜙 on the set theoretic difference 𝐴⧵𝐼∶={𝑎∈𝐴∶𝑎∉𝐼}.

Proposition 2 (see [6, Theorem 1]). Let 𝐼 be a closed ideal of 𝐴.(i)If 𝐴/𝐼 is amenable and 𝐼²=𝐼, then 𝐴 is amenable modulo 𝐼.(ii)If 𝐴 is amenable modulo 𝐼, then 𝐴/𝐼 is amenable.(iii)If 𝐴 is amenable modulo 𝐼 and 𝐼 is amenable, then 𝐴 is amenable.

Blanket Assumption. All over this paper we fix 𝐴 and 𝐼 as above, unless they are otherwise specified.

Definition 3. Let 𝐴 be a Banach algebra. A bounded net {𝑢_𝛼}_𝛼⊆𝐴 is approximate identity modulo 𝐼 if lim_𝛼𝑢_𝛼⋅𝑎=lim_𝛼𝑎⋅𝑢_𝛼=𝑎(𝑎∈𝐴⧵𝐼). (1)

Theorem 4. Let 𝐴 be amenable modulo 𝐼. Then 𝐴 has an approximate identity modulo 𝐼.

Proof. Set 𝑋=𝐴/𝐼; then 𝑋 is a Banach 𝐴-bimodule where it acts on 𝐴 via 𝑎⋅𝜋(𝑏)∶=𝜋(𝑎)𝜋(𝑏),𝜋(𝑏)⋅𝑎∶=0(𝑎,𝑏∈𝐴). (2) Clearly 𝐼⋅𝑋^∗=𝑋^∗⋅𝐼=0. Suppose 𝐷=𝜄∘𝜋∶𝐴→𝑋^∗∗, where 𝜄∶𝑋→𝑋^∗∗ is a natural inclusion map and 𝜋∶𝐴→𝑋 is a natural quotient map. Since 𝐷 is a derivation, there exists 𝜙∈𝑋^∗∗ such that 𝐷(𝑎)=𝑎⋅𝜙(𝑎∈𝐴⧵𝐼). Goldstine's theorem provides a bounded net {𝑢_𝛼}_𝛼 in 𝐴 such that 𝜙=𝑤^∗−lim_𝛼𝑢_𝛼. Then 𝜙=𝑤^∗−lim_𝛼𝐷(𝑢_𝛼). Now 𝑤^∗−lim_𝛼𝐷𝑎⋅𝑢_𝛼=𝑤^∗−lim_𝛼𝐷(𝑎)⋅𝑢_𝛼+𝑎⋅𝐷𝑢_𝛼=𝑎⋅𝑤^∗−lim_𝛼𝐷𝑢_𝛼=𝑎⋅𝜙=𝐷(𝑎)(𝑎∈𝐴⧵𝐼) (3) which is equivalent to 𝑤−lim_𝛼𝑎⋅𝑢_𝛼=𝑎(𝑎∈𝐴⧵𝐼). Passing to convex combinations, we can suppose that 𝑎=lim_𝛼𝑎⋅𝑢_𝛼(𝑎∈𝐴⧵𝐼). In an analogous fashion, we obtain a net {𝑣_𝛽}_𝛽 in 𝐴 such that 𝑎=𝑤−lim_𝛽𝑣_𝛽⋅𝑎(𝑎∈𝐴⧵𝐼). (4) Define 𝑒_𝛼,𝛽∶=𝑢_𝛼+𝑣_𝛽−𝑢_𝛼𝑣_𝛽. Then 𝑒_𝛼,𝛽 is an approximate identity modulo 𝐼.

We now express the version of Cohen's factorization theorem [8], where 𝐴 has an approximate identity modulo 𝐼. Since the proof is similar, we omit the proof.

Theorem 5. If 𝐴 has a bounded approximate identity modulo 𝐼, 𝑧∈𝐴⧵𝐼, and 𝛿>0, then there exist 𝑎,𝑦∈𝐴 such that 𝑧=𝑎𝑦 and ‖𝑧−𝑦‖≤𝛿.

Let 𝐴 be a Banach algebra and let 𝐼 be a closed ideal of 𝐴; then the corresponding diagonal operator of 𝐴/𝐼 is defined as 𝜋_𝐴/𝐼∶𝐴𝐼⊗𝐴𝐼⟶𝐴𝐼,(𝑎+𝐼)⊗(𝑏+𝐼)⟶𝑎𝑏+𝐼. (5) Clearly, 𝜋_𝐴/𝐼 is a bimodule homomorphism.

Definition 6. (i) An element 𝑀∈((𝐴/𝐼)⊗(𝐴/𝐼))^∗∗ is a virtual diagonal modulo 𝐼 if 𝑎⋅𝜋^∗∗_𝐴/𝐼(),().𝑀−𝑎=0𝑎∈𝐴,𝑎=𝑎+𝐼𝑎⋅𝑀−𝑀⋅𝑎=0𝑎∈𝐴⧵𝐼 (6)
(ii) A bounded net (𝑚_𝛼)_𝛼⊂(𝐴/𝐼)⊗(𝐴/𝐼) is an approximate diagonal modulo 𝐼 if 𝑎⋅𝜋_𝐴/𝐼𝑚_𝛼−𝑎⟶0(𝑎∈𝐴,𝑎=𝑎+𝐼),𝑎⋅𝑚_𝛼−𝑚_𝛼⋅𝑎⟶0(𝑎∈𝐴⧵𝐼). (7)
(iii) An element 𝑚∈(𝐴/𝐼)⊗(𝐴/𝐼) is a diagonal modulo 𝐼 if 𝑎⋅𝜋_𝐴/𝐼(),().𝑚−𝑎=0𝑎∈𝐴,𝑎=𝑎+𝐼𝑎⋅𝑚−𝑚⋅𝑎=0𝑎∈𝐴⧵𝐼 (8)

Theorem 7. The following conditions are equivalent:(i)𝐴 is amenable modulo 𝐼,(ii)there is an approximate diagonal modulo 𝐼,(iii)there is a virtual diagonal modulo 𝐼.

Proof. (i) ⇒ (iii) Since 𝐴 is amenable modulo 𝐼, 𝐴/𝐼 is amenable. Thus 𝐴/𝐼 has a bounded approximate identity (𝑒_𝛼)_𝛼. Let 𝐸∈((𝐴/𝐼)⊗(𝐴/𝐼))^∗∗ be a 𝑤^∗-accumulation point of (𝑒_𝛼⊗𝑒_𝛼)_𝛼. Put 𝑎𝑑_𝐸∶𝐴→((𝐴/𝐼)⊗(𝐴/𝐼))^∗∗,𝑎↦𝑎⋅𝐸−𝐸⋅𝑎; then 𝜋^∗∗_𝐴/𝐼(𝑎⋅𝐸−𝐸⋅𝑎)=𝑤^∗−lim_𝛼𝜋_𝐴/𝐼𝑎⋅𝑒_𝛼⊗𝑒_𝛼−𝑒_𝛼⊗𝑒_𝛼⋅𝑎=lim_𝛼𝑎𝑒²_𝛼−𝑒²_𝛼𝑎=0(∀𝑎∈𝐴). (9) Theorem 5 implies that the map 𝜋_𝐴/𝐼 is surjective; then (ker𝜋_𝐴/𝐼)^∗∗=ker𝜋^∗∗_𝐴/𝐼. Furthermore (𝐴/𝐼)⊗(𝐴/𝐼) can be made into Banach 𝐴-bimodule by defining ()()𝐴𝑎⋅𝑒=𝑎+𝐼⋅𝑒,𝑒⋅𝑎=𝑒⋅𝑎+𝐼𝑎∈𝐴,𝑒∈𝐼⊗𝐴𝐼. (10) Since 𝜋_𝐴/𝐼 is a closed submodule of ((𝐴/𝐼)⊗(𝐴/𝐼)), ker𝜋^∗∗_𝐴/𝐼 is a dual Banach 𝐴-bimodule. Let 𝑋∶=((𝐴/𝐼)⊗(𝐴/𝐼))^∗ and 𝑀∶=(𝜋^∗_𝐴/𝐼(𝐴/𝐼)^∗); then ker𝜋^∗∗_𝐴/𝐼=𝑀^⟂=(𝑋/𝑀)^∗. Since 𝐼⋅(𝑋/𝑀)=(𝑋/𝑀)⋅𝐼=0, amenability modulo 𝐼 of 𝐴 implies that there exists 𝐹∈ker𝜋^∗∗_𝐴/𝐼 such that 𝑎𝑑_𝐸=𝑎𝑑_𝐹 on 𝐴⧵𝐼. Define 𝑀^′∶=𝐸−𝐹; then 𝑎⋅𝜋^∗∗_𝐴/𝐼𝑀^′=𝑎⋅𝜋^∗∗_𝐴/𝐼(𝐸−𝐹)=𝑎⋅𝜋^∗∗_𝐴/𝐼(𝐸)−𝑎⋅𝜋^∗∗_𝐴/𝐼(𝐹)=lim_𝛼𝑎𝑒²_𝛼−0=𝑎+𝐼(𝑎∈𝐴),𝑎⋅𝑀^′−𝑀^′⋅𝑎=𝑎⋅(𝐸−𝐹)−(𝐸−𝐹)⋅𝑎=𝑎𝑑_𝐸(𝑎)−𝑎𝑑_𝐹(𝑎)=0(𝑎∈𝐴⧵𝐼). (11)
(ii) ⇒ (i) Let 𝑚_𝛼=∑^∞_𝑛=1𝑎_𝑛^(𝛼)⊗𝑏_𝑛^(𝛼), where 𝑎_𝑛^(𝛼)=𝑐_𝑛^(𝛼)+𝐼 and 𝑏_𝑛^(𝛼)=𝑑_𝑛^(𝛼)+𝐼(𝛼∈𝐼,𝑛∈ℕ) with ∑^∞_𝑛=1‖𝑐_𝑛^(𝛼)‖‖𝑑_𝑛^(𝛼)‖<∞. Suppose 𝐸 is a Banach 𝐴-bimodule such that 𝐸⋅𝐼=𝐼⋅𝐸=0 and 𝐷∶𝐴→𝐸^∗ is a bounded derivation. Then (∑^∞_𝑛=1𝑐_𝑛^(𝛼)⋅𝐷𝑑_𝑛^(𝛼))_𝛼 is a bounded net in 𝐸^∗, which has a 𝑤^∗-accumulation point; say 𝜙∈𝐸^∗. Let 𝐹∶𝐴→𝐸^∗ by 𝐹(𝑎)=𝑎⋅𝜙−𝜙⋅𝑎(𝑎∈𝐴). Clearly, 𝐹 is an inner derivation and 𝐹(𝑎)=0(𝑎∈𝐼). Let 𝑎∈𝐴⧵𝐼 and 𝑥∈𝐸; then 𝑥,𝑎⋅𝜙=lim_𝛼𝑥,^∞_𝑛=1𝑎𝑐_𝑛^(𝛼)⋅𝐷𝑑_𝑛^(𝛼)=lim_𝛼𝑥,^∞_𝑛=1𝑐_𝑛^(𝛼)⋅𝐷𝑑_𝑛^(𝛼)⋅𝑎=lim_𝛼𝑥,^∞_𝑛=1𝑐_𝑛^(𝛼)𝑑_𝑛^(𝛼)⋅𝐷(𝑎)+𝑐_𝑛^(𝛼)𝐷𝑑_𝑛^(𝛼)⋅𝑎=lim_𝛼𝑥,^∞_𝑛=1𝑐_𝑛^(𝛼)𝑑_𝑛^(𝛼)(𝑎+=⟨⟩+.⋅𝐷)𝑥,𝜙⋅𝑎𝑥,𝐷𝑎𝑥,𝜙⋅𝑎 (12) So 𝑎⋅𝜙=𝐷𝑎+𝜙⋅𝑎 and 𝐷=𝑎𝑑_𝜙 on 𝐴⧵𝐼.
(iii) ⇒ (ii) Let 𝑀 be a virtual diagonal modulo 𝐼 and let (𝑚_𝛼)_𝛼 be a bounded net in (𝐴/𝐼)⊗(𝐴/𝐼) with 𝑀=𝑤^∗−lim_𝛼𝑚_𝛼. Then 𝑤−lim_𝛼𝑎⋅𝜋_𝐴/𝐼𝑚_𝛼=𝑎+𝐼(𝑎∈𝐴),𝑤−lim_𝛼𝑎⋅𝑚_𝛼−𝑚_𝛼⋅𝑎=0(𝑎∈𝐴⧵𝐼). (13) By passing to convex combinations, we obtain an approximate diagonal.

Theorem 8. Let 𝐴 be amenable modulo 𝐼, let 𝐵 be a Banach algebra, and let 𝐽 be a closed ideal of 𝐵. Let 𝜃∶𝐴→𝐵 be a continuous homomorphism with dens range such that 𝜃(𝐼)⊆𝐽. Then 𝐵 is amenable modulo 𝐽.

Proof. Suppose 𝐸 is a Banach 𝐵-bimodule such that 𝐽⋅𝐸=𝐸⋅𝐽=0 and 𝐷∶𝐵→𝐸^∗ is a bounded derivation. Then 𝐸 becomes a Banach 𝐴-bimodule via 𝑎⋅𝑥∶=𝜃(𝑎)⋅𝑥,𝑥⋅𝑎∶=𝑥⋅𝜃(𝑎)(𝑎∈𝐴,𝑥∈𝐸). (14) Clearly, 𝐼⋅𝐸=𝐸⋅𝐼=0 and 𝐷∘𝜃∶𝐴→𝐸^∗ is a bounded derivation. Since 𝐴 is amenable modulo 𝐼, there exists 𝜙∈𝐸^∗ such that 𝐷∘𝜃=𝑎𝑑_𝜙 on 𝐴⧵𝐼. Let 𝑏∈𝐵⧵𝐽; then there is a net (𝑎_𝛼)_𝛼⊂𝐴 such that 𝑏=lim_𝛼𝜃(𝑎_𝛼). Since 𝜃(𝐼)⊆𝐽, we may assume that (𝑎_𝛼)_𝛼⊂𝐴⧵𝐼. We have 𝐷(𝑏)=lim_𝛼(𝐷∘𝜃)𝑎_𝛼=lim_𝛼𝑎_𝛼⋅𝜙−𝜙⋅𝑎_𝛼=lim_𝛼𝜃𝑎_𝛼⋅𝜙−𝜙⋅𝜃𝑎_𝛼=𝑏⋅𝜙−𝜙⋅𝑏=𝑎𝑑_𝜙(𝑏). (15) Thus 𝐵 is amenable modulo 𝐽.

3. Contractibility Modulo an Ideal of Banach Algebra

We recall that a Banach algebra 𝐴 is contractible (super-amenable) if 𝐻¹(𝐴,𝑋)=0 for every Banach 𝐴-bimodule 𝑋, where the left hand side is the first cohomology group of 𝐴 with coefficient in 𝑋 (see [3, 6]).

Definition 9. A Banach algebra 𝐴 is contractible modulo 𝐼 if for every Banach 𝐴-bimodule 𝑋 such that 𝐼⋅𝑋=𝑋⋅𝐼=0, every bounded derivation 𝐷 from 𝐴 into 𝑋 is an inner derivation on the set theoretic difference 𝐴⧵𝐼∶={𝑎∈𝐴∶𝑎∉𝐼}.

Clearly, if 𝐴 is contractible modulo 𝐼, then 𝐴 is amenable modulo 𝐼.

Definition 10. An element 𝑒∈𝐴 is an identity modulo 𝐼 if 𝑎𝑒−𝑎∈𝐼,𝑒𝑎−𝑎∈𝐼(𝑎∈𝐴). A Banach algebra 𝐴 is unital modulo 𝐼 if 𝐴 has an identity modulo 𝐼.

Lemma 11. 𝐴 has an identity modulo 𝐼 if and only if 𝐴/𝐼 has an identity.

Proof. The element 𝑒 is an identity modulo 𝐼 of 𝐴 if and only if 𝑎𝑒−𝑎∈𝐼,𝑒𝑎−𝑎∈𝐼. These relations are equivalent to (𝑎+𝐼)(𝑒+𝐼)=𝑎+𝐼,(𝑒+𝐼)(𝑎+𝐼)=𝑎+𝐼 or 𝑒+𝐼 is an identity of 𝐴/𝐼.

Theorem 12. The following assertions hold.(i)If 𝐴/𝐼 is contractible and 𝐼²=𝐼, then 𝐴 is contractible modulo 𝐼.(ii)If 𝐴 is contractible modulo 𝐼, then 𝐴/𝐼 is contractible.(iii)If A is contractible modulo 𝐼 and 𝐼 is contractible, then 𝐴 is contractible.(iv)If 𝐴 is contractible modulo 𝐼, then 𝐴/𝐼 is unital.(v)If 𝐴 is contractible modulo 𝐼, then 𝐴 is unital modulo 𝐼.

Proof. (i) Let 𝑋 be a Banach 𝐴-bimodule such that 𝑋⋅𝐼=𝐼⋅𝑋=0 and let 𝐷∶𝐴→𝑋 be a bounded derivation. Since 𝐼⋅𝑋=𝑋⋅𝐼=0, 𝑋 with module action (𝑎+𝐼)⋅𝑥=𝑎⋅𝑥 can be regarded as 𝐴/𝐼-bimodule. Let 𝐷₁∶𝐴/𝐼→𝑋 by 𝐷₁(𝑎+𝐼)=𝐷(𝑎)(𝑎∈𝐴). Since 𝐼²=𝐼, 𝐷(𝑎)=0(𝑎∈𝐼). Hence 𝐷₁ is well-defined bounded derivation. Since 𝐴/𝐼 is contractible, there exits 𝜙∈𝑋 such that 𝐷₁=𝑎𝑑_𝜙. We have 𝐷(𝑎)=𝐷₁(𝑎+𝐼)=(𝑎+𝐼)⋅𝜙−𝜙⋅(𝑎+𝐼)=𝑎⋅𝜙−𝜙⋅𝑎=𝑎𝑑_𝜙(𝑎)(𝑎∈𝐴⧵𝐼). (16)
(ii) Let 𝑋 be a Banach 𝐴/𝐼-bimodule and let 𝐷∶𝐴/𝐼→𝑋 be a bounded derivation. Now 𝑋 can be made into Banach 𝐴-bimodule by defining 𝑎⋅𝑥=𝜋(𝑎)⋅𝑥,𝑥⋅𝑎=𝑥⋅𝜋(𝑎)(𝑎∈𝐴,𝑥∈𝑋), (17) where 𝜋∶𝐴→𝐴/𝐼 is the natural quotient map. Define 𝐷∘𝜋∶𝐴→𝑋. Clearly, 𝐷∘𝜋 is bounded derivation. Since 𝐼⋅𝑋=𝑋⋅𝐼=0 and 𝐴 is contractible modulo 𝐼, there exists 𝜙∈𝑋 such that 𝐷∘𝜋=𝑎𝑑_𝜙 on 𝐴⧵𝐼. Now if 𝑎∈𝐼, clearly 𝐷(𝑎+𝐼)=𝑎𝑑_𝜙(𝑎) and if 𝑎∈𝐴⧵𝐼 then 𝐷(𝑎+𝐼)=𝐷∘𝜋(𝑎)=𝑎𝑑_𝜙(𝑎)=𝑎⋅𝜙−𝜙⋅𝑎. (18) Thus 𝐷=𝑎𝑑_𝜙.
(iii) It follows from (ii).
(iv) This follows from (ii) and Exr 4.1.1 [9].
(v) It follows from (iv) and Lemma 11.

Theorem 13. 𝐴 is contractible modulo 𝐼 if and only if 𝐴 has a diagonal modulo 𝐼.

Proof. Since 𝐴 is contractible modulo 𝐼, 𝐴/𝐼 has an identity 𝑒̃ (by Theorem 12). Let 𝐸∶=𝑒̃⊗𝑒̃∈(𝐴/𝐼)⊗(𝐴/𝐼) and 𝑎𝑑_𝐸∶𝐴→(𝐴/𝐼)⊗(𝐴/𝐼) by 𝑎↦𝑎⋅𝐸−𝐸⋅𝑎; then 𝜋_𝐴/𝐼(𝑎⋅𝐸−𝐸⋅𝑎)=0(𝑎∈𝐴). (19) So 𝑎𝑑_𝐸(𝑎)⊆ker𝜋_𝐴/𝐼(𝑎∈𝐴). Since 𝐼⋅ker𝜋_𝐴/𝐼=ker𝜋_𝐴/𝐼⋅𝐼=0, contractibility of 𝐴/𝐼 implies that there exists 𝑁∈ker𝜋_𝐴/𝐼 such that 𝑎𝑑_𝐸=𝑎𝑑_𝑁 on 𝐴⧵𝐼. Define 𝑀∶=𝐸−𝑁; then 𝑎⋅𝜋_𝐴/𝐼(𝑀)=𝑎⋅𝑒²̃(),()−()−0=𝑎+𝐼𝑎∈𝐴𝑎⋅𝑀−𝑀⋅𝑎=𝑎⋅𝐸−𝑁𝐸−𝑁⋅𝑎=𝑎𝑑_𝐸(𝑎)−𝑎𝑑_𝑁(𝑎)=0(𝑎∈𝐴⧵𝐼). (20)
Conversely, Let 𝑀 be a diagonal modulo 𝐼, let 𝐸 be a Banach 𝐴-bimodule such that 𝐼⋅𝐸=𝐸⋅𝐼=0, and let 𝐷∶𝐴→𝐸 be a bounded derivation. Put 𝑀∶=∑^∞_𝑛=1𝑎^′_𝑛⊗𝑏^′_𝑛, where 𝑎^′_𝑛=𝑎_𝑛+𝐼 and 𝑏^′_𝑛=𝑏_𝑛+𝐼 and ∑^∞_𝑛=1‖𝑎_𝑛‖‖𝑏_𝑛‖<∞. Set 𝑥∶=∑^∞_𝑛=1𝑎_𝑛⋅𝐷(𝑏_𝑛); then 𝑥∈𝐸. Let 𝐹∶𝐴→𝐸 by 𝐹(𝑎)=𝑎⋅𝑥−𝑥⋅𝑎. Then 𝐹 is an inner derivation and 𝐹(𝑎)=0(𝑎∈𝐼). Let 𝑎∈𝐴⧵𝐼,𝜙∈𝐸^∗; then 𝑎⋅𝑥,𝜙=^∞_𝑛=1𝑎𝑎_𝑛𝐷𝑏_𝑛=,𝜙_∞𝑛=1𝑎_𝑛𝐷𝑏_𝑛𝑎=,𝜙_∞𝑛=1𝑎_𝑛𝐷𝑏_𝑛⋅𝑎,𝜙+^∞_𝑛=1𝑎_𝑛𝑏_𝑛𝐷(𝑎)=+𝐷(𝑎).,𝜙𝑥⋅𝑎,𝜙,𝜙 (21) Thus 𝑎⋅𝑥=𝑥⋅𝑎+𝐷(𝑎), so 𝐷=𝐹 on 𝐴⧵𝐼.

By replacing 𝑋 instead of 𝑋^∗ in the Theorem 8, we have the following theorem.

Theorem 14. Let 𝐴 be contractible modulo 𝐼, let 𝐵 be a Banach algebra, and let 𝐽 be a closed ideal of 𝐵. Let 𝜃∶𝐴→𝐵 be a continuous homomorphism with dens range such that 𝜃(𝐼)⊆𝐽. Then 𝐵 is contractible modulo 𝐽.

4. Contractibility Modulo an Ideal of Semigroup Algebra

In this section we state a version of Selivanov's theorem [9] for a large class of semigroups, characterizing contractibility modulo an ideal for the semigroups. Let 𝑆 be a semigroup and let 𝐸(𝑆) be the set of idempotents of 𝑆. A congruence 𝜌 on semigroup 𝑆 is called a group congruence if 𝑆/𝜌 is a group. We denote the least group congruence on 𝑆 by 𝜎, as used in [10, 11]. We now recall the following lemma of [6].

Lemma 15. Let 𝑆 be a semigroup:(i)for a group congruence 𝜌 on 𝑆, 𝑙¹(𝑆/𝜌)≃𝑙¹(𝑆)/𝐼_𝜌, where 𝐼_𝜌 is a closed ideal of 𝑙¹(𝑆),(ii)if 𝜎 is the least group congruence on 𝑆 and 𝑒𝑓=𝑓𝑒(𝑒,𝑓∈𝐸(𝑆)), then 𝐼_𝜎=𝐼²_𝜎.

A semigroup 𝑆 is called 𝐸-inversive if for all 𝑥∈𝑆 there exists 𝑦∈𝑆 such that 𝑥𝑦∈𝐸(𝑆). Let 𝑆 be a semigroup, let Reg(𝑆)={𝑎∈𝑆∶𝑎∈𝑎𝑆𝑎} be the set of regular elements of 𝑆, and let 𝑉(𝑎)={𝑥∈𝑆∶𝑎=𝑎𝑥𝑎,𝑥=𝑥𝑎𝑥} be the set of inverses of 𝑎∈𝑆. When 𝑉(𝑎)≠𝜙, for every 𝑎∈𝑆, 𝑆 is called regular. Let 𝑊(𝑎)={𝑥∈𝑆∶𝑥=𝑥𝑎𝑥} be the set of all weak inverses of element 𝑎∈𝑆. Then 𝑆 is 𝐸-inversive if and only if 𝑊(𝑎)≠𝜙 for all 𝑎∈𝑆 [12]. In particular, regular semigroups are 𝐸-inversive. We recall a semigroup 𝑆 is called an 𝐸-semigroup if 𝐸(𝑆) forms a subsemigroup of 𝑆. We now recall the following preposition of [13].

Proposition 16. If 𝑆 is an 𝐸-inversive 𝐸-semigroup with commutative idempotents, then the relation 𝜎={(𝑎,𝑏)∈𝑆×𝑆∣𝑒𝑎=𝑓𝑏𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑒,𝑓∈𝐸_𝑆} is the least group congruence on S.

A semigroup 𝑆 is called eventually regular if every element of 𝑆 has some power that is regular and 𝐸(𝑆) is semilattice. It is shown that if 𝑆 is an eventually semigroup then the relation 𝜎={(𝑠,𝑡)∶𝑒𝑠=𝑒𝑡,forsome𝑒∈𝐸(𝑆)} is the least group congruence on 𝑆 [14].

Theorem 17. If 𝑆 is either(i)𝐸-inverse 𝐸-semigroup with commutative idempotents or(ii)eventually inverse semigroup with commutative idempotents,then 𝑙¹(𝑆) is contractible modulo 𝐼_𝜎 if and only if 𝑆/𝜎 is finite.

Proof. By Selivanov's theorem [7], 𝑆/𝜎 is finite if and only if 𝑙¹(𝑆/𝜎)≃𝑙¹(𝑆)/𝐼_𝜎 is contractible. Now by using Theorem 12 proof is complete.

Example 18. Suppose ℕ is commutative semigroup of positive integers with maximum operation, then 𝐸(ℕ)=ℕ. Suppose 𝑚𝜎𝑛 if and only if 𝑘𝑚=𝑘𝑛, for some 𝑘∈𝐸(ℕ)(𝑛,𝑚∈ℕ). Then 𝜎 is the least group congruence on ℕ and ℕ/𝜎≃𝐺_ℕ is the trivial maximum group image of ℕ. Since 𝐺_ℕ≃ℕ/𝜎 is finite, 𝑙¹(ℕ/𝜎) is contractible and by Lemma 15 and Theorem 12,  𝑙¹(ℕ) is contractible modulo 𝐼_𝜎. We note that 𝑙¹(ℕ) is not contractible because 𝑙¹(ℕ) has no diagonal. In fact, if 𝑙¹(ℕ) has a diagonal, so it should be form 𝑚=𝛿₁⊗𝛿₁. Then 𝛿_𝑘⋅𝑚=𝑚⋅𝛿_𝑘(𝑘∈ℕ). But this equality holds if and only if 𝛿_𝑘⊗𝛿₁=𝛿₁⊗𝛿_𝑘 which is a contradiction.

Example 19. Let 𝒞={𝑝^𝑚𝑞^𝑛∶𝑚,𝑛≥0} be the bicyclic semigroup generated by 𝑝,𝑞. Clearly, 𝒞 is an 𝐸-unitary semigroup with 𝐸(𝒞)={𝑝^𝑛𝑞^𝑛∶𝑛=0,1,2,…}. Then 𝒞/𝜎=𝐺_𝒞=ℤ, where 𝜎 is defined by 𝑥𝜎𝑦 if and only if 𝑒𝑥=𝑒𝑦 for some 𝑒∈𝐸(𝒞). Since 𝐺_𝒞=ℤ is infinite, 𝑙¹(𝐺_𝒞)≅𝑙¹(𝒞)/𝐼_𝜌 is not contractible. Then 𝑙¹(𝒞) is not contractible modulo 𝐼_𝜎 (by Theorem 12 (ii)). But 𝑙¹(𝒞) is amenable modulo 𝐼_𝜎 [6].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.