Morita Equivalence of Brandt Semigroup Algebras

Abstract

We show that Banach semigroup algebras of any two Brandt semigroups over a fixed group are Morita equivalence with respect to the Morita theory of self-induced Banach algebras introduced by Grønbæk. As applications, we show that the bounded Hochschild (co)homology groups of Brandt semigroup algebras over amenable groups are trivial and prove that the notion of approximate amenability is not Morita invariant.

1. Introduction

Morita theory is a very useful tool in the study of rings and algebras. In the area of topological algebras, there are different notions of Morita theory in the literature, but all of these notions are simple variants of the original one, defined by Kiiti Morita, in the pure algebraic case. Niels Grønbæk defined a Morita theory for Banach algebras with bounded approximate identities in [1]. Then he extended his theory to the larger class of self-induced Banach algebras in [2]. In the first theory, the only class of algebras that are Morita equivalent to the algebra of complex numbers ℂ is the class of finite dimensional matrix algebras, but in the second one we find many infinite dimensional Banach algebras that are Morita equivalent to ℂ. In this paper we construct a new class of infinite dimensional self-induced Banach algebras Morita equivalent to ℂ. Then by this construction, we show that for a discrete group 𝐺, and every two nonempty sets 𝐼,𝐽, the Banach convolution algebras ℓ¹(𝐵(𝐼,𝐺)) and ℓ¹(𝐵(𝐽,𝐺)) are Morita equivalent, where 𝐵(𝐼,𝐺) denotes the Brandt semigroup over 𝐺 with index set 𝐼, [3]. Brandt semigroups are one of the most important classes of inverse semigroups paid. Some authors have studied the bounded Hochschild cohomology and amenability properties of inverse semigroup algebras, [4–9]. As a corollary of Morita equivalence of Brandt semigroup algebras and a strong result of [2], we show that if 𝐺 is an amenable semigroup and 𝐼 is an arbitrary nonempty set, then the topological Hochschild homology and cohomology groups ℋ_𝑛(ℓ¹(𝐵(𝐼,𝐺)),𝐸) and ℋ^𝑛(ℓ¹(𝐵(𝐼,𝐺)),𝐸^∗), for any 𝑛>0 and every ℓ¹(𝐵(𝐼,𝐺))-induced Banach bimodule 𝐸, are trivial. Also by a specific example, we show that the notion of approximate amenability of Banach algebras, introduced by Ghahramani and Loy [10], is not Morita invariant. This result is in contrast to Grønbæk's corollary on amenability [1, Corollary 6.5] that says for Banach algebras with bounded approximate identities the notion of amenability is Morita invariant.

2. Preliminaries

Throughout this paper, for an element 𝑥 of a set 𝑋, 𝛿_𝑥 is its point mass measure in ℓ¹(𝑋). Let 𝐸 and 𝐹 be Banach spaces. The Banach space which is the completed projective tensor product of 𝐸 and 𝐹 is denoted by 𝐸⊗𝐹; for 𝑧∈𝐸⊗𝐹, there are sequences (𝑥_𝑛)∈𝐸 and (𝑦_𝑛)∈𝐹 such that ∑^∞_𝑛=1‖𝑥_𝑛‖‖𝑦_𝑛‖<∞ and 𝑧=∑^∞_𝑛=1𝑥_𝑛⊗𝑦_𝑛. Analogous to the pure algebraic case, the key property of 𝐸⊗𝐹 is that, for each continuous bilinear map 𝐵∶𝐸×𝐹→𝐷, where 𝐷 is a Banach space, there is a unique continuous linear map 𝑇∶𝐸⊗𝐹→𝐷 with ‖𝑇‖=‖𝐵‖ and 𝑇(𝑥⊗𝑦)=𝐵(𝑥,𝑦) (𝑥∈𝐸,𝑦∈𝐹), see [11] for more details.

It is well known that for nonempty sets 𝑋 and 𝑌, the map 𝛿_𝑥⊗𝛿_𝑦↦𝛿_(𝑥,𝑦) (𝑥∈𝑋,𝑦∈𝑌) defines an isometric isomorphism between Banach spaces ℓ¹(𝑋)⊗ℓ¹(𝑌) and ℓ¹(𝑋×𝑌); we use frequently this identification.

Let 𝐴,𝐵, and 𝐶 be Banach algebras. A left Banach 𝐴-module 𝐸 is an ordinary left 𝐴-module which is a Banach space and there is a constant 𝑀>0 such that ‖𝑎⋅𝑥‖≤𝑀‖𝑎‖‖𝑥‖ (𝑎∈𝐴,𝑥∈𝐸). Similarly, right Banach modules and Banach bimodules are defined. The category of left Banach 𝐴-modules and bounded module homomorphisms is denoted by 𝐴-mod. Similarly, one can define the category of right Banach 𝐴-modules mod-𝐴, and the category of left-𝐴 right-𝐵 Banach bimodules 𝐴-mod-𝐵. The notations _𝐴𝐸, 𝐸_𝐴, and _𝐴𝐸_𝐵 are shorthand indications that 𝐸 is in 𝐴-mod, mod-𝐴, and 𝐴-mod-𝐵, respectively.

For 𝐸 in mod-𝐴 and 𝐹 in 𝐴-mod, let 𝐸⊗_𝐴𝐹 be the universal object for 𝐴-balanced bounded bilinear maps from 𝐸×𝐹. This can be realized as the Banach space 𝐸⊗𝐹/𝑁 where 𝑁 is the closed linear span of {𝑥⋅𝑎⊗𝑦−𝑥⊗𝑎⋅𝑦∶𝑥∈𝐸,𝑦∈𝐹,𝑎∈𝐴} [11, 12]. For _𝐵𝐸_𝐴 and _𝐴𝐹_𝐶 the tensor product 𝐸⊗_𝐴𝐹 is in 𝐵-mod-𝐶. For _𝐴𝐸, define a left module homomorphism 𝜇_𝐸∶𝐴⊗_𝐴𝐸→𝐸 by 𝜇_𝐸(𝑎⊗𝑥)=𝑎⋅𝑥 (𝑎∈𝐴,𝑥∈𝐸).

The Banach algebra 𝐴 is called self-induced if the multiplication map 𝑎⊗𝑏→𝑎𝑏, from 𝐴⊗_𝐴𝐴 to 𝐴, is an isomorphism (between 𝐴 bimodules). More generally, a left Banach 𝐴-module 𝐸, is 𝐴 induced if 𝜇_𝐸 is an isomorphism [13]. Similarly, right-and two-sided-induced modules are defined. The category of left Banach 𝐴-induced modules is denoted by ind-𝐴-mod.

Let 𝐴 and 𝐵 be two self-induced Banach algebras. Then 𝐴 and 𝐵 are Morita equivalent if ind-𝐴-mod and ind-𝐵-mod are equivalent, i.e., there are covariant functors Φ∶ind-𝐴-mod⟶ind-𝐵-modΨ∶ind-𝐵-mod⟶ind-𝐴-mod (2.1) such that ΨΦ and ΦΨ are natural isomorphic to the identity functors on ind-𝐴-mod and ind-𝐵-mod, respectively. For complete definitions see the original paper [2]. We only need the following characterization of Morita equivalence [2].

Theorem 2.1. Let 𝐴 and 𝐵 be self-induced Banach algebras. Then 𝐴 and 𝐵 are Morita equivalent if and only if there are two-sided-induced modules 𝑃∈𝐵-mod-𝐴 and 𝑄∈𝐴-mod-𝐵 such that 𝑃⊗_𝐴𝑄≅𝐵 and 𝑄⊗_𝐵𝑃≅𝐴, where ≅ denotes topological isomorphism of bimodules.

3. A Banach Algebra Morita Equivalent to ℂ

In this section, for any set 𝐼, we define a matrix-like Banach algebra ℳ_𝐼 and prove that ℳ_𝐼 is Morita equivalent to the algebra of complex numbers ℂ.

Let 𝐼 be a nonempty set. Let the underlying Banach space of ℳ_𝐼 be ℓ¹(𝐼×𝐼) and let its multiplication be the convolution product (𝑎𝑏)𝑖,𝑗=_𝑘∈𝐼𝑎(𝑖,𝑘)𝑏𝑘,𝑗𝑎,𝑏∈ℳ_𝐼,𝑖,𝑗∈𝐼. (3.1) Note that if 𝐼 is a finite set, then ℳ_𝐼 is isomorphic to an ordinary matrix algebra. Also, for any 𝑖,𝑝,𝑞,𝑗∈𝐼, we have the following identity in ℳ_𝐼: 𝛿_(𝑖,𝑝)𝛿_(𝑞,𝑗)=𝛿_(𝑖,𝑗)if𝑝=𝑞,0if𝑝≠𝑞. (3.2) Define a two-sided Banach module action of ℳ_𝐼 on ℓ¹(𝐼) by (𝑎⋅𝑏)(𝑖)=_𝑘∈𝐼𝑎(𝑖,𝑘)𝑏(𝑘),(𝑏⋅𝑎)(𝑖)=_𝑘∈𝐼𝑏(𝑘)𝑎(𝑘,𝑖), (3.3) for 𝑎∈ℳ_𝐼,𝑏∈ℓ¹(𝐼),𝑖∈𝐼.

Lemma 3.1. The map 𝜈∶ℓ¹(𝐼)⊗_ℳ𝐼ℓ¹(𝐼)→ℂ, defined by 𝜈(𝑎⊗𝑏)=∑_𝑖∈𝐼𝑎(𝑖)𝑏(𝑖), is an isomorphism of Banach spaces.

Proof. By definition, it is enough to prove that the map 𝜈∶ℓ¹(𝐼)⊗ℓ¹(𝐼)⟶ℂ, (3.4) defined by 𝜈(𝑎⊗𝑏)=∑_𝑖∈𝐼𝑎(𝑖)𝑏(𝑖), is nonzero and 𝑁, the closed linear span of {𝛿_𝑖⋅𝛿_(𝑗,𝑘)⊗𝛿_𝑖′−𝛿_𝑖⊗𝛿_(𝑗,𝑘)⋅𝛿_𝑖′∶𝑖,𝑖′,𝑗,𝑘∈𝐼}, is equal to ker𝜈.
If 𝑘₀∈𝐼 is arbitrary, then 𝜈(𝛿_𝑘0⊗𝛿_𝑘0)=1. This shows that 𝜈 is not zero.
A simple computation shows that for every 𝑖,𝑖′,𝑗,𝑘∈𝐼, 𝜈(𝛿_𝑖⋅𝛿_(𝑗,𝑘)⊗𝛿_𝑖′)=𝜈(𝛿_𝑖⊗𝛿_(𝑗,𝑘)⋅𝛿_𝑖′). This implies that 𝑁⊆ker𝜈.
For the converse, we have 𝛿_(𝑖,𝑗)∈𝑁,if𝑖≠𝑗, (3.5) since 𝛿_𝑖⋅𝛿_(𝑗,𝑘)⊗𝛿_𝑘−𝛿_𝑖⊗𝛿_(𝑗,𝑘)⋅𝛿_𝑘=−𝛿_𝑖⊗𝛿_𝑗. Also for every 𝑖,𝑗∈𝐼, we have 𝛿_(𝑗,𝑗)−𝛿_(𝑖,𝑖)∈𝑁, (3.6) since 𝛿_𝑖⋅𝛿_(𝑖,𝑗)⊗𝛿_𝑗−𝛿_𝑖⊗𝛿_(𝑖,𝑗)⋅𝛿_𝑗=𝛿_𝑗⊗𝛿_𝑗−𝛿_𝑖⊗𝛿_𝑖. Now suppose that 𝑐=∑_{𝑖,𝑗∈𝐼}𝑐(𝑖,𝑗)𝛿_(𝑖,𝑗) is in ker𝜈. Thus we have 𝜈(𝑐)=_𝑖∈𝐼𝑐(𝑖,𝑖)=0. (3.7) Consider the following decomposition of 𝑐: 𝑐=_{𝑖,𝑗∈𝐼,𝑖≠𝑗}𝑐𝑖,𝑗𝛿_(𝑖,𝑗)+_𝑖∈𝐼𝑐(𝑖,𝑖)𝛿_(𝑖,𝑖)=𝑎+𝑏. (3.8) Then by (3.5), 𝑎 is in 𝑁. Let 𝑘₀∈𝐼 be arbitrary and fixed, then by (3.7), we have 𝑏=∑_𝑖∈𝐼𝑐(𝑖,𝑖)𝛿_(𝑖,𝑖)−∑_𝑖∈𝐼𝑐(𝑖,𝑖)𝛿_{(𝑘0,𝑘0)}. Thus by (3.6), 𝑏 is also in 𝑁. Therefore 𝑐 is in 𝑁 and ker𝜈⊆𝑁.

Proposition 3.2. (i) ℓ¹(𝐼) is a two-sided ℳ_𝐼-induced module.
(ii) ℳ_𝐼 is a self-induced Banach algebra.

Proof. The canonical map 𝛿_𝑖⊗𝛿_𝑗↦𝛿_(𝑖,𝑗) (𝑖,𝑗∈𝐼), from ℓ¹(𝐼)⊗ℓ¹(𝐼) to ℳ_𝐼, is an isomorphism of Banach ℳ_𝐼 bimodules. Thus we have, ℳ_𝐼⊗_ℳ𝐼ℓ¹(𝐼)≅ℓ¹(𝐼)⊗ℓ¹(𝐼)⊗_ℳ𝐼ℓ¹(𝐼)≅ℓ¹(𝐼)⊗ℓ¹(𝐼)⊗_ℳ𝐼ℓ¹(𝐼)≅ℓ¹(𝐼)⊗ℂbyLemma3.1≅ℓ¹(𝐼). (3.9) This proves ℓ¹(𝐼) is left ℳ_𝐼 induced. Similarly, it is proved that ℓ¹(𝐼) is right ℳ_𝐼 induced. For (ii), we have ℳ_𝐼⊗_ℳ𝐼ℳ_𝐼≅ℓ¹(𝐼)⊗ℓ¹(𝐼)⊗_ℳ𝐼ℳ_𝐼≅ℓ¹(𝐼)⊗ℓ¹(𝐼)⊗_ℳ𝐼ℳ_𝐼≅ℓ¹(𝐼)⊗ℓ¹(𝐼)by(i)≅ℳ_𝐼. (3.10) Thus ℳ_𝐼 is self-induced.

Theorem 3.3. ℳ_𝐼 is Morita equivalent to ℂ.

Proof. By Lemma 3.1 and Proposition 3.2, the Banach algebras 𝐴=ℳ_𝐼, 𝐵=ℂ and Banach bimodules 𝑃=_ℂℓ¹(𝐼)_ℳ𝐼, 𝑄=_ℳ𝐼ℓ¹(𝐼)_ℂ satisfy conditions of Theorem 2.1. Thus ℳ_𝐼 is Morita equivalent to ℂ.

Remark 3.4. (I) It is proved in [2] that for any Banach space 𝐸, the tensor algebra 𝐸⊗𝐸^∗ is Morita equivalent to ℂ. Also, it is well known that if 𝐸 has bounded approximate property, then the algebra 𝒩(𝐸) of nuclear operators on 𝐸 and 𝐸⊗𝐸^∗ is isomorphic. Thus by Theorem 3.3, 𝒩(ℓ¹(𝐼))≅ℓ¹(𝐼)⊗ℓ^∞(𝐼) and ℳ_𝐼 are Morita equivalent, but clearly these are not isomorphic if 𝐼 is an infinite set.
(II) For Morita theory of some other Matrix-like algebras, see [1, 2, 14].

4. The Main Result

Let 𝐼 be a nonempty set and let 𝐺 be a discrete group. Consider the set 𝑇=𝐼×𝐺×𝐼, add an extra element ø to 𝑇 and define a semigroup multiplication on 𝑆=𝑇∪{ø}, as follows. For 𝑖,𝑖′,𝑗,𝑗′∈𝐼 and 𝑔,𝑔′∈𝐺, let 𝑖,𝑔,𝑗𝑖^′,𝑔^′,𝑗^′=𝑖,𝑔𝑔^′,𝑗^′if𝑗=𝑖′øif𝑗≠𝑖′, (4.1) also let ø(𝑖,𝑔,𝑗)=(𝑖,𝑔,𝑗)ø=ø and øø=ø. Then 𝑆 becomes a semigroup that is called the Brandt semigroup over 𝐺 with index set 𝐼 and usually denoted by 𝐵(𝐼,𝐺). For more details see [3].

The Banach space ℓ¹(𝑇), with the convolution product (𝑎𝑏)𝑖,𝑔,𝑗=_{𝑘∈𝐼,ℎ∈𝐺}𝑎𝑖,𝑔ℎ⁻¹,𝑘𝑏𝑘,ℎ,𝑗, (4.2) for 𝑎,𝑏∈ℓ¹(𝑇),𝑖,𝑗∈𝐼,𝑔∈𝐺 becomes a Banach algebra.

Lemma 4.1. The Banach algebras ℓ¹(𝑆) and ℓ¹(𝑇)⊕ℂ are homeomorphically isomorphic, where the multiplication of ℓ¹(𝑇)⊕ℂ is coordinatewise.

Proof. Consider the following short exact sequence of Banach algebras and continuous algebra homomorphisms. 0⟶ℓ¹(𝑇)⟶ℓ¹(𝑆)⟶ℂ⟶0, (4.3) where the second arrow 𝑢∶ℓ¹(𝑇)→ℓ¹(𝑆) is defined by 𝑢(𝑏)(𝑡)=𝑏(𝑡) and 𝑢(𝑏)(ø)=−∑_𝑠∈𝑇𝑏(𝑠), for 𝑏∈ℓ¹(𝑇) and 𝑡∈𝑇⊂𝑆, and the third arrow 𝑣∶ℓ¹(𝑆)→ℂ is the integral functional, 𝑣(𝑎)=∑_𝑠∈𝑆𝑎(𝑠) (𝑎∈ℓ¹(𝑆)). Now, let 𝑤∶ℓ¹(𝑆)→ℓ¹(𝑇) be the restriction map, 𝑤(𝑎)∶=𝑎|_𝑇. Then 𝑤 is a continuous algebra homomorphism and 𝑤𝑢=𝐼𝑑_ℓ1(𝑇). Thus the exact sequence splits and we have ℓ¹(𝑆)≅ℓ¹(𝑇)⊕ℂ.

The following Theorem is our main result.

Theorem 4.2. Let 𝐼 and 𝐽 be nonempty sets and let 𝐺 be a discrete group. Then ℓ¹(𝐵(𝐼,𝐺)) and ℓ¹(𝐵(𝐽,𝐺)) are Morita equivalent self-induced Banach algebras.

Proof. Let 𝑇 be as above. It is easily checked that the map 𝛿_(𝑖,𝑗)⊗𝛿_𝑔↦𝛿_{(𝑖,𝑔,𝑗)} (𝑖,𝑗∈𝐼,𝑔∈𝐺) is an isometric isomorphism from the Banach algebra ℳ_𝐼⊗ℓ¹(𝐺) onto ℓ¹(𝑇). Thus ℓ¹(𝑇) is self-induced, since ℳ_𝐼 and ℓ¹(𝐺) are self-induced. Also, since ℳ_𝐼≈ℂ, we have ℓ¹(𝑇)≈ℂ⊗ℓ¹(𝐺)≈ℓ¹(𝐺). By Lemma 4.1, we have ℓ¹(𝐵(𝐼,𝐺))≅ℓ¹(𝑇)⊕ℂ thus ℓ¹(𝐵(𝐼,𝐺)) is self-induced and Morita equivalent to ℓ¹(𝐺)⊕ℂ. Similarly ℓ¹(𝐵(𝐽,𝐺))≈ℓ¹(𝐺)⊕ℂ, therefore we have ℓ¹(𝐵(𝐼,𝐺))≈ℓ¹(𝐵(𝐽,𝐺)).

5. Some Applications

For the topological Hochschild (co)homology of Banach algebras, we refer the reader to [11]. Recall that a Banach algebra 𝐴 is amenable if for every Banach 𝐴-bimodule 𝐸, the first-order bounded Hochschild cohomology group of 𝐴 with coefficients in the dual Banach bimodule 𝐸^∗ vanishes, ℋ¹(𝐴,𝐸^∗)=0, or equivalently any bounded derivation 𝐷∶𝐴→𝐸^∗ is inner. A famous Theorem proved by Johnson [15] says that for any locally compact group 𝐺, amenability of 𝐺 is equivalent to the amenability of the convolution group algebra 𝐿¹(𝐺). For a modern account on amenability see [16].

Proposition 5.1. Let 𝐴 and 𝐵 be Morita equivalent self-induced Banach algebras. Suppose that 𝐴 is amenable. Then for every two-sided 𝐵-induced module 𝐸∈𝐵-mod-𝐵, and 𝑛≥1, ℋ_𝑛(𝐵,𝐸)=0, and the complete quotient seminorm of ℋ₀(𝐵,𝐸) is a norm.

Proof. Corollary IV.10 of [2].

Theorem 5.2. Let 𝐺 be an amenable discrete group, 𝐼 be a nonempty set and 𝐴=ℓ¹(𝐵(𝐼,𝐺)). Then for any two-sided 𝐴 induced Banach 𝐴 bimodule E and every 𝑛>0, the topological Hochschild homology groups ℋ_𝑛(𝐴,𝐸) are trivial and ℋ₀(𝐴,𝐸) is a Banach space.

Proof. It was proved in the preceding section that ℓ¹(𝐵(𝐼,𝐺))≈ℓ¹(𝐺)⊕ℂ. By Johnson's Theorem, ℓ¹(𝐺) is an amenable Banach algebra and thus so is ℓ¹(𝐺)⊕ℂ. Now, apply Proposition 5.1.

Note that for any self-induced Banach algebra 𝐴, the class of two sided 𝐴 induced modules is very wide, since for any Banach 𝐴 bimodule 𝐹, the module 𝐴⊗_𝐴𝐹⊗_𝐴𝐴 is two-sided 𝐴 induced. In fact, any 𝐴-induced bimodule is of this form.

The following Theorem directly follows from duality between definitions of Hochschild homology and cohomology, Theorem 5.2, and general results of homology theory in the category of Banach spaces and continuous linear maps, see for instance [11] or Theorem 4.8 of [17].

Theorem 5.3. Let 𝐺 be an amenable discrete group, 𝐼 be a nonempty set and 𝐴=ℓ¹(𝐵(𝐼,𝐺)). Then for any two-sided 𝐴 induced Banach 𝐴 bimodule 𝐸 and every 𝑛>0, the bounded Hochschild cohomology groups ℋ^𝑛(𝐴,𝐸^∗) are trivial.

A Banach algebra 𝐴 is called approximately amenable [8, 10], if for any Banach 𝐴 bimodule 𝐸, every bounded derivation 𝐷∶𝐴→𝐸^∗ is approximately inner, that is, for a net (𝑓_𝜆)∈𝐸^∗ and every 𝑎∈𝐴, 𝐷(𝑎)=norm-lim_𝜆𝑎⋅𝑓_𝜆−𝑓_𝜆⋅𝑎. The following Theorem (that corrects! some preceding results on amenability properties of Brandt semigroup algebras) is proved in [9].

Theorem 5.4. Let 𝑆=𝐵(𝐼,𝐺) be a Brandt semigroup. Then the following are equivalent. (1)ℓ¹(𝑆) is amenable. (2)ℓ¹(𝑆) is approximately amenable. (3)𝐼 is finite and 𝐺 is amenable.

Theorem 5.5. The notion of approximate amenability of self-induced Banach algebras is not a Morita invariant.

Proof. Let 𝐺 be an amenable group, 𝐼 be a finite nonempty set, and 𝐽 be an infinite set. Then by Theorem 5.4, ℓ¹(𝐵(𝐼,𝐺)) is approximately amenable and ℓ¹(𝐵(𝐽,𝐺)) is not approximately amenable. But by Theorem 4.2, we have ℓ¹(𝐵(𝐼,𝐺))≈ℓ¹(𝐵(𝐽,𝐺)).