# Morita Equivalence of Brandt Semigroup Algebras

Authors: Maysam Maysami Sadr
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## 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.


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## Body

## 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 ℋn(ℓ1(B(I,G)),E) and ℋn(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 elementx of a set X, δx 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 (xn)∈E and (yn)∈F such that ∑n=1∞‍∥xn∥∥yn∥<∞ and z=∑n=1∞‍xn⊗yn. 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 setsX and Y, the map δx⊗δy↦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.LetA,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, EA, and AEB are shorthand indications that E is in A-mod, mod-A, and A-mod-B, respectively.ForE 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 BEA and AFC the tensor product E⊗^AF is in B-mod-C. For AE, define a left module homomorphism μE:A⊗^AE→E by μE(a⊗x)=a·x (a∈A,x∈E).The Banach algebraA 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 μE 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.LetA 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
(2.1)Φ:ind-A-mod⟶ind-B-modΨ:ind-B-mod⟶ind-A-mod
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.
LetA 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 toℂ

In this section, for any setI, we define a matrix-like Banach algebra ℳI and prove that ℳI is Morita equivalent to the algebra of complex numbers ℂ.LetI be a nonempty set. Let the underlying Banach space of ℳI be ℓ1(I×I) and let its multiplication be the convolution product
(3.1)(ab)(i,j)=∑k∈I‍a(i,k)b(k,j)(a,b∈MI,i,j∈I).
Note that if I is a finite set, then ℳI is isomorphic to an ordinary matrix algebra. Also, for any i,p,q,j∈I, we have the following identity in ℳI:
(3.2)δ(i,p)δ(q,j)={δ(i,j)ifp=q,0ifp≠q.
Define a two-sided Banach module action of ℳI on ℓ1(I) by
(3.3)(a⋅b)(i)=∑k∈I‍a(i,k)b(k),(b⋅a)(i)=∑k∈I‍b(k)a(k,i),
for a∈ℳI,b∈ℓ1(I),i∈I.Lemma 3.1.
The mapν: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(3.4)ν¯:l1(I)⊗^l1(I)⟶C,
defined by ν¯(a⊗b)=∑i∈I‍a(i)b(i), is nonzero and N, the closed linear span of {δi·δ(j,k)⊗δi′-δi⊗δ(j,k)·δi′:i,i′,j,k∈I}, is equal to kerν¯.
Ifk0∈I is arbitrary, then ν¯(δk0⊗δk0)=1. This shows that ν¯ is not zero.
A simple computation shows that for everyi,i′,j,k∈I, ν¯(δi·δ(j,k)⊗δi′)=ν¯(δi⊗δ(j,k)·δi′). This implies that N⊆kerν¯.
For the converse, we have(3.5)δ(i,j)∈N,ifi≠j,
since δi·δ(j,k)⊗δk-δi⊗δ(j,k)·δk=-δi⊗δj. Also for every i,j∈I, we have
(3.6)δ(j,j)-δ(i,i)∈N,
since δi·δ(i,j)⊗δj-δi⊗δ(i,j)·δj=δj⊗δj-δi⊗δi. Now suppose that c=∑i,j∈I‍c(i,j)δ(i,j) is in kerν¯. Thus we have
(3.7)ν¯(c)=∑i∈I‍c(i,i)=0.
Consider the following decomposition of c:
(3.8)c=∑i,j∈I,i≠j‍c(i,j)δ(i,j)+∑i∈I‍c(i,i)δ(i,i)=a+b.
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 ℳI-induced module.
(ii)ℳI is a self-induced Banach algebra.Proof.
The canonical mapδi⊗δj↦δ(i,j) (i,j∈I), from ℓ1(I)⊗^ℓ1(I) to ℳI, is an isomorphism of Banach ℳI bimodules. Thus we have,
(3.9)MI⊗^MIl1(I)≅(l1(I)⊗^l1(I))⊗^MIl1(I)≅l1(I)⊗^(l1(I)⊗^MIl1(I))≅l1(I)⊗^C(by  Lemma  3.1)≅l1(I).
This proves ℓ1(I) is left ℳI induced. Similarly, it is proved that ℓ1(I) is right ℳI induced. For (ii), we have
(3.10)MI⊗^MIMI≅(l1(I)⊗^l1(I))⊗^MIMI≅l1(I)⊗^(l1(I)⊗^MIMI)≅l1(I)⊗^l1(I)(by(i))≅MI.
Thus ℳI is self-induced.Theorem 3.3.
ℳ
I is Morita equivalent to ℂ.Proof.
By Lemma3.1 and Proposition 3.2, the Banach algebras A=ℳI, B=ℂ and Banach bimodules P=ℂℓ1(I)ℳI, Q=ℳIℓ1(I)ℂ satisfy conditions of Theorem 2.1. Thus ℳI 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 ℳI 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

LetI 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
(4.1)(i,g,j)(i′,g′,j′)={(i,gg′,j′)ifj=i′øifj≠i′,
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 spaceℓ1(T), with the convolution product
(4.2)(ab)(i,g,j)=∑k∈I,h∈G‍a(i,gh-1,k)b(k,h,j),
for a,b∈ℓ1(T),i,j∈I,g∈G becomes a Banach algebra.Lemma 4.1.
The Banach algebrasℓ1(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.(4.3)0⟶l1(T)⟶l1(S)⟶C⟶0,
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=Idℓ1(T). Thus the exact sequence splits and we have ℓ1(S)≅ℓ1(T)⊕ℂ.The following Theorem is our main result.Theorem 4.2.
LetI 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.
LetT be as above. It is easily checked that the map δ(i,j)⊗δg↦δ(i,g,j) (i,j∈I,g∈G) is an isometric isomorphism from the Banach algebra ℳI⊗^ℓ1(G) onto ℓ1(T). Thus ℓ1(T) is self-induced, since ℳI and ℓ1(G) are self-induced. Also, since ℳI≈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.
LetA 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, ℋn(B,E)=0, and the complete quotient seminorm of ℋ0(B,E) is a norm.Proof.
Corollary IV.10 of [2].Theorem 5.2.
LetG 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 ℋn(A,E) are trivial and ℋ0(A,E) is a Banach space.Proof.
It was proved in the preceding section thatℓ1(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 algebraA, 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, Theorem5.2, 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.
LetG 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 ℋn(A,E*) are trivial.A Banach algebraA 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 (fλ)∈E* and every a∈A, D(a)=norm-limλa·fλ-fλ·a. The following Theorem (that corrects! some preceding results on amenability properties of Brandt semigroup algebras) is proved in [9].Theorem 5.4.
LetS=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.
LetG 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)).



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