Morita Equivalence of Brandt Semigroup Algebras
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.
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 . Then he extended his theory to the larger class of self-induced Banach algebras in . In the first theory, the only class of algebras that are Morita equivalent to the algebra of complex numbers 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 . In this paper we construct a new class of infinite dimensional self-induced Banach algebras Morita equivalent to . Then by this construction, we show that for a discrete group , and every two nonempty sets , the Banach convolution algebras and are Morita equivalent, where denotes the Brandt semigroup over with index set , . 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 , we show that if is an amenable semigroup and is an arbitrary nonempty set, then the topological Hochschild homology and cohomology groups and , for any and every -induced Banach bimodule , are trivial. Also by a specific example, we show that the notion of approximate amenability of Banach algebras, introduced by Ghahramani and Loy , 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.
Throughout this paper, for an element of a set , is its point mass measure in . Let and be Banach spaces. The Banach space which is the completed projective tensor product of and is denoted by ; for , there are sequences and such that and . Analogous to the pure algebraic case, the key property of is that, for each continuous bilinear map , where is a Banach space, there is a unique continuous linear map with and (), see  for more details.
It is well known that for nonempty sets and , the map () defines an isometric isomorphism between Banach spaces and ; we use frequently this identification.
Let , and be Banach algebras. A left Banach -module is an ordinary left -module which is a Banach space and there is a constant such that (). Similarly, right Banach modules and Banach bimodules are defined. The category of left Banach -modules and bounded module homomorphisms is denoted by -. Similarly, one can define the category of right Banach -modules -, and the category of left- right- Banach bimodules --. The notations , , and are shorthand indications that is in -, -, and --, respectively.
For in - and in -, let be the universal object for -balanced bounded bilinear maps from . This can be realized as the Banach space where is the closed linear span of [11, 12]. For and the tensor product is in --. For , define a left module homomorphism by ().
The Banach algebra is called self-induced if the multiplication map , from to , is an isomorphism (between bimodules). More generally, a left Banach -module , is induced if is an isomorphism . Similarly, right-and two-sided-induced modules are defined. The category of left Banach -induced modules is denoted by ind--.
Let and be two self-induced Banach algebras. Then and are Morita equivalent if ind-- and ind-- are equivalent, i.e., there are covariant functors such that and are natural isomorphic to the identity functors on ind-- and ind--, respectively. For complete definitions see the original paper . We only need the following characterization of Morita equivalence .
Theorem 2.1. Let and be self-induced Banach algebras. Then and are Morita equivalent if and only if there are two-sided-induced modules -- and -- such that and , where denotes topological isomorphism of bimodules.
3. A Banach Algebra Morita Equivalent to
In this section, for any set , we define a matrix-like Banach algebra and prove that is Morita equivalent to the algebra of complex numbers .
Let be a nonempty set. Let the underlying Banach space of be and let its multiplication be the convolution product Note that if is a finite set, then is isomorphic to an ordinary matrix algebra. Also, for any , we have the following identity in : Define a two-sided Banach module action of on by for .
Lemma 3.1. The map , defined by , is an isomorphism of Banach spaces.
Proof. By definition, it is enough to prove that the map
defined by , is nonzero and , the closed linear span of , is equal to .
If is arbitrary, then . This shows that is not zero.
A simple computation shows that for every , . This implies that .
For the converse, we have since . Also for every , we have since . Now suppose that is in . Thus we have Consider the following decomposition of : Then by (3.5), is in . Let be arbitrary and fixed, then by (3.7), we have . Thus by (3.6), is also in . Therefore is in and .
Proposition 3.2. (i) is a two-sided -induced module.
(ii) is a self-induced Banach algebra.
Proof. The canonical map (), from to , is an isomorphism of Banach bimodules. Thus we have, This proves is left induced. Similarly, it is proved that is right induced. For (ii), we have Thus is self-induced.
Theorem 3.3. is Morita equivalent to .
Remark 3.4. (I) It is proved in  that for any Banach space , the tensor algebra is Morita equivalent to . Also, it is well known that if has bounded approximate property, then the algebra of nuclear operators on and is isomorphic. Thus by Theorem 3.3, and are Morita equivalent, but clearly these are not isomorphic if is an infinite set.
(II) For Morita theory of some other Matrix-like algebras, see [1, 2, 14].
4. The Main Result
Let be a nonempty set and let be a discrete group. Consider the set , add an extra element ø to and define a semigroup multiplication on , as follows. For and , let also let and . Then becomes a semigroup that is called the Brandt semigroup over with index set and usually denoted by . For more details see .
The Banach space , with the convolution product for becomes a Banach algebra.
Lemma 4.1. The Banach algebras and are homeomorphically isomorphic, where the multiplication of is coordinatewise.
Proof. Consider the following short exact sequence of Banach algebras and continuous algebra homomorphisms. where the second arrow is defined by and , for and , and the third arrow is the integral functional, (). Now, let be the restriction map, . Then is a continuous algebra homomorphism and . Thus the exact sequence splits and we have .
The following Theorem is our main result.
Theorem 4.2. Let and be nonempty sets and let be a discrete group. Then and are Morita equivalent self-induced Banach algebras.
Proof. Let be as above. It is easily checked that the map () is an isometric isomorphism from the Banach algebra onto . Thus is self-induced, since and are self-induced. Also, since , we have . By Lemma 4.1, we have thus is self-induced and Morita equivalent to . Similarly , therefore we have .
5. Some Applications
For the topological Hochschild (co)homology of Banach algebras, we refer the reader to . Recall that a Banach algebra is amenable if for every Banach -bimodule , the first-order bounded Hochschild cohomology group of with coefficients in the dual Banach bimodule vanishes, , or equivalently any bounded derivation is inner. A famous Theorem proved by Johnson  says that for any locally compact group , amenability of is equivalent to the amenability of the convolution group algebra . For a modern account on amenability see .
Proposition 5.1. Let and be Morita equivalent self-induced Banach algebras. Suppose that is amenable. Then for every two-sided -induced module --, and , , and the complete quotient seminorm of is a norm.
Proof. Corollary IV.10 of .
Theorem 5.2. Let be an amenable discrete group, be a nonempty set and . Then for any two-sided induced Banach bimodule E and every , the topological Hochschild homology groups are trivial and is a Banach space.
Proof. It was proved in the preceding section that . By Johnson's Theorem, is an amenable Banach algebra and thus so is . Now, apply Proposition 5.1.
Note that for any self-induced Banach algebra , the class of two sided induced modules is very wide, since for any Banach bimodule , the module is two-sided induced. In fact, any -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  or Theorem 4.8 of .
Theorem 5.3. Let be an amenable discrete group, be a nonempty set and . Then for any two-sided induced Banach bimodule and every , the bounded Hochschild cohomology groups are trivial.
A Banach algebra is called approximately amenable [8, 10], if for any Banach bimodule , every bounded derivation is approximately inner, that is, for a net and every , . The following Theorem (that corrects! some preceding results on amenability properties of Brandt semigroup algebras) is proved in .
Theorem 5.4. Let be a Brandt semigroup. Then the following are equivalent. (1) is amenable. (2) is approximately amenable. (3) is finite and is amenable.
Theorem 5.5. The notion of approximate amenability of self-induced Banach algebras is not a Morita invariant.
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