Biflat-like Banach algebras
Sanaz Haddad sabzevar, Amin Mahmoodi

TL;DR
This paper introduces the concept of $\sigma$-biflatness in Banach algebras, explores its differences from biflatness, and examines its relationships with other properties like $\sigma$-biprojectivity and $\sigma$-amenability, including tensor products.
Contribution
It defines $\sigma$-biflatness for Banach algebras, distinguishes it from biflatness, and investigates its connections with related concepts and tensor product behavior.
Findings
$\sigma$-biflatness is a proper generalization of biflatness.
$\sigma$-biflatness and biflatness are shown to be distinct properties.
Relations between $\sigma$-biflatness, $\sigma$-biprojectivity, and $\sigma$-amenability are established.
Abstract
Given a Banach algebra and a continuous homomorphism on it, the notion of -biflatness for is introduced. This is a generalization of biflatness and it is shown that they are distinct. The relations between -biflatness and some other close concepts such as -biprojectivity and -amenability are studied. The -biflatness of tensor product of Banach algebras are also discussed.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Advanced Banach Space Theory
Biflat-like Banach algebras
Sanaz Haddad sabzevar
Department of Mathematics, Central Tehran Branch, Islamic Azad University,Tehran, Iran, e-mail: [email protected]
and
Amin Mahmoodi
Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran, e-mail: [email protected]
Abstract.
Given a Banach algebra and a continuous homomorphism on it, the notion of -biflatness for is introduced. This is a generalization of biflatness and it is shown that they are distinct. The relations between -biflatness and some other close concepts such as -biprojectivity and -amenability are studied. The -biflatness of tensor product of Banach algebras are also discussed.
Keywords: -amenable, -biflat, -virtual diagonal, -biprojective, -derivation.
MSC 2010: Primary: 46H25; Secondary: 16E40, 43A20.
1. Introduction
Biprojectivity and biflatness for Banach algebras, as introduced and studied in the works of Helemskii (see for instance [5]), have proved to be important and fertile notions. There are close relationship between these notions and some other concepts of Banach algebras such as amenability. It is known that every biprojective Banach algebra with a bounded approximate identity is amenable and in the presence of a bounded approximate identity, biflatness and amenability are the same notions [5].
Before preceding further, we recall some preliminaries. Let be a Banach algebra. Then its projective tensor product is a Banach -bimodule through
[TABLE]
For a Banach -bimodule , the dual space becomes a Banach -bimodule in a natural manner. Let and be Banach -bimodules. A bounded linear map is an -bimodule homomorphism if and , for , . It is obvious that the diagonal operator given by , is an -bimodule homomorphism. If it is clear to which algebra we refer, we simply write .
A Banach algebra is biprojective if has a right inverse which is an -bimodule homomorphism. If there is an -bimodule homomorphism which is a left inverse of , then we say that is biflat.
Let be a Banach algebra. We write for the set of all continuous homomorphisms on . Let and be Banach -bimodule, and let . A bounded linear map is a --bimodule homomorphism if and where and . A Banach algebra is -biprojective if there exists a --bimodule homomorphism such that [13].
The purpose of this paper is to study the concept of -biflatness for Banach algebras. We have to stress that our definition is completely different from what have introduced in [4, Definition 2.11]. For comparison, unlike our definition, -biflatness in [4] is not a generalization of the notion of biflatness [4, Remark 2.13].
The organization of the paper is as follows. Firstly, in section 2 we investigate some basic properties of -biflat Banach algebras. We find an equivalent condition to -biflatness (Theorem 2.3). We prove that every -biprojective Banach algebra is -biflat (Proposition 2.4). However biflat Banach algebras are -biflat (Proposition 2.5), we give an example to show that the class of -biflat Banach algebras is larger than that for biflat Banach algebras (Example 2.1).
In section 3, we find the relations between -biflatness and both -amenability and the existence of some certain -diagonals. There are examples of -biflat Banach algebras which are not -amenable (Examples 3.1 and 3.2).
In section 4, we deal with the short exact sequence , where is a Banach algebra. We prove that if is -amenable, then and behave like splitting sequences (Proposition 4.2).
Finally in section 5, we generalize Ramsden’s theorem [11, Proposition 2.5] related to biflatness of tensor product of Banach algebras to the -case (Theorem 5.1).
2. Basic properties
Definition 2.1**.**
Let be a Banach algebra and let . Then is -biflat if there exists a bounded linear map satisfying
[TABLE]
for such that .
Lemma 2.2**.**
Let be a Banach algebra, let and let and be Banach -bimodules. If is a --bimodule homomorphism, then satisfies , that is, and , for .
Proof.
For every , and , we have
[TABLE]
so . Similarly . ∎
The following characterization is useful.
Theorem 2.3**.**
Let be a Banach algebra and let . Then, the following are equivalent:
- (i)
is -biflat; 2. (ii)
There is a --bimodule homomorphism such that , where is the canonical embedding of into .
Proof.
(i)(ii) Since is -biflat, there exists a bounded linear map satisfying () and . We set to be the restriction of to . Then, for every and we have
[TABLE]
Next, for every and
[TABLE]
so . A Similar argument shows that .
(ii)(i) Let be as specified in the clause (ii). Suppose that is the restriction of into . Clearly is a bounded linear map and satisfies , by Lemma 2.2. For every and
[TABLE]
showing that . ∎
It is well-known that every biprojective Banach algebra is biflat. The next proposition gives a generalization of this fact.
Proposition 2.4**.**
Let be a Banach algebra and let . If is -biprojective, then is -biflat.
Proof.
If is a -biprojective Banach algebra, then there exists a bounded --bimodule homomorphism such that . For every and we have
[TABLE]
Since is a --bimodule homomorphism, satisfies by Lemma 2.2, so is -biflat. ∎
The relation between biflatness and -biflatness appears as follows.
Proposition 2.5**.**
Let be a Banach algebra and let . Then:
- (i)
If is biflat, then is -biflat. 2. (ii)
If is -biflat and has a bounded approximate identity, and if has a dense range, then is biflat.
Proof.
(i) There exists a bounded -bimodule homomorphism such that . Define . Then, for every and , we have
[TABLE]
and analogously, . It is clear that , and hence is -biflat.
(ii) By Proposition 3.3 below, is amenable and then by [3, Theorem 2.9.65], is biflat. ∎
Now, we give a -biflat Banach algebra which is not biflat.
Example 2.1**.**
It is known that is not amenable, and so is not amenable by [12, Corollary 2.3.11]. Therefore, is not biflat by [5]. We consider the homomorphism defined by , . An argument similar to [4, Example 2.7], shows that is -amenable. Then by Remark 3.4 below, is -biflat.
3. Relation to -amenability
Let be a Banach algebra, let , and let be a Banach -bimodule. A bounded linear map is a -derivation if
[TABLE]
and it is -inner derivation if there is an element such that for all . A Banach algebra is -amenable if for every Banach -bimodule , every -derivation is -inner [8, 9]. An element is said to be a -virtual diagonal for , if and for all . A bounded net is said to be a -approximate diagonal for if and for all [10]. In [4, Proposition 2.4] it is shown that if has a -virtual diagonal, then it has a -approximate diagonal. An easy verification shows that the converse is also true.
In the following two propositions, we establish a connection between -biflatness and existence of -virtual diagonals.
Proposition 3.1**.**
Let be a Banach algebra with a bounded approximate identity and . If is -biflat, then has a -virtual diagonal.
Proof.
Let be a bounded approximate identity for . Since is -biflat there exists a bounded linear map satisfying such that . We may suppose that converges in the weak* topology to an element of , say . Then for each and , we have
[TABLE]
and similarly . Thus . An application of Theorem 2.3 shows that
[TABLE]
hence is a -virtual diagonal for . ∎
Proposition 3.2**.**
Let be a Banach algebra and . If has a -virtual diagonal, then is -biflat.
Proof.
Let be a -virtual diagonal. Define by . Clearly is bounded, linear and --bimodule homomorphism. Also
[TABLE]
Thus is -biflat. ∎
Now, we describe the relation between -biflatness and -amenability.
Proposition 3.3**.**
Let be a -biflat Banach algebra with a bounded approximate identity. If has a dense range, then is amenable so is -amenable.
Proof.
Let be a -biflat Banach algebra. By Proposition 3.1 has a -virtual diagonal, equivalently, has a -approximate diagonal . We show that has an approximate diagonal, so it is amenable by [3, Theorem 2.9.65] and thus by [10, Corollay 2.2] it is -amenable, as required. Suppose that and . There exists such that for every , then . There is such that for all , . Hence for , we have , where is a bounded of . This shows that and similarly, . ∎
Remark 3.4**.**
There is a converse for Proposition 3.3. Indeed if is -amenable with a bounded approximate identity, then it has -virtual diagonal [4, Theorem 2.2], and whence by Proposition 3.2, is -biflat.
We conclude the current section with two examples of -biflat Banach algebras which are not -amenable. We refer the reader to [12, Definition 3.1.8 and Definition C.1.1] for the definitions of property () and approximation property for Banach spaces. We also write for the space of approximable operators on a Banach space .
Example 3.1**.**
let be a Banach space with property () such that does not have the bounded approximation property. Then is biflat while it is not amenable [12, Theorem 4.3.24]. Therefore, is -biflat for each . On the other hand, One may check that every -amenable Banach algebra for which has a dense range, is amenable. Hence, choosing a homomorphism with a dense range, it is readily seen that is not -amenable.
Example 3.2**.**
let be an infinite dimensional Hilbert space. It was shown in [12, Example 4.3.25] that is biflat but not amenable. Then, an argument similar to Example 3.1 shows that is -biflat which is not -amenable, whereas is a homomorphism in with a dense range.
4. The role of sequences
We start with the following which is similar to Lemma 2.2.
Lemma 4.1**.**
Let be a Banach algebra, and be Banach -bimodule and let . If is a bounded linear map satisfying and (), then is a --bimodule homomorphism.
For a Banach algebra , we consider the short exact sequence
[TABLE]
and its duals and , where is the natural injection. It is known that a Banach algebra with a bounded approximate identity is amenable if and only if (and so ) split [2].
Proposition 4.2**.**
Let be a Banach algebra with a bounded approximate identity and . Suppose that is -amenable. Then we have the following statements:
Regarding the sequence , there is a bounded linear map such that , and
[TABLE]
Regarding the sequence , there is a --bimodule homomorphism such that .
Proof.
Let be a -virtual diagonal for . We define the map via
[TABLE]
Then, for and
[TABLE]
so that . Next for and , we have
[TABLE]
whence . The equality is even easier.
Take , where is given by the clause . Then, it is immediate by Lemma 4.1. ∎
Finally, we generalize [12, Proposition 4.3.23] where the proof reads somehow the same lines.
Theorem 4.3**.**
Let be Banach algebra, let be a closed subalgebra of and for which with the followings properties:
- (i)
is -amenable; 2. (ii)
is a left ideal of ; 3. (iii)
has a bounded approximate identity which is also a bounded left approximate identity for .
Then is -biflat.
Proof.
Since is -amenable, Proposition 4.2(ii) yields the existence of a --bimodule homomorphism such that . Set , where is the canonical embedding. Let be a bounded approximate identity for which is a left bounded approximate identity for . For each , is a bounded net in , and so without loss of generality we may suppose that it is convergence. Therefore we obtain a bounded linear map , defined by . It is immediate that is a right --module homomorphism. To check that is a left --module homomorphism, we first notice that
[TABLE]
Let . By Cohen’s factorization theorem, for some and . Then, it follows from the above observation that
[TABLE]
Now, let . Again, by Cohen’s factorization theorem, there are such that . Then
[TABLE]
So is --bimodule homomorphism. It remains to prove that . For each we have
[TABLE]
So by Theorem 2.3, is -biflat. ∎
5. Application for Tensor products
Let be a Banach algebra, let and let . We say that is --biflat, If there exists a map , satisfying such that and .
Let and be Banach algebras and and . Let be a Banach -bimodule, and let be a Banach -bimodule. We regard as a Banach -bimodule with the actions
[TABLE]
for every and .
Using Ramsden’s notation [11], we construct and as follows.
Let be the space with the following module actions:
[TABLE]
for every , , , . Consider the map defined by . By [1, 42. Proposition 13], is an isometric isomorphism of Banach spaces. We claim that satisfies (1). Indeed
[TABLE]
and on the other hand
[TABLE]
Let be with the following module actions
[TABLE]
for all , , , . Consider the map defined by . A similar argument as we use for , shows that is an isometric isomorphism of Banach spaces satisfying (1)
Now, we are ready to prove the main goal of the current section.
Theorem 5.1**.**
Let be --biflat Banach algebra and let be --biflat Banach algebra and if both and are idempotents. Then is --biflat.
Proof.
There exists a bounded linear map such that , satisfying (1) and and also a bounded linear map such that , satisfying (1) and . Consider the composition
[TABLE]
All the maps satisfy the equation (1), and . For , we follow under the sequence of composition . We have
[TABLE]
So
[TABLE]
hence .
Note that the last equality comes from the following isomorphisms
[TABLE]
∎
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Varlag, Berlin Heidelberg New York, 1973.
- 2[2] P. C. Curtis, R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. , (2) 40 (1989), 89-104.
- 3[3] H. G. Dales, Banach Algebras And Automatic continuity, London Mathematical Society Monographs 24, Clarendon Press, Oxford, 2000.
- 4[4] Z. Ghorbani and M. L. Bami, φ 𝜑 \varphi -amenable and φ 𝜑 \varphi -biflat Banach algebras, Bull. Iranian Math. Soc. 39 (3) (2013), 507-515.
- 5[5] A. Y. Helemskii, The Homology of Banach and Topological Algebras, Dordrecht, Netherlands, Kluwer, 1989.
- 6[6] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).
- 7[7] B. E. Johnson, Approximate diagonals and Cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685-698.
- 8[8] M. Mirzavaziri and M. S. Moslehian, σ 𝜎 \sigma -derivation in Banach algebras, Bull. Iranian Math. Soc. 32 (1) (2006), 65-78.
