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Multinorms and Approximate Amenability of Weighted Group Algebras
Let G be a locally compact group, and take with . We prove that, for any left -multiinvariant functional on and for any weight function on , the approximate amenability of the Banach algebra implies the left -amenability of , but in general the opposite is not true. Our proof uses the notion of multinorms. We also investigate the approximate amenability of .
Dedicated with respect to my mother and my father
Ghahramani and Loy in  introduced the notion of approximately amenable Banach algebras and, among other interesting results, they proved that for a locally compact group , the group algebra is approximately amenable if and only if is amenable.
Let be a locally compact group. Recently Dales et al. in  defined another generalized notion of amenability, called left -amenability of , for any , such that . The aim of the present paper is to show that, for any left -multiinvariant function on and for any weight function on , the approximate amenability of the Banach algebra implies the left -amenability of , but in general the opposite is not true.
2. Notation and Preliminaries
In this section, we will recall various notations that we will use and give the definitions and some conventions.
2.1. Weighted Group Algebras and Approximate Amenability of Banach Algebras
Let denote a locally compact group with a fixed left Haar measurable and a weight function on ; that is, a Borel measurable function such that The weighted group algebra is the space of all measurable complex-valued functions on such that and is equipped with the convolution product of functions; that is, for and , and the norm Also, let be the space of all measurable complex-valued functions on , such that is essentially bounded, and for define The spaces and are in duality by and if , , then . Let be the Banach space of all complex-valued, regular Borel measures on such that Note that . If , , then is a subalgebra of .
Let be a Banach algebra and a Banach -bimodule. A bounded linear map is called a derivation if For every we define by It is easily seen that is a derivation. Derivation of this form is called inner derivation.
Let be a Banach algebra and a Banach -bimodule. Then , the dual space of , is a Banach -bimodule for operations given by is the dual module of ; and in particular is the dual module of .
A Banach algebra is called amenable if, for any -bimodule , any derivation is inner. This definition of amenability was introduced by Johnson in . A Banach algebra is called weakly amenable if any derivation is inner. Trivially, any amenable Banach algebra is weakly amenable.
Let be a locally compact group; a mean on is a positive functional such that A mean on is called left invariant if A locally compact group is called amenable if there exists a left invariant mean on . Note that is amenable if and only if is an amenable Banach algebra.
A derivation is called approximately inner if there exists net such that, for every , Recall from  that a Banach algebra is called approximately amenable if, for any -bimodule , any derivation is approximately inner.
2.2. Banach Spaces
For , we set . Let be a normed space. For each , we denote by the linear space direct product of copies of . The closed unit ball of is denoted by . We denote the dual of by ; the action of on an element is written as .
Following the notation of [4, 5] we define the weak -summing norm (for ) on by where . See also [6, 7]. Notice that, by the weak* density of in , the weak -summing norm on can also be computed as where .
2.3. Multinormed Spaces
The following definition is by Dales and Polyakov. For a full account of the theory of multinormed spaces, see .
Definition 1. Let be a normed space, and let be a sequence such that is a norm on for each , with on . Then the sequence is a multinorm if the following axioms hold (where in each case the axiom is required to hold for all and all ): (A1) for each permutation of ;(A2) ;(A3);(A4).
The normed space equipped with a multinorm is a multinormed space, denoted in full by . We say that such a multinorm is based on .
Definition 2. Let be a multinormed space. A subset is multibounded if The constant is the multibound of .
The following easy remark is from [4, Proposition ].
Remark 3. Let be a multinormed space. Then the absolutely convex hull of a multibounded set is multibounded, with the same multibound.
2.4. The -Multinorm
Following , we now introduce an important class of multinorms. Let be a normed space, and take with , . For each and each , we define It is clear that is a norm on . As proved in , in the case where , the sequence is a multinorm based on .
Definition 4. Let be a normed space, and take with . Then the multinorm described above is the -multinorm over .
A subset of is -multibounded if it is multibounded with respect to the -multinorm. The -multibound of such a set is denoted by .
Lemma 5. Let be a normed space, and take with , . Then, for each and , we have
Proof. This is proved in [4, Proposition 4.10]; it follows from the Principal of Local Reflexivity.
2.5. Left -Multiinvariant Means
Definition 6. Let be a locally compact group, and take with . A functional is left -multiinvariant if the set is multibounded in the -multinorm. The group is left -amenable if there exists a left -multiinvariant mean on .
The idea behind this definition is to attempt to measure the “left invariance” of a mean by measuring the growth of the sets as ranges through all the finite subsets of . See  for more details.
3. Multinorms and Approximate Amenability of Weighted Group Algebras
Ghahramani and Loy in  proved that for a locally compact group the Banach algebra is approximately amenable if and only if is amenable. In this section we prove that, if is any weight function on and is approximately amenable, then is left -amenable. Through an example we show is not valid in general.
Theorem 7. Let be a locally compact group, and let be a weight function on such that . If is approximately amenable, then is left -amenable.
Proof. By the following operations, is a Banach -bimodule; where , , and . Note that we have for , , and . For every and ,
Thus ; and by definition we have ; that is, the space is a submodule of . If we set , then is a -bimodule, with . By Hahn Banach Theorem there is such that . It is easy to prove that the mapping , defined by is a well-defined derivation. If the restrict ion to is denoted by , then from the approximate amenability of it follows that there is a net in such that for every we have For with and , . Hence for every , , we have Thus . Therefore for every and with we have Thus Since , it follows that , so there exists a subnet of such that . For every , if we set then . If for each the restriction of to is denoted by (26) we have norm-, for every . Since the space is a commutative -algebra with identity , there is a compact space such that . Since, for each and , we have , therefore and since , thus, for each , Let be a weak*-cluster point of . Clearly is positive; take , with . Recall that can be identified isometrically as a Banach lattice with for some measure space . Set . Since , it is clear that is a mean on . Since for every and every with [5, 2.6], we see that Now note that for every , and so is multibounded in the -multinorm. The result follows, and is left -amenable.
Lemma 8. Let be a commutative Banach algebra. If is approximately amenable, then is weakly amenable.
Proof. Suppose that is a derivation we show that is inner. Since is approximately amenable and commutative, so there exists a net in such that for each On the other hand since is commutative, every inner derivation on is zero, so is inner. Therefore is weakly amenable.
The Example 6.2 of  shows that an approximately amenable Banach algebra need not be weakly amenable.
The following example shows that the opposite of Theorem 7 is not true in general.
Example 9. If we define the weight function on by for every , then the Banach algebra is not approximately amenable.
Proof. Let Then is a commutative Banach algebra with respect to convolution multiplication, and the norm We show that is not approximately amenable. To see this note that So by Theorem 2.3 of , is not weakly amenable. On the other hand, since is a commutative Banach algebra and if it is approximately amenable, so by Lemma 8 it is weakly amenable, and this is a contradiction. So is not approximately amenable.
Let be a weight function on ; for , we define . In the following theorem we prove that if is bounded on then the opposite of Theorem 7 is true.
Theorem 10. Suppose that is a weight function on a locally compact group such that and is bounded. Then is left -amenable if and only if is approximately amenable.
Proof. is left -amenable, so suppose that is a left -invariant mean on ; that is, is -multibounded on . By Remark 3 or the Krein-Smulian theorem, the closed convex hull of is weakly compact. For each , consider that the map defined by We obtain a group of isometric affine maps. By Ryll-Nardzewski fixed point theorem, given in [9, 10], there exists which is a common fixed point for the set . Obviously, must be a left-invariant mean on . Hence the group is amenable and is bounded on ; by Theorem 0 of  is amenable, so is approximately amenable.
4. Approximate Amenability of a Locally Compact Group
It is standard that always has a bounded approximate identity which is a net consisting of continuous functions of compact support, and this net is clearly also a bounded approximate identity for .
Let be a weight function on with (); then, with the convolution product given by the Banach space defines a unital convolution Banach algebra for which is a closed ideal.
In the following lemma we show that if is approximately amenable then is left -amenable.
Theorem 11. Let be a locally compact group. If is approximately amenable then is left -amenable.
It is well known (c.f. ) that if is amenable and is bounded on , then there is a continuous positive character on such that In particular,
Theorem 12. If is approximately amenable and is bounded on , then is a discrete group.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This research was done while the author was at the Department of Pure Mathematics, Isfahan University of Technology, during the years 2012-2013. The author would like to express his thanks to Professor Rasoul Nasr-Isfahani and Professor Mehdi Nemati in the Banach algebras and Operator Theory for warm hospitality and great scientific atmosphere. Last but not least, He thanks his family for all their support.
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Copyright © 2014 Saman Ghaderkhani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.