Closed ideals with bounded Δ -weak approximate identities in some certain Banach algebras
aa r X i v : . [ m a t h . F A ] J a n CLOSED IDEALS WITH BOUNDED ∆ -WEAKAPPROXIMATE IDENTITIES IN SOME CERTAIN BANACHALGEBRAS J. LAALI AND M. FOZOUNI
Abstract.
It is shown that a locally compact group G is amenable ifand only if some certain closed ideals of the Fig`a-Talamanca-Herz algebra A p ( G ) admit bounded ∆-weak approximate identities. Also, similar resultsare obtained for the function algebras L A ( G ) and C w ( G ). Introduction and preliminaries
Let A be a Banach algebra, ∆( A ) be the character space of A , that is,the space of all non-zero homomorphisms from A into C and A ∗ be the dualspace of A consisting of all bounded linear functionals on A . Throughout thepaper, A is a commutative and semi-simple Banach algebra, hence ∆( A ) isnon-empty.Let { e α } be a net in Banach algebra A . The net { e α } is called,(1) an approximate identity if for each a ∈ A , k ae α − a k → weak approximate identity if for each a ∈ A , | f ( ae α ) − f ( a ) | → f ∈ A ∗ ,(3) a ∆ -weak approximate identity if for each φ ∈ ∆( A ), | φ ( e α ) − | → Definition 1.
Let A be a Banach algebra. A bounded ∆-weak approximateidentity for subspace E ⊆ A is a bounded net { e α } in E such that for each a ∈ E , lim α | φ ( ae α ) − φ ( a ) | = 0 ( φ ∈ ∆( A )) . For simplicity of notation, let b.∆-w.a.i stand for a bounded ∆-weak approx-imate identity.It was proved that every Banach algebra A with a bounded ∆-weak approx-imate identity has a bounded approximate identity (b.a.i) and conversely [4,Proposition 33.2]. Mathematics Subject Classification.
Key words and phrases.
Character space, amenable group, Fig`a-Talamanca-Herz algebra,Lebesgue-Fourier algebra.
The notion of a ∆-weak approximate identity introduced and studied in[15] where an example of a Banach algebra which has a ∆-weak approximateidentity but does not have an approximate identity, was given. Indeed, if S = Q + is the semigroup of positive rationales under addition, it was shownthat the semigroup algebra l ( S ) has a b.∆-w.a.i, but it does not have anybounded or unbounded approximate identity.As the second example to see the difference between bounded approximateand bounded ∆-weak approximate identities, let R be the additive real linegroup and 1 < p ≤ ∞ . Put S p ( R ) = L ( R ) ∩ L p ( R ) and define the followingnorm; k f k S p = max {k f k , k f k p } ( f ∈ S p ( R )) . Using [14, Theorem 2.1], we can see that S p ( R ) has a b.∆-w.a.i, but it hasno b.a.i. Because we know that S p ( R ) is a Segal algebra and it is well-knownthat a Segal algebra S in L ( R ) has a b.a.i if and only if S = L ( R ). But it isclear that S p ( R ) = L ( R ).These type of approximate identities have some interesting applications, forexample; see [9,16,22]. In the past decades, B. E. Forrest studied the relationsbetween the amenability of a group G and closed ideals of A ( G ) and A p ( G )with a b.a.i; see [5–8], and the relations between some properties of G andclosed ideals of A ( G ) with a b.∆-w.a.i; see [9].In this paper, we try to improve some of the theorems in [5, 6, 8, 10, 16]with changing b.a.i by b.∆-w.a.i. As an application, we give the converseof [8, Corollary 4.2] due to B. Forrest, E. Kaniuth, A. T. Lau and N. Spronk.2. Main results
Let G be a locally compact group. For 1 < p < ∞ , let A p ( G ) denote thesubspace of C ( G ) consisting of all functions of the form u = P ∞ i =1 f i ∗ e g i where f i ∈ L p ( G ), g i ∈ L q ( G ), 1 /p + 1 /q = 1, P ∞ i =1 k f i k p k g i k q < ∞ and e g ( x ) = g ( x − ) for all x ∈ G . With the pointwise operation and the followingnorm, k u k A p ( G ) = inf { ∞ X i =1 k f i k p k g i k q : u = ∞ X i =1 f i ∗ e g i } ,A p ( G ) is a Banach algebra called the Fig`a-Talamanca-Herz algebra. It isclear that k u k ≤ k u k A p ( G ) where k u k is the uniform norm of u ∈ C ( G ).By [12, Theorem 3], we know that∆( A p ( G )) = { φ x : x ∈ G } = G, where φ x is defined by φ x ( f ) = f ( x ) for each f ∈ A p ( G ).The dual of the Banach algebra A p ( G ) is the Banach space P M p ( G ) con-sisting of all limits of convolution operators associated to bounded measures. LOSED IDEALS AND BOUNDED ∆-WEAK APPROXIMATE IDENTITIES 3
Indeed,
P M p ( G ) is the w ∗ -closure of λ p ( L ( G )) in B ( L p ( G )) where λ p is theleft regular representation of G on L p ( G ); see [3] for more details.The group G is said to be amenable if, there exists an m ∈ L ∞ ( G ) ∗ suchthat m ≥ m (1) = 1 and m ( L x f ) = m ( f ) for each x ∈ G and f ∈ L ∞ ( G )where L x f ( y ) = f ( x − y ). Theorem 1. (Leptin-Herz) Let G be a locally compact group and < p < ∞ .Then A p ( G ) has a b.a.i if and only if G is amenable The proof of the above theorem in the case p = 2 is due to Leptin [19] andin general is due to Herz [12].Forrest and Skantharajah in [9] showed that if G is a discrete group, then A ( G ) = A ( G ) has a b.∆-w.a.i if and only if G is amenable. Kaniuth and¨Ulger in [18, Theorem 5.1], for the first time in our knowledge, announcedthat A ( G ) has a b.∆-w.a.i if and only if G is an amenable group, but thesame result holds for the Fig`a-Talamanca-Herz algebras as follows. The proofis similar to the Fourier algebra case, but we give it for the convenience ofreader. Theorem 2.
Let G be a locally compact group and < p < ∞ . Then A p ( G ) has a b. ∆ -w.a.i if and only if G is amenable.Proof. Let { e α } be a b.∆-w.a.i for A p ( G ) and e ∈ A p ( G ) ∗∗ be a w ∗ -clusterpoint of { e α } . So, for each φ ∈ ∆( A p ( G )) = G , we have e ( φ ) = lim α φ ( e α ) = 1 , because { e α } is a b.∆-w.a.i for A p ( G ). Therefore, by [23, Proposition 2.8] G is weakly closed in P M p ( G ) = A p ( G ) ∗ . Now, by [2, Corollary 2.8] we concludethat G is an amenable group. (cid:3) The following corollary immediately follows from the Leptin-Herz Theoremand Theorem 2.
Corollary 1.
Let G be a locally compact group and < p < ∞ . Then A p ( G ) has a b. ∆ -w.a.i. if and only if it has a b.a.i. The following theorem is a key tool in the sequel.
Theorem 3.
Let A be a Banach algebra, I be a closed two-sided ideal of A which has a b. ∆ -w.a.i and the quotient Banach algebra A/I has a b.l.a.i. Then A has a b. ∆ -w.a.i.Proof. Let { e α } be a b.∆-w.a.i for I and { f δ + I } be a b.l.a.i for A/I . We canassume that { f δ } is bounded. Indeed, since { f δ + I } is bounded, there existsa positive integer K with k f δ + I k < K for each δ . So, there exists y δ ∈ I such J. LAALI AND M. FOZOUNI that k f δ + I k < k f δ + y δ k < K . Put f ′ δ = f δ + y δ . Clearly, { f ′ δ + I } is a b.l.a.ifor A/I which { f ′ δ } is bounded.Now, consider the bounded net { e α + f δ − e α f δ } ( α,δ ) . For each φ ∈ ∆( A )we have φ ( e α + f δ − e α f δ ) = φ ( e α ) + φ ( f δ )(1 − φ ( e α )) ( α,δ ) −−−→ . Therefore, A has a b.∆-w.a.i. (cid:3) Let G be a locally compact group, E be a closed non-empty subset of G and 1 < p < ∞ . Define I p ( E ) = { u ∈ A p ( G ) : u ( x ) = 0 for all x ∈ E }· The following result improves [5, Theorem 3.9].
Theorem 4.
Let G be a locally compact group. Then the following assertionsare equivalent. (1) G is an amenable group. (2) ker( φ ) has a b. ∆ -w.a.i for each φ ∈ ∆( A p ( G )) . (3) I p ( H ) has a b. ∆ -w.a.i for some closed amenable subgroup H of G .Proof. (1) ⇒ (2): Let G be an amenable group. Then A p ( G ) has a b.a.i byLeptin-Herz’s Theorem. Now, the result follows from [17, Corollary 2.3].(2) ⇒ (3): Just take H = { e } , because we know that I p ( { e } ) = ker( φ e ).(3) ⇒ (1): Suppose that I p ( H ) for a closed amenable subgroup H of G has a b.∆-w.a.i. By [21, Lemma 3.19] we know that A p ( H ) is isometricallyisomorphic to A p ( G ) /I p ( H ). But A p ( H ) has a b.a.i, since H is an amenablegroup. Therefore, A p ( G ) /I p ( H ) also has a b.a.i. Now, the result follows fromTheorems 3 and 2. (cid:3) Forrest in [6, Lemma 3.14] improved [5, Theorem 3.9]. Indeed he showedthat G is an amenable group if for some closed proper subgroup H of G , I ( H )has a b.a.i. Also, using the operator space structure of A ( G ), it was shownin [8, Theorem 1.5] that G is an amenable group only if I ( H ) for some closedsubgroup H of G has a b.a.i.Now, we give the following result which improves [8, Corollary 1.6] andTheorem 4. Our proof is a mimic of [6, Lemma 3.14]. Theorem 5.
Let G be a locally compact group and < p < ∞ . Then thefollowing are equivalent. (1) G is an amenable group. (2) I p ( H ) has a b. ∆ -w.a.i for some proper closed subgroup H of G .Proof. In view of [8, Corollary 4.2], only (2) ⇒ (1) needs proof. LOSED IDEALS AND BOUNDED ∆-WEAK APPROXIMATE IDENTITIES 5
Let H be a proper closed subgroup of G such that I p ( H ) has a b.∆-w.a.i.We will show that H is an amenable group and this completes the proof byTheorem 4.Since H is a proper subgroup, there exists x ∈ G \ H . On the other hand, themapping I p ( H ) → I p ( xH ) defined by u → L x u is an isometric isomorphism,because for each t ∈ G and f ∈ A p ( G ), we have, L t f ∈ A p ( G ) , k L t f k A p ( G ) = k f k A p ( G ) . Therefore, I p ( xH ) has a b.∆-w.a.i which we denote it by ( u α ). For each α ,let v α be the restriction of u α to H . Using [12, Theorem 1a], we conclude that( v α ) is a bounded net in A p ( H ).Let ν ∈ A p ( H ) ∩ C c ( H ) and K = supp ν ⊆ H . Then there exists aneighborhood V of K in G such that V ∩ xH = ∅ , because K ∩ xH = ∅ (otherwise we conclude that x is in H ) and G is completely regular by [13,Theorem 8.4] and hence it is a regular topological space. Indeed, for each y ∈ xH , let V y be a neighborhood of K such that y / ∈ V y . So, V = ∩ y ∈ xH V y satisfies V ∩ xH = ∅ .By [3, Proposition 1, pp.34] there is u ∈ A p ( G ) such that u ( x ) = 1 for each x ∈ K and supp u ⊆ V , and by [12, Theorem 1b], there exists a v ∈ A p ( G )such that v | H = ν . Now, put w = vu . Since V ∩ xH = ∅ and supp u ⊆ V ,we have w ∈ I p ( xH ), and since u ( x ) = 1 for each x ∈ K and K = supp ν , wehave w | H = ν .Now, for each x ∈ H , we havelim α | φ x ( v α ν ) − φ x ( ν ) | = lim α | v α ( x ) ν ( x ) − ν ( x ) | = lim α | u α ( x ) w ( x ) − w ( x ) | = 0 . Therefore, ( v α ) is a b.∆-w.a.i for A p ( H ), since by [3, Corollary 7, pp. 38], A p ( H ) ∩ C c ( H ) is dense in A p ( H ). Hence, by Theorem 2, H is amenable. (cid:3) As an application of the above theorem, we give the following corollarywhich is the converse of [8, Corollary 4.2].
Corollary 2.
Let G be a locally compact group, < p < ∞ and H be a properclosed subgroup of G . If I p ( H ) has a b.a.i, then G is amenable. Ghahramani and Lau in [10] introduced and studied a new closed ideal of A ( G ). Indeed, let G be a locally compact group and put L A ( G ) = L ( G ) ∩ A ( G )with the norm |k f k| = k f k + k f k A ( G ) . J. LAALI AND M. FOZOUNI
Clearly L A ( G ) with pointwise multiplication is a commutative Banach al-gebra with ∆( L A ( G )) = G and it is called the Lebesgue-Fourier algebra of G .It was shown that L A ( G ) has a b.a.i if and only if G is a compact group [10,Proposition 2.6]. Now, we give the following result concerning the b.∆-w.a.identities of this Banach algebra. Theorem 6.
Let G be a locally compact group. (1) for each x ∈ G , ker( φ x ) ⊆ L A ( G ) has a b. ∆ -w.a.i. (2) L A ( G ) has a b. ∆ -w.a.i. (3) G is amenable. (4) G is compact.Then (1) ⇒ (2) ⇒ (3) and (4) ⇒ (1) hold.Proof. (1) ⇒ (2) : Follows from Theorem 3.(2) ⇒ (3) : Let ( u α ) be a b.∆-w.a.i for L A ( G ), Then ( u α ) is a boundednet in A ( G ). Now, the result follows from Theorem 2 and this fact that∆( L A ( G )) = G = ∆( A ( G )).(4) ⇒ (1) : Let G be a compact group. Then by [10, Proposition 2.6],we know that L A ( G ) = A ( G ). On the other hand, by [17, Example 2.6] foreach x ∈ G , A ( G ) is φ x -amenable. Therefore, the result follows from [17,Proposition 2.2]. (cid:3) We do not know whether the implication (3) ⇒ (2) in Theorem 6 remainstrue? Remark . In [11], Granirer gave a ( p, r )-version of the Lebesgue-Fourier al-gebra. Indeed, let 1 < p < ∞ , ≤ r ≤ ∞ and put A rp ( G ) = A p ( G ) ∩ L r ( G )with the norm, k u k A rp ( G ) = k u k r + k u k A p ( G ) . It was shown that A rp ( G ) with pointwise multiplication is a commutativesemi-simple Banach algebra such that ∆( A rp ( G )) = G and for all 1 ≤ r ≤ ∞ , A rp ( G ) = A p ( G ) if G is a compact group; see [11, Theorem 1, Theorem 2].Therefore, in view of Theorems 2 and 3, Theorem 6 remains true if wereplace L A ( G ) with A rp ( G ). Remark . Runde in [20], by using the canonical operator space structureof L p ( G ), introduced and studied the algebra O A p ( G ) for 1 < p < ∞ , the operator Fig`a-Talamanca-Herz algebra . It was shown that A p ( G ) ⊆ O A p ( G )[20, Remark 4. pp. 159] and O A p ( G ) has a b.a.i if and only if G is anamenable group [20, Theorem 4.10]. That would be an interesting question:Are the preceding results remain true if we replace A p ( G ) by O A p ( G )? LOSED IDEALS AND BOUNDED ∆-WEAK APPROXIMATE IDENTITIES 7
Now, let G be a locally compact group and w : G → R be an upper semi-continuous function such that w ( x ) ≥ x ∈ G . Put C w ( G ) = { f ∈ C ( G ) : f w ∈ C ( G ) } . It is clear that C w ( G ) with pointwise operation and weighted supremum normdefined by k f k w = sup x ∈ G | f ( x ) | w ( x ) ( f ∈ C w ( G )) , is a commutative Banach algebra such that ∆( C w ( G )) = ∆( C ( G )) = G ;see [16, Section 4.3]. Theorem 7.
Let G be a locally compact group and t ∈ G . Then the followingare equivalent. (1) ker( φ t ) has a b.a.i. (2) ker( φ t ) has a b. ∆ -w.a.i. (3) C w ( G ) has a b. ∆ -w.a.i. (4) w is bounded.Proof. (1) ⇒ (2) : This part is clear. Applying Theorem 3, we conclude(2) ⇒ (3). In view of [16, Corollary 4.7], we have (3) ⇒ (4). Therefore, weonly show (4) ⇒ (1).Suppose that w is bounded by M >
0. Let V = { V α } α ∈ Γ be a neighborhoodbase at t directed by the reverse inclusion. For each α ∈ Γ, by the UrysohnLemma, there exists a continuous function f α : X → [0 ,
1] such that f α ( t ) = 1and supp f α ⊆ V α .Let ǫ > g ∈ C w ( G ). For ǫ ′ = ǫ/ M , there exists a neighborhood V of t such that, | g ( y ) − g ( t ) | < ǫ ′ ( y ∈ V ) . On the other hand, let α ∈ Γ be such that V α ⊆ V . Therefore, there exists x ∈ G such that k gf α − φ t ( g ) f α k w = sup x ∈ G | g ( x ) f α ( x ) − g ( t ) f α ( x ) | w ( x ) < | g ( x ) f α ( x ) − g ( t ) f α ( x ) | w ( x ) + ǫ/ < M ǫ ′ + ǫ/ ǫ. Therefore, for each g ∈ C w ( G ), k gf α − g ( t ) f α k w −→
0. Hence, by [17, Theorem1.4, Proposition 2.2] and [1, Corollary 3.6] we conclude that ker( φ t ) has ab.a.i. (cid:3) Acknowledgement
The authors are grateful to the referee of the paper for his (her) invaluablecomments which improved the presentation of the paper, especially for givinga shorter proof for Theorem 3.
J. LAALI AND M. FOZOUNI
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Javad Laali , Kharazmi University, Faculty of Mathematical and ComputerScience, Department of Mathematics, 50 Taleghani Avenue, 15618, Tehran, Iran
E-mail address : [email protected] Mohammad Fozouni , Gonbad Kavous University, Faculty of Sciences and En-gineering, Department of Mathematics, Gonbad Kavous, Iran
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