ϕ APPROXIMATE BIFLAT AND ϕ AMENABLE BANACH ALGEBRAS

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMNIN CDEMY, Series, OF THE ROMNIN CDEMY Volume 13, Number 1/2012, pp 3 10 PPROXIMTE BIFLT ND MENBLE BNCH LGEBRS Zahra GHORBNI, Mahmood Lashkarizadeh BMI Department of Mathematics, Faculty of Science, University of Isfahan, Isfahan, Iran horbani@sciuiacir Department of Mathematics, Faculty of Science, University of Isfahan, Isfahan, Iran lashkari@sciuiacir In this paper we introduce and study the concept of a -approximate biflat and -pseudo contractible Banach alebra, where is a continuous homomorphism on We show that is - pseudo contractible if and only if is - approximate biflat and has a central approximate identity We also introduce the notion of -amenability of a locally compact roup G, where is a continuous homomorphism on G We prove that if the roup alebra L 1 ( G ) is -amenable then G is -amenable, where is the extension of to M ( G ) In the case where is an isomorphism on G it is shown that the converse is also valid Indeed, we have eneralized a well-known result due to B E Johnson Key words: Banach alebra, biprojective, amenable 1 INTRODUCTION ND PRELIMINRIES Banach alebra is called amenable if for each Banach -module X, every bounded derivation from into the dual -module X is an inner derivation The Banach alebra is called biprojective if there exists a bounded -bimodule map θ: ˆ such that π θ = id, where π denots the product morphism from ˆ into iven by π( a b) = ab for all ab, The notion of a biprojective Banach alebra was introduced by Helemskii [9] Recently, some authors have added a kind of twist to the amenability definition Given a continuous homomorphism from into, they defined and studied -derivations and -amenability (see [5],[15] and [17]) Suppose that is a Banach alebra, let Hom( ) denote the set of all continuous homomorphisms from into itself lso let X be a Banach -bimodule linear operator D: X is called a -derivation if D( ab) = D( a) ( b) + ( a) D( b) for all ab, -derivation D is called -inner derivation if there is x X such that Da ( ) = ( ax ) x ( a) for all a Let Ζ 1 ( X, ) denote the set of all continuous -derivations and Ν 1 ( X, ) be the set of all -inner derivations from into X The first cohomoloy roup Η ( X, ) is defined to be the quotient space Ζ (, X)/ Ν (, X) Banach alebra is called 1 -amenable if Η ( X, )={0} for all -bimodules X Note that every derivation of a Banach alebra into an -bimodule X is an id-derivation, where id is the identity operator on Let G be a locally compact roup and M ( G ) be the Banach space of complex-valued, reular Borel measure on G Recall that the space M ( G ) is a unital Banach alebra with the convolution multiplication and L 1 ( G ), the roup alebra on G, is a closed ideal in M ( G ) We write for the point mass at G, the element e is the identity of M ( G )

2 4 Zahra Ghorbani, Mahmood Lashkarizadeh Bami 2 The aim of the present paper is to introduce and investiate approximate biflat Banach alebras with Hom( ) In particular, we prove that if the roup alebra L 1 ( G ) is -amenable then G is -amenable In the case where is an isomorphism on G the converse is also valid 2 THE RESULTS We start this section by introducin the followin: Definition 21 Let be a Banach alebra and Hom( ) We say that is pseudo amenable if admit a approximate virtual diaonal, ie, there is a net ( m ) ˆ (not necessary bounded), such that m ( a) ( a) m 0 and π( m ) ( a) ( a) where π denots the product homomorphism from ˆ into iven by π( a b) = ab for all ab, ( a ) Definition 22 Let be a Banach alebra and Hom( ) We say that is approximate biflat if there is a net θ : ( ˆ ) ( I) of bounded -bimodule morphisms such that π θ ( a) ( a) THEOREM 23 Let be a Banach alebra with an approximate identity Then is - pseudo amenable ( Hom( ) ) if and only if is - approximate biflat Proof Let ( e ) β β I be an approximate identity for and suppose that θ : ( ˆ ) ( ) satisfies π θ () a ()( a a ) Then for every a and f ( ˆ ) limlim f, θ ( ( e )) ( a) ( a) θ ( ( e )) = limlim f, θ ( e ) ( a) ( a) ( e ) = 0 ( ) β β β β β β lso, for a and ψ, limlim ψ, ( a) π θ ( ( e ) = lim ψ, ( a) e β β β β = ( a) I Let E = I be directed by the product orderin and for each =( β, ) E, define m = θ( ( eβ)) Usin the iterated limit theorem [6, Theorem 24], the above calculation ives w lim ( m ( a) ( a) m) =0 ( a ), and w lim( a) π ( m )= ( a) ( a ) By Goldestine's theorem we can assume that ( m ) ˆ and we can replace weak converence in equations by weak converence pplyin Mazur's theorem, we then obtain a net ( m ) ˆ of convex combinations of ( m ) such that m ( a) ( a) m 0, and ( a) π ( m ) ( a) ( a ) That is is pseudo amenable Conversely, let ( m β ) be a approximate virtual diaonal for and define θ : ( ˆ β ) by a a m β Then for every a we have π θ ( a)= π ( ( a) m )= ( a) π ( m ) ( a) β β β

3 3 pproximate biflat 5 PROPOSITION 24 Let be a amenable Banach alebra and be an idempotent homomorphism on Then is -approximate biflat Proof By [8, Proposition 41] has a bounded approximate identity ( e ) I Let E be a w cluster point of ( ( e ) ( e )) in ( ˆ ) We define a derivation D: ( ˆ ) by Da ( ) = ( a) E E ( a) ( a ) Then for every a π ( Da ( ))=0 Therefore D ( ) ker( π ) = =( kerπ ) Thus there exists N ( kerπ ) such that D= ad ; N Put M = E N Then for every a π ( M ) ( a)= ( a) Let ( m ) be a net in ˆ such that M = w lim m w lim m ( a) ( a) m =0 and w lim ( π( m ) ( a) ( a))=0 ( a ) Followin the arument iven in the proof of [2, Lemma 2964] we can show that there exists a net ( m β ) β in ˆ such that each m β is a convex combination of m 's with Then ( ) mβ ( a) ( a) mβ 0 and π( mβ ) ( a) ( a) ( a ) Thus is pseudo amenable and so by Theorem 23 is -approximate biflat From the proof of the above proposition we obtain the followin corollaries COROLLRY 25 Let be a amenable Banach alebra ( Hom( )) with a bounded approximate identity Then is -approximate biflat COROLLRY 26 Let 1 1 L ( G ) be a amenable Banach alebra ( Hom( L )) Then L 1 ( G ) is -approximate biflat Definition 27 Let be a Banach alebra with the norm Then a Banach alebra B with the norm B is said to be an abstract Seal alebra with respect to if: (i) B is a dense left ideal in ; (ii) there exists M >0 such that b M b for all b B ; (iii) there exists C >0 such that ab C a b for all a, b B B B THEOREM 28 Let be a Banach alebra and B be an abstract Seal alebra with respect to Suppose that Hom( ) and B contains a net ( e ) such that ( e ) 2 is an approximate identity for B and a( e )= ( e ) a for all a If ( B) B and is -approximate biflat then B is -approximate biflat Proof Let T : ˆ B ˆ B be defined by a b a( e ) b( e ) Since is -approximate biflat there is a net ( θ β) βwith : ( ˆ θβ ) such that π θβ ( a) ( a) ( a ) For =( γ, ) define : ( ˆ θ B B B) by θ := T θ j where j : B is the inclusion map Then θ is a bounded β B bimodul map Note that because ( e ) lies in the center of, πb T = R π, where R : B 2 is defined by a a( e ) ( a ) Let b B, f B By the iterated limit theorem we have lim f, π θ ( b) = limlim f, π T θ j ( b) B B β β limlim f R β β = lim f, R ( ( b)) 2 =, π θ ( ( b)) = lim f, ( b) ( e ) = ( b)

4 6 Zahra Ghorbani, Mahmood Lashkarizadeh Bami 4 Hence πb θ () b ()( b b B) Definition 29 Let be a Banach alebra and Hom( ) We say that is pseudo contractible if it has a central -approximate diaonal, ie, a -approximate diaonal ( m ) satisfyin ( am ) = m ( a) for all a and all PROPOSITION 210 For a Banach alebra the followin two statements are equivalent i) is pseudo contractible ii) is approximate biflat and has a central approximate identity Proof (i) (ii) Suppose that ( m ) ˆ is a central -approximate diaonal for Define ˆ θ : ( ) by θ( a):= a m Then for every a we have limπ θ ( a)= limπ ( ( a) m ) = ( a) So π ( m ) is a central approximate identity for (ii) (i) Since is approximate biflat, there is a net θ ˆ : ( ) ( ) such that limπ θ ( a)= ( a) ( a ) Let ( e ) I β β I be a central approximate identity for Let E = I be directed by the product orderin and for each =( β, ) E define m = θ ( e ) Then ( m ) is a central -approximate diaonal for β Definition 211 Let be a Banach alebra and Hom( ) We say that is biprojective if there exists a bounded -bimodule map θ: ˆ such that π θ = id where id is the identity operator on Remark 212 (i) Let be a biprojective Banach alebra Then is id biprojective # # # # # (ii) Let be a biprojective Banach alebra Then is ( : ) biprojective Banach alebra with # = PROPOSITION 212 Let be a biprojective Banach alebra, and let B be a ψ biprojective Banach alebra with Hom( ) and ψ Hom( B) Then ˆ B is ψ biprojective Proof There exist an -bimodule map θ ˆ 1 : with π θ1 = id and B -bimodule map θ ˆ 2 : B ( B B) with π θ2 = id B Let θ ˆ ˆ ˆ ˆ ˆ ˆ 0 :( ) ( B B) ( B) ( B) be the isometric isomorphism iven by ( a1 a2) ( b1 b2) ( a1 b1) ( a2 b2) ( a1, a2, b1, b2 B) We let θ= θ ˆ ˆ ˆ ˆ 0 ( θ1 θ2): ( B) ( B) Then for a b B we have π θ ( ψ)( a b)= π θ0 ( θ1 θ2) ( ψ)( a b)= =π θ0 ( θ1 θ2)( ( a) ψ ( b)) = =π θ0( θ1( ( a)) θ2( ψ ( b))) = = π θ ( a) π θ ( ψ ( b)) = a b Therefore ˆ B is ψ biprojective 1 2 The proof of the followin result is similar to that of Proposition 213 PROPOSITION 213 Let be a -biprojective Banach alebra, and let B be a ψ -biprojective Banach alebra with Hom( ) and ψ Hom( B) Then B is ψ biprojective

5 5 pproximate biflat 7 3 MENBLE BNCH LGEBRS PROPOSITION 31 Let be a Banach alebra with a bounded approximate identity which is a closed ideal of a Banach alebra B Let E be a pseudo-unital Banach -bimodule, and : B B be a continuous homomorphism such that := and be a continuous epimorphism on Let 1 D Ζ (, E ), then E is a Banach B -bimodule in a canonical fashion, and there is a unique 1 D Ζ ( BE, ) for which the followin are valid (i) D = D ; (ii) D is continuous with respect to the strict topoloy on B and the w -topoloy on E Proof For x E, let ( a) and y E be such that x = ( a) y For b B, define b x:= ( ba) y We claim that b x is well defined, ie independent of the choices a and y Let ( a0 ) and y0 E be such that x = ( a0) y0, and let ( e ) be a bounded approximate identity for Then ( ba) y= ( b) ( a) y= lim ( b) e ( a) y= lim ( b) e x= ( ba ) y 0 0 It is obvious that this operation of B on E makes E into a left Banach B -module Similarly, one defines a riht Banach B -module structure on E, so that E becomes a Banach B -bimodule Now we define D : B E by b w lim D ( be ) We claim that D is well-defined, ie, the w lim D( be ) does exist Let x E, and let ( a) and y E be such that x = y ( a) Then Dbe ( ), x = Dbe ( ), y ( a) = Dbe ( ) ( a), y = Dbea ( ), y Da ( ), y ( be) Dba ( ), y Da ( ), y ( b) ( b B) So the w limd( be ) exists Moreover, for every a, D ( a)= w limd( ae )= D( a) That is D extends D For every b B and a we have D( ba)= ( b) Da + D ( b) ( a) It is clear that for every b B, ( be ) ( b) strictly Let bc, B, Then D ( bc)= ( b) D ( c) + D ( b) ( c) So D is a derivation Finally, for every b B and a lim lim ( Db ) ( a)= w ( D( be ) ( a)= w ( D( be a) ( be ) D( a)= D( ba) ( b) Da It follows that D is continuous with respect to the strict topoloy on B and the w -topoloy on E To see this, let ( b ) B such that lim = (strict-limit) Then for every a, ε > 0 there exists β such that for every β, ab ( b) + ( b ba ) < ε So ba ba ( a ) For x E, let ( a) and y E be such that x = ( a) y Then ( Db )( x)=( Db )( ( a) y)= D( ba)( y) ( b) Da( y) D( ba)( y) ( b) Da( y) = D ( b)( ( a) y) = D ( b)( x) Before turnin our next result, we note that if G is a locally compact roup, and is a continuous homomorphism on G and : M M is defined by ( µ ), f = f dµ ( f C 0 ) then 1 1 = and ( L ) L G

6 8 Zahra Ghorbani, Mahmood Lashkarizadeh Bami 6 Definition 32 Let G be a locally compact roup, and be a continuous homomorphism on G, and let E be a subspace of L containin the constant functions -mean on E is a functional m E such that ( m)(1) = 1 Definition 33 locally compact roup G is called -amenable if there is a -mean on L that is left invariant, ie, for all G and σ L we have m( σ)= m( σ ) (Note that the latter equation makes sense since σ L ) THEOREM 34 Let G be a locally compact roup and be a homomorphism on G, if L 1 ( G ) is -amenable then G is -amenable Proof Define 1 L -bimodule actions on L by G G G 1 ( G ) ( ) f ψ:= f( ( )) ψ( )d m ( ) and ψ f := f( ( ))d m ( ) ψ f L, ψ L 1 Choose n L with n (1) = 1, and define D : L L by f ( f) n n ( f) Then D( f )(1) = 0 Let E:= L /C1, then D( L 1 ) E Since 1 L ( G ) is -amenable, there is n E such that D= adn Let m:= n n 1, and p := { f L, f 0, f d mg = 1} For every ψ L and G f p, we have m( f ψ)== m( ψ ) It is clear that if f p then f p We conclude that m( σ)= m( f σ)= m( σ ) for f p, σ L Thus G is -amenable THEOREM 35 Let G be a locally compact roup and be an isomorphism on G If G is -amenable then, L 1 ( G ) is -amenable Proof Let X be a Banach L 1 -bimodule By [8, Proposition 45] there is no loss of enerality if we suppose that X is pseudo-unital Let D Ζ ( L, X ), and by Proposition 31, let D Ζ ( M, X ) be the extension of D For every x X we define ψ x : G C by ( ( ) D 1)( x) Let m be a - mean on L and let the functional F be defined on X by x ( m)( ψx ) It is obvious the F is bounded We prove that D = adf To see this we first prove that D = F F ( G) () Let x X, put z= x ( ) ( ) x For h G we have ψz( h )=( ( h) D 1 )( z )= ( h) D 1 ( x ( ) ( ) x ) h h Since D ( 1 )= ( 1 ) ( ) ( 1 ( ), h D D h + h ) it follows that ( D )( )= D D ( ) = ( h) h 1 ( ) h h 1 ( h) h 1 = D ( 1 ) ( ) ( ( 1 ) ( h) D( 1 h ) ) Takin y= x ( ), we obtain ψ + = ( ) = ( 1 ( ) 1 )( ) ( ) ( ( 1 ( ) 1 ) Z h h D h )( ) h y D x D y h 1 = ψy( h) ψ y( h) + ( D ) x So we have ψ = ψ ψ + D ( x) Thus Z y y

7 7 pproximate biflat 9 m( ψz)= m( ψy) m( ψ y) + m( D )( x)= m( ψy) m( ψ y) + m( D )( x) = = ( D )( x) m(1), and ( m)( ψz) = D ( x)( m)(1) = D ( x) Thus D ( x)= F( z) So ( D )( x)==( F F )( x) We now prove that is an epimorphism on L 1 ( G ) Let 1 1 µ L, define µ ( B)= µ ( ( B)) ( B is borel set on G ) It is clear that µ ( )= µ Since every measure µ in M ( G ) is the s- lim (strict-lim ) of a net ( µ i ) such that each µ i a linear combination of point masses [9, Exercise 24 ], then by () we have D ( µ ) = µ F µ F ( µ M), as required PROPOSITION 36 Let be a Banach alebra and B an abstract Seal alebra with respect to, and suppose that : B B is an idempotent homomorphism such that ( B)= B If B is -amenable, then so is Proof By [8, Proposition 41] B has a bounded approximate identity ( e ) I, let M =sup{ e : I} Since B is an abstract Seal alebra, there exists C >0 such that ab C a b B B for all ab, B So, for each b B, b = lim be C b M B Thus the norms and B are equivalent on B Since B is dense in, it follows that = is -amenable B B Hence COROLLRY 37 Let S ( G ) be a Seal alebra on G and : S S be an idempotent 1 1 homomorphism such that ( S )= S If S 1 ( G ) is -amenable, then so is L 1 ( G ) CKNOWLEDGEMENTS The authors would like to express their deep ratitude to the referees for their careful readin of the earlier version of the manuscript and several insihtful comments We are rateful to the office of Graduate Studies of the University of Isfahan for their support REFERENCES 1 CURTIS, P C, LOY, R J, The structure of amenable Banach alebras, J London Math Soc, 2, pp , DLES, H G, Banach alebras and automatic continuity, London Mathematical Society Monoraphs, 24, Clarendon Press, Oxford, HELEMSKII, Ya, The Homoloy of Banach and Topoloical lebras, 41, Mathematics and its pplications Soviet Series, Kluwer cademic Publishers Group, Dordrecht, JOHNSON, B E, pproximate diaonals and cohomoloy of certain annihilator Banach alebras, mer J Math, 94, pp , 1972

8 10 Zahra Ghorbani, Mahmood Lashkarizadeh Bami 8 5 KNIUTH, E, LU,, PYM, J, On -amenability of Banach alebras, Math Proc Camb Phil Soc, 144, pp 85 96, KELLEY, J L, General topoloy, D Van Nostrand Company, Inc, New Yprk, MIRZVZIRI, M, MOSLEHIN, S M, σ -derivations in Banach alebras, Bull Iranian Math Soc, 2006, pp MOSLEHIN, S M, MOTLGH, N, Some notes on ( σ, τ) -amenability of Banach alebras, Stud Univ Babes-Bolyai Math, 53, pp 57 68, RUNDE, V, Lectures on menability, Lecture Notes in Mathematics, 1774, Spriner, 2002 Received July 1, 2011