Lie Weak Amenability of Triangular Banach Algebra

Transcription

1 Journal of Mathematical Research with Applications Sept 217 Vol 37 No 5 pp DOI:1377/jissn: ie Weak Amenaility of Triangular anach Algera in CHEN 1 Fangyan U 2 1 Department of Mathematics and Physics Anshun University Guizhou 561 P R China; 2 Department of Mathematics Soochow University Jiangsu 2156 P R China Astract et A and e unital anach algera and M e anach A -module Then T A M ecomes a triangular anach algera when equipped with the anach space norm a m a A m M A anach algera T is said to e ie n-weakly amenale if all ie derivations from T into its n th dual space T n are standard In this paper we investigate ie n-weak amenaility of a triangular anach algera T in relation to that of the algeras A and their action on the module M Keywords triangular anach algera; weak amenaility; ie derivation MR21 Suject Classification 4747; Introduction et A and e anach algeras and suppose M is a anach A -module; that is M is a anach space a left A-module and a right -module and the action of A and are jointly continuous et M e the dual space of M We can define a right action of A on M and a left action on M via m f a am f m f m f for all a A and f M Then M is a anach A-module Similarly the second dual M of M ecomes a anach A -module under the actions f a ϕ f a ϕ f ϕ f ϕ for all a A and f M and ϕ M y continuing this process to higher order dual space of M it is seen that for each n 1 M 2n is a anach A -module and M 2n 1 is a anach A-module et M e a anach A-imodule a derivation is a continuous linear map δ : A M such that δa δa a δ holds for all a A If m M we define δ m a a x x a for each a A Then δ m is a derivation Such derivations are called inner derivations We use the notation Z 1 A M for the space of all continuous derivations from A into M and N 1 A M for the space of all inner derivations from A into M The quotient space H 1 A X Z 1 A M/N 1 A M is called the first Hochschild cohomology group of A with Received Novemer 2 216; Accepted May Supported y the National Natural Science Foundation of China Grant Nos ; * Corresponding author address: linchen198112@163com in CHEN; fylu@sudaeducn Fanyan U

2 64 in CHEN and Fangyan U coefficients in M The anach algera A is amenale if H 1 A M {} for every anach A- imodule M This notion was first defined y Johnson in [1] who proved that the amenaility of a locally compact group G is equivalent to amenaility of the group algera 1 G Haagerup [2] proved that all nuclear C -algeras are amenale ater ade Curtis and Dales [3] introduced the concept of weak amenaility who showed that a commutative anach algera is weakly amenale if and only if every ounded derivation δ : A A is inner and hence zero This motivated Johnson to introduce the notion of weak amenaility for general anach algeras and proved the group algera 1 G is weakly amenale for every locally compact group G in [4] More recently Dales et al [5] defined the concept of n-weak amenaility and permanent weak amenaility for anach algera A as follows: for every positive integer n anach algera A is said to e n-weakly amenale if H 1 A A n {} and A is permanently weakly amenale if it is n-weakly amenale for all n 1 and showed every C -algera is permanently weakly amenale Zhang [6] investigated n-weak amenaility of module extension anach algera esides derivation on operator algera we also consider ie derivation A ie derivation is a continuous linear map δ : A M such that δ[a ] [δa ] [a δ] holds for all a A here [a ] a a is the usual ie product A ie derivation δ is standard if it can e decomposed as δ d τ where d is a derivation from A into M and τ is a linear map from A into the center of M vanishing on each commutator In [7] Alaminos re sar and Villena showed that all ie derivations from a von Neumann algera A into any anach A-imodule are standard Cheung [8] investigated the ie derivation from triangular anach algera into itself u [9] showed every ie derivation from nest algera into itself is standard Naturally for every positive integer n we can define ie n-weak amenaility of anach algera A A is said to e ie n-weakly amenale if every ie derivation from A into A n is standard et A and e two anach algeras and M e a anach A -module We define A M T Then T is a complex algera with usual multiplication and addition actions in the space of 2 2 matrices T ecomes a anach algera with the following norm for every a A and m M a m : a A m M This algera T is called the triangular anach algera et t a m T and τ f hg T Then T acts on T as follows: τt fa hm g Now the left module action of T on T is give y a m f hg afmh h g and the right module action of T on T is as follows: a m f h a f m h h g g f h a m f a h a g h m g

3 ie weak amenaility of triangular anach algera 65 Thus T ecomes a anach T -imodule We remove the dot for simplicity This process may e repeated to define the module actions of T on T n for every n N as follows [1]: a m a 2n 1 m 2n 1 aa 2n 1 mm 2n 1 m 2n 1 2n 1 2n 1 a 2n 1 m 2n 1 2n 1 a m a 2n 1 a m 2n 1 a m 2n 1 m 2n 1 and a m a 2n m 2n 2n aa 2n am 2n m 2n 2n a 2n m 2n 2n a m a 2n a a 2n m m 2n 2n for all a m T a 2n 1 m 2n 1 T 2n 1 and a 2n m 2n T 2n 2n 1 2n The n-weak amenaility of triangular anach algera was first investigated y Forrest and Marcoux in [1] and it was proved that triangular anach algera T is weakly amenale if and only if A and are weakly amenale Since then amenaility of triangular anach algera was studied y many authors [11 14] In [14] it was shown that the triangular anach algera T is amenale if and only if oth A and satisfy M In this paper we consider ie weak amenaility of triangular anach algera 2 ie 2n 1-weak amenaility For triangular anach algera T the projections π A : T A and π : T are defined y π A x a and π x for any x a m T For a anach A-imodule X the centre of X denoted y ZX is the set { X X : AX XA for all A A } The following lemma gives the structure of the dual of triangular anach algera emma 21 et A and e anach algeras and M e a anach A -module et T A M e the corresponding triangular anach algera The centre of T n is given y { ZT 2n 1 f } : f ZA 2n 1 g Z 2n 1 g and { ZT 2n } : am m m M 2n a ZA 2n Z 2n Moreover if M is faithful then there exists a unique algera isomorphism γ from π A ZT 2n to π ZT 2n such that am mγa for every m M

4 66 in CHEN and Fangyan U Proof y the module action of T on T 2n 1 it is easy to show that { ZT 2n 1 f } : f ZA 2n 1 g Z 2n 1 g Suppose a A 2n 2n and am m for all m M 2n it is easy to verify ZT 2n Conversely if a n ZT 2n we have a n I A a n a n I A so n Since am m m m then am m for any m M 2n Since aa a a a we have aa a a and for any a A and and hence a ZA 2n and Z 2n have In the following we suppose that M is a faithful imodule If am m for any a A we aa a am aa m aa am a m a m m M 2n Hence aa a a as M 2n is faithful Therefore a ZA 2n and similarly we have Z 2n So ZT 2n Thus for any a π A ZT 2n ZA 2n there exists π ZT 2n such that ZT 2n Suppose there exists another satisfying ZT 2n then we have m am m for all m M 2n and hence as M 2n is faithful Therefore there exists a unique map γ : π A ZT 2n π ZT 2n satisfying am mγa for all a A and m T 2n Next we show that γ is an algera isomorphism If γa then am for every m M 2n and thus a Therefore γ is injective That γ is surjective follows from the definition of π ZT 2n It is clear that γ is linear For any a a π A ZT 2n from aa m aa m a mγa a mγa mγaγa it follows γaa γaγa proving that γ is an algera isomorphism

5 ie weak amenaility of triangular anach algera 67 emma 22 et A and e unital anach algeras and M e a anach A -module A continuous linear map : T T is a ie derivation if and only if is of the form a m g A a h mm m a m g h A a m m where m M g A : A A is a ie derivation g : is a ie derivation h A : A Z is a linear map satisfying h A [a a ] and h : ZA is a linear map satisfying h [ ] Proof Suppose is a ie derivation on T Write as a m g A a h k 1 m r 1 a r 2 fm g h A a k 2 m where g A : A A h : A k 1 : M A g : h A : A k 2 : M r 1 : A M r 2 : M f : M M are all continuous linear maps For identity I A A let I A i m j for some i A n M and j Since [ IA ] we have [ I A ] [ I A ] [ I A ] [ g A a r 1 a I A ] [ i m ] h A a j ai ia r 1 a m a Thus for all a A we can get ai ia r 1 a m a Similarly since [ IA ] we can get [ I A ] [ h r 2 I A g r 2 m j j ] [ Thus j j r 2 m for all For m M then m k 1 m fm k 2 m [ I A m [ I A ] [ I A i m j m m ] ] ]

6 68 in CHEN and Fangyan U [ i m m ] [ I A k 1 m fm ] j k 2 m mm fm m m Thus k 1 m mm k 2 m m m fm for all m M So for each a A ga a m a h A a For a A [aa ] ga [aa ] [aa ]m h A [aa ] On the other hand [ a ] [ [ g A a m a h A a a a ] [ [g A a a ] [a g A a ] [a a ]m ] [ a ] g A a m a h A a Thus g A [a a ] [g A a a ][a g A a ] and h A [a a ] Similarly we can show g [ ] [g ] [ g ] and h [ ] Finally we have [ a ] [ ] [ [ g A a m ] [ h A a [a h ] [h A a ] ] h m g Thus [a h ] and [h A a ] Since a and are aritrary we conclude that h A A Z and h ZA Conversely if is of the aove form it is routine to calculate and verify that is a ie derivation Theorem 23 et A and e unital anach algeras and M e a anach A -module et T A M e the corresponding triangular anach algera Then T is ie weakly amenale if and only if oth A and are ie weakly amenale Proof Suppose T is ie weakly amenale Then ie derivation : T T is written as D τ where D is a derivation and τ is a linear map from T into the center of T vanishing on each ] ]

7 ie weak amenaility of triangular anach algera 69 commutator y [1 emma 32] there exist derivations δ 1 : A A δ 3 : and an element n M such that a m D δ 1 a mn n a n δ 3 n m y emma 21 we know ZT ZA Z Thus we can suppose a m i 1 a i 2 m i 3 τ j 1 a j 2 m j 3 where i l and j l l are continuous linear maps from A M into ZA and Z respectively y emma 22 h A A Z and h ZA hence we can get h A j 1 and h i 3 Since D τ we have n m y taking a I A and m i 2 j 2 y taking a m m and g A δ 1 i 1 g δ 2 j 3 y taking a a m Since τ vanishes on commutators so do i 1 and j 3 Therefore A and are ie weakly amenale y [1 emma 32] and emma 22 it is easy to verify that if A and are ie weakly amenale then T is ie weakly amenale Recall that the action of T on T 3 is a restriction of T 2 on T 3 y repeating the same process as aove we can show T is ie 3-weakly amenale if and only if oth A and are ie 3-weakly amenale In fact this argument can extend to any dual of the form T 2n 1 with n 1 So we have the following result Theorem 24 et A and e unital anach algeras and M e a anach A -module et T A M e the corresponding triangular anach algera et n e a positive integer Then T is ie 2n 1-weakly amenale if and only if oth A and are ie 2n 1-weakly amenale 3 ie 2n-weak amenaility The action of T upon T 2n for each n 1 is a restriction of the Arens product to the canonical emedding of T into T 2n We oserve that the action coincide with the standard matrix multiplications Similar to the proof of emma 22 through elementary matrix calculations we have following lemma emma 31 et A and e unital anach algeras and M e a anach A -module A continuous linear map : T T 2n is a ie derivation if and only if is of the form a m g A a h am m fm g h A a where m M 2n g A : A A 2n g : 2n h A : A Z 2n with h A [a a ] h : ZA 2n with h [ ] and f : M M 2n are continuous linear maps satisfying i g A is a ie derivation on A fam g A am mh A a afm ii g is a ie derivation on fm mg h m fm Theorem 32 et A and e unital anach algeras and M e a faithful anach A -

8 61 in CHEN and Fangyan U module et T A M e the corresponding triangular anach algera et n e a positive integer Then T is ie 2n-weakly amenale if and only if h A A π ZT 2n and h π A ZT 2n Proof Suppose T is ie 2n-weakly amenale Then ie derivation : T T is written as D τ where D is a derivation and τ is a linear map from T into the center of T vanishing on each commutator y [1 Proposition 39] there exist continuous derivations δ 1 : A A 2n δ 3 : 2n and an element n M 2n such that a m δ 1 a an n ρm D δ 3 n m with ρam δ 1 am aρm and ρm ρm mδ 3 y emma 21 we can assume that a m τ i 1 a i 2 m i 3 j 1 a j 2 m j 3 where i l and j l l are continuous linear maps from A M into ZA and Z respectively Since Dτ we have n m y taking a I A and m i 2 j 2 y taking a m m and g A δ 1 i 1 g δ 2 j 3 h A j 1 and h i 3 y taking a a m Hence h A a j 1 a π τ π ZT 2n and h i 3 π A τ π A ZT 2n Conversely suppose that h A A π ZT 2n and h π A ZT 2n y emma 21 there exists the unique group isomorphism γ from π A ZT 2n to π ZT 2n Define a m g A a γ 1 h A a an n fm D g γh and a m h γ 1 h A a h h A a γh It is easy to verify that h a m ZT 2n vanishing on commutators y emma 31 we have and similarly fam g A am mh A a afm g A a γ 1 h A am afm fm mg h m fm mg γh fm et p A a g A a γ 1 h A a Then faa m p A aa m aa fm Also faa m p A aa m afa m p A aa m ap A a m aa fm

9 ie weak amenaility of triangular anach algera 611 Thus p A aa m p A aa ap A a m Hence since M 2n is faithful we have p A aa p A aa ap A a p A is a derivation Similarly g γh is also a derivation Therefore y [1 Proposition 39] D is a derivation 4 ie weak amenaility of finite dimension nest algera Recall that a nest on a Hilert space H is a chain N of closed suspaces of H which contains {} and H and which is closed under the operations of taking intersections and closed spans with the corresponding nest algera T N {T H : T N N for all N N } Our main result here is the following theorem Theorem 41 et T N e a nest algera on a finite dimensional complex Hilert space H Then T N is ie permanently weakly amenale Proof For each n 1 since T N is finite dimensional it follows that T N 2n is isomorphic to T N as a anach algera y [9] T N is ie 2n-weak amenale When the nest is trivial and T N H y [15] As for odd weak amenaility the proof works y induction upon the dimension m of the space H upon which T N acts When n 1 T N C It is clear that the conclusion holds Suppose the result holds for all nest algeras acting upon Hilert spaces of dimension less than k et T N e a nest algera acting on a space of dimension k Then either the nest N is trivial in which case T N H is a C -algera y [7 Theorem 611] it is ie amenale and hence is permanently ie weakly amenale or there exists an element {} N H in N and we may decompose T N as T N A M where A is a nest algera upon N is a nest algera upon N and M is the space of all operators taking N into N y our induction hypothesis A and are ie permanently weakly amenale y Theorem 24 we conclude that T N is ie 2n 1-weakly amenale for all n 1 Comining these two results yields our theorem Acknowledgements The authors would like to thank the referee for his valuale comments and corrections which have improved the quality of this paper References [1] E JOHNSON Cohomology in anach algeras American Mathematical Society Providence RI 1972 [2] U HAAGERUP All nuclear C -algera are amenale Invent Math : [3] W G ADE P C CURTIS H G DAES Amenaility and weak amenaility for eurling and ipschitz algeras Proc ondon Math Soc : [4] E JOHNSON Weak amenaility of group algeras ull ondon Math Soc : [5] H G DAES F GHAHRAANI N GRONAEK Derivations into iterated duals of anach algeras Studia Math : [6] Yong ZHANG Weak amenaility of module extensions of anach algeras Trans Amer Math Soc : [7] J AAMINOS M RE SAR A R VIENA The strong degree of von Neumann algeras and the structure of ie and Jordan derivations Math Proc Camridge Philos Soc : [8] W CHEUNG ie derivations of triangular algeras inear Multilinear Algera :

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