Stability of derivations under weak-2-local continuous perturbations
aa r X i v : . [ m a t h . OA ] M a y STABILITY OF DERIVATIONS UNDER WEAK-2-LOCALCONTINUOUS PERTURBATIONS
ENRIQUE JORD ´A AND ANTONIO M. PERALTA
Abstract.
Let Ω be a compact Hausdorff space and let A be a C ∗ -algebra. We provethat if every weak-2-local derivation on A is a linear derivation and every derivationon C (Ω , A ) is inner, then every weak-2-local derivation ∆ : C (Ω , A ) → C (Ω , A ) isa (linear) derivation. As a consequence we derive that, for every complex Hilbertspace H , every weak-2-local derivation ∆ : C (Ω , B ( H )) → C (Ω , B ( H )) is a (linear)derivation. We actually show that the same conclusion remains true when B ( H ) isreplaced with an atomic von Neumann algebra. With a modified technique we provethat, if B denotes a compact C ∗ -algebra (in particular, when B = K ( H )), then everyweak-2-local derivation on C (Ω , B ) is a (linear) derivation. Among the consequences,we show that for each von Neumann algebra M and every compact Hausdorff spaceΩ, every 2-local derivation on C (Ω , M ) is a (linear) derivation. Introduction
We recall that a derivation from a Banach algebra A into a Banach A -bimodule X isa linear map D : A → X satisfying D ( ab ) = D ( a ) b + aD ( b ) , (1.1)for every a, b in A . Given x ∈ X , the operator ad x : A → X , a ad x ( a ) = [ x , a ] = x a − ax , is a derivation. Derivations of this form are termed inner derivations . Alinear mapping T : A → X is called a local derivation if for each a ∈ A there exits aderivation D a : A → X , depending on a , satisfying T ( a ) = D a ( a ).In the setting above, the dual space X ∗ can be equipped with a natural structure ofBanach A -bimodule with respect to the products( aφ )( x ) = φ ( xa ) , and, ( φa )( x ) = φ ( ax ) ( a ∈ A, x ∈ X, φ ∈ X ∗ ) . Date : April 12, 2018.2010
Mathematics Subject Classification.
Key words and phrases. derivation; 2-local linear map; 2-local symmetric maps; 2-local ∗ -derivation;2-local derivation; weak-2-local derivation.First author partially supported by the Spanish Ministry of Economy and Competitiveness ProjectMTM2013-43540-P. Second author partially supported by the Spanish Ministry of Economy andCompetitiveness and European Regional Development Fund project no. MTM2014-58984-P and Juntade Andaluc´ıa grant FQM375. Derivations whose domain is a C ∗ -algebra or a von Neumann algebra are, by far, themost studied and best understood class of derivations. S. Sakai sets some of the mostinfluencing results by showing that every derivation on a C ∗ -algebra is automaticallycontinuous (see [18] or [19, Lemma 1.4.3]), and every derivation on a von Neumannalgebra or on a unital simple C ∗ -algebra is inner (cf. [19, Theorems 4.1.6 and 4.1.11]).There are examples of derivations on a C ∗ -algebra which are not inner (see [19, Example1.4.8]). Subsequently, J. Ringrose established in [16] that every derivation from a C ∗ -algebra A into a Banach A -bimodule is continuous. A derivation D on a C ∗ -algebra A iscalled a ∗ -derivation if D ( a ∗ ) = D ( a ) ∗ for all a ∈ A . An inner derivation ad x : A → A is a ∗ -derivation if and only if x ∗ = − x . Let ∆ be a mapping from a C ∗ -algebra A into a C ∗ -algebra B . ∆ ♯ : A → B is the map defined by ∆ ♯ ( a ) = ∆( a ∗ ) ∗ ( a ∈ A ). ∆ iscalled symmetric if ∆ ♯ = ∆, i.e., ∆( a ∗ ) = ∆( a ) ∗ , for every a ∈ A .Researchers belonging to different generations have been striving to explore the sta-bility of the set of derivations under weaker and weaker hypothesis since 1990. Thepioneering contribution of R.V. Kadison in [12] asserts that every continuous localderivation from a von Neumann algebra M into a dual Banach M -bimodule X is aderivation. B.E. Johnson shows in [10] that the conclusion in Kadison’s theorem holdsfor local derivations from a C ∗ -algebra A into a Banach A -bimodule. These results canbe interpreted as properties of stability of derivations under local perturbations in theset of linear maps from A into X .A weaker stability property derives from the notion of 2-local derivation. Let X bea Banach A -bimodule over a Banach algebra A . Following P. ˇSemrl [20], a mapping∆ : A → X is said to be a if for every a, b ∈ A , there existsa derivation D a,b : A → X , depending on a and b , satisfying ∆( a ) = D a,b ( a ) and∆( b ) = D a,b ( b ). In a recent contribution, Sh. Ayupov and K. Kudaybergenov provethat every 2-local derivation on a von Neumann algebra M is a derivation, that is,derivations are stable under 2-local perturbations in the set of mappings on M (see[5]). As long as we know the case of 2-local derivations on a general C ∗ -algebra remainsas an open problem.More recent studies are leading the mathematical community to the notion of weak-2-local S -maps between Banach spaces (see [14, 15, 6, 7]). According to the notationin the just quoted references, given a subset S of the space L ( X, Y ), of all linear mapsbetween Banach spaces X and Y , a (non-necessarily linear nor continuous) mapping∆ : X → Y is said to be a weak-2-local S map (respectively, a S -map ) if forevery x, y ∈ X and φ ∈ Y ∗ (respectively, for every x, y ∈ X ), there exists T x,y,φ ∈ S ,depending on x , y and φ (respectively, T x,y ∈ S , depending on x and y ), satisfying φ ∆( x ) = φT x,y,φ ( x ) , and φ ∆( y ) = φT x,y,φ ( y )(respectively, ∆( x ) = T x,y ( x ) , and ∆( y ) = T x,y ( y )). If we take S = K ( X, Y ) the spaceof compact linear mappings from X to Y then it is straightforward to check that if ∆is any non-linear, 1-homogeneous map, i.e. ∆( αx ) = α ∆( x ) for each α ∈ C , then ∆ isa 2-local S -map. Some particular cases receive special names. When S is the set of all TABILITY OF DERIVATIONS UNDER WEAK-2-LOCAL CONTINUOUS PERTURBATIONS 3 derivations on a C ∗ -algebra A (respectively, the set of all ∗ -derivations on A , or, moregenerally, the set of all symmetric maps from A into a C ∗ -algebra B ), weak-2-local S -maps are called weak-2-local derivations (respectively, weak-2-local ∗ -derivations or weak-2-local symmetric maps ).The first results in this line prove that every weak-2-local derivation on a finite di-mensional C ∗ -algebra is a linear derivation (see [15, Corollary 2.13]), and for a separablecomplex Hilbert space H every (non-necessarily linear nor continuous) weak-2-local ∗ -derivation on B ( H ) is linear and a ∗ -derivation [15, Theorem 3.10]. Symmetric mapsbetween C ∗ -algebras A and B are very stable under weak-2-local perturbations inthe space L ( A, B ), more concretely, every weak-2-local symmetric map between C ∗ -algebras is a linear map [6, Theorem 2.5]. Fruitful consequences are derived from thisresult, for example, every weak-2-local ∗ -derivation on a general C ∗ -algebra is a (lin-ear) ∗ -derivation, and every 2-local ∗ -homomorphism between C ∗ -algebras is a (linear) ∗ -homomorphism (see [6, Corollary 2.6]). Weak-2-local derivations on von Neumannalgebras and on general C ∗ -algebras remain unknowable. In [7], J.C. Cabello and thesecond author of this note prove that every weak-2-local derivation on B ( H ) or on K ( H ) is a linear derivation, where H is an arbitrary complex Hilbert space, and con-sequently, every weak-2-local derivation on an atomic von Neumann algebra or on acompact C ∗ -algebra is a linear derivation.Another attempt to study 2-local derivations on C ∗ -algebras has been conducted bySh. Ayupov and F.N. Arzikulov [4]. The main result in the just quoted paper provesthat, for every compact Hausdorff space Ω, every 2-local derivation on C (Ω , B ( H )) isa derivation.In this note we continue with the study of weak-2-local derivations in new classesof C ∗ -algebras. In the first main result (Corollary 2.5) we prove that, for everyHilbert space H , every weak-2-local ∆ : C (Ω , B ( H )) → C (Ω , B ( H )) is a (linear)derivation. This result is a consequence of a technical theorem in which we estab-lish that if A is a C ∗ -algebra such that every weak-2-local derivation on A is a linearderivation and every derivation on C (Ω , A ) is inner, then every weak-2-local derivation∆ : C (Ω , A ) → C (Ω , A ) is a (linear) derivation (see Theorem 2.4). Actually this tech-nical result, combined with recent results on weak-2-local derivations on atomic vonNeumann algebras in [7], implies that every weak-2-local C (Ω , A ) is a linear deriva-tion whenever A is an atomic von Neumann algebra. We particularly show that theresult of Sh. Ayupov and F.N. Arzikulov remains true if we replace “2-local” with“weak-2-local”.In order to extend our study to weak-2-local derivations on C ∗ -algebras of the form C (Ω , K ( H )), where K ( H ) is the C ∗ -algebra of compact linear operators on a Hilbertspace H , or to C (Ω , B ) where B is a compact C ∗ -algebra, we represent, in Proposi-tion 2.10, every derivation on C (Ω , K ( H )) as an “inner derivations” associated witha mapping Z : Ω → B ( H ) which is, in general, τ -weak ∗ -continuous, where τ is thetopology of Ω. The τ -norm continuity of the mapping Z cannot be, in general, pur-sued (compare Remark 2.11). The paper culminates with a result asserting that, for E. JORD ´A AND A.M. PERALTA a compact C ∗ -algebra B , every weak-2-local derivation ∆ : C (Ω , B ) → C (Ω , B ) is a(linear) derivation (see Theorem 2.15). Weak-2-local derivations on C (Ω) ⊗ A Let A and B be C ∗ -algebras. Henceforth A ⊙ B will denote the algebraic tensorproduct of A and B . A norm α on A ⊙ B is said to be a C ∗ -norm if α ( xx ∗ ) = α ( x )and α ( xy ) ≤ α ( x ) α ( y ), for every x, y ∈ A ⊙ B . It is known that there exists a leastC ∗ -norm α among all C ∗ -norms α on A ⊙ B such that α ∗ is finite. It is further knownthat α is a cross norm and λ ≤ α ≤ γ , where λ and ε denote the injective and theprojective tensor norm on A ⊙ B , respectively (see [19, Propsition 1.22.2]). Along thisnote, the symbol A ⊗ B will denote the C ∗ -algebra obtained as the completion of A ⊙ B with respect to α .For each C ∗ -algebra A and every locally compact Hausdorff space L , the Banachspace C ( L, A ), of all A -valued continuous functions on L vanishing at infinite, admitsa natural structure of C ∗ -algebra with respect to the “pointwise” operations and the supnorm. It is also known that C ( L, A ) and C ( L ) ⊗ A = C ( L ) ⊗ λ A are isometricallyC ∗ -isomorphic as C ∗ -algebras (see [19, Proposition 1.22.3]). When Ω is a compactHausdorff space, we have C (Ω , A ) ∼ = C (Ω) ⊗ A = C (Ω) ⊗ λ A .Another influential result due to S. Sakai, apart from those commented in the intro-duction, asserts that every von Neumann algebra admits a unique (isometric) predualand its product is separately weak ∗ -continuous (see [19, Theorem 1.7.8]). It is knownthat, for a C ∗ -algebra A , its second dual, A ∗∗ , is a von Neumann algebra [19, Theorem1.17.2]. Combining these facts with the identity in (1.1), we can see that for everyderivation D : A → A , its bitransposed map D ∗∗ : A ∗∗ → A ∗∗ is a derivation on A ∗∗ .Therefore, there exists z in A ∗∗ satisfying D ∗∗ ( x ) = [ z , x ], for every x ∈ A ∗∗ . We havealready commented that we cannot assume that z lies in A . The following lemma,which was originally proved by R.V. Kadison in [11, Theorem 2], can be also derivedfrom the above results: Lemma 2.1. [11, Theorem 2]
Let D : A → A be a derivation on a C ∗ -algebra A whosecenter is denoted by Z ( A ) . Then D ( c ) = 0 , for every c ∈ Z ( A ) . ✷ A Banach algebra A is said to be super-amenable if every derivation from A intoa Banach A -bimodule is inner. The Banach algebra A is called amenable if for everyBanach A -bimodule X , every derivation from A into X ∗ is inner. Finally if everyderivation from A into A ∗ is inner we say that A is weakly amenable. Every super-amenable (respectively, amenable) Banach algebra is amenable (respectively, weaklyamenable), and the three classes are mutually different. We refer to the monographs[9, 17] for a detailed account on the theory of super-amenable, amenable and weaklyamenable Banach algebras.The problem of determining those C ∗ -algebras admitting only inner derivations hasbeen considered by a wide number of researchers. Although there are some basic TABILITY OF DERIVATIONS UNDER WEAK-2-LOCAL CONTINUOUS PERTURBATIONS 5 unanswered questions along these lines, a partial result obtained by C.A. Akemannand B.E. Johnson (see [1]) will be very useful for our purposes.
Theorem 2.2. [1, Theorem 2.3]
Let M be a von Neumann algebra, and let B be aunital abelian C ∗ -algebra. Then every derivation of the C ∗ -tensor product B ⊗ M isinner. Equivalently, given a compact Hausdorff space Ω , every derivation on C (Ω , M ) is inner. ✷ Let X and Y be Banach A -bimodules over a Banach algebra A . A linear mappingΦ : X → Y is called a module homomorphism if Φ( ax ) = a Φ( x ) and Φ( xa ) = Φ( x ) a ,for every a ∈ A , x ∈ X. The next lemma, whose proof is left to the reader, gatherssome basic properties.
Lemma 2.3.
Let A be a Banach algebras and let Φ : X → Y be a module homomor-phism. ( a ) If D : A → X is a derivation, then Φ D : A → Y is a derivation; ( b ) If ∆ : A → X is a weak-2-local derivation and Φ is continuous, then Φ∆ : A → Y is a weak-2-local derivation; ( c ) Suppose B is another Banach algebra such that X is a Banach B -bimodule andthere is a homomorphism Ψ : B → A satisfying Ψ( b ) x = bx and xb = x Ψ( b ) , forevery x ∈ X , b ∈ B . Then for each derivation ( respectively, for each weak-2-localderivation ) D : A → X the composition D Ψ : B → X is a derivation ( respectively,a weak-2-local derivation ) . ✷ Following standard notation, given t ∈ Ω, δ t : C (Ω , A ) → A will denote the ∗ -homomorphism defined by δ t ( X ) = X ( t ). The space C (Ω , A ) also is a Banach A -bimodule with products ( aX )( t ) = aX ( t ) and ( Xa )( t ) = X ( t ) a , for every a ∈ A , X ∈ C (Ω , A ). The mapping δ t : C (Ω , A ) → A is an A -module homomorphism.Given a compact Hausdorff space Ω and a C ∗ -algebra A , the ∗ -homomorphism map-ping each element a in A to the constant function Ω → A , t a will be denoted by b a . The mapping Γ : A → C (Ω , A ) = C (Ω) ⊗ A , Γ( a ) = 1 ⊗ a = b a , is an A -modulehomomorphism. Theorem 2.4.
Let Ω be a compact Hausdorff space and let A be a C ∗ -algebra. Supposethat every weak-2-local derivation on A is a linear derivation, every derivation on C (Ω , A ) is inner. Then every weak-2-local derivation ∆ : C (Ω , A ) → C (Ω , A ) is a ( linear ) derivation.Proof. By [8, Theorem 3.4] it is enough to prove that ∆ is linear. This linearity certainlyholds if and only if δ t ∆ is linear for all t ∈ Ω.Fix an arbitrary t ∈ Ω. We claim that δ t ∆Γ δ t ( X ) = δ t ∆( X ) , (2.1)for every X ∈ C (Ω , A ), where for each a ∈ A , Γ( a ) = b a is the constant function withvalue a on Ω. Indeed, by hypothesis, every derivation D : C (Ω , A ) → C (Ω , A ) is inner,and hence of the form D ( X ) = [ Z, X ] ( ∀ X ∈ C (Ω , A )), where Z ∈ C (Ω , A ). Thus, E. JORD ´A AND A.M. PERALTA for each t ∈ Ω we have δ t D Γ δ t ( X ) = δ t D ( X ), for every X ∈ C (Ω , A ). Let φ be anyelement in A ∗ and consider the functional φ ⊗ δ t : C (Ω , A ) → C , X → φ ( X ( t )). Bythe weak-2-local property of ∆, for each X ∈ C (Ω , A ), there exists a derivation D = D X, Γ( X ( t )) ,φ ⊗ δ t : C (Ω , A ) → C (Ω , A ), depending on X , Γ( X ( t )) = Γ δ t ( X ) , and φ ⊗ δ t ,such that φ ( δ t ∆( X ) − δ t ∆Γ δ t ( X )) = φ ⊗ δ t (∆( X ) − ∆Γ δ t ( X )) = φ ⊗ δ t ( D ( X ) − D Γ δ t ( X )) = 0 , because δ t D Γ δ t ( X ) = δ t D ( X ). The arbitrariness of φ ∈ A ∗ proves (2.1).Since the operators δ t : C (Ω , A ) → A is a continuous A -module homomorphism, andΓ : A → C (Ω , A ) is a homomorphism satisfying Γ( a ) A = aA and A Γ( a ) = Aa , forevery a ∈ A , A ∈ C (Ω , A ), Lemma 2.3( c ) implies that δ t ∆Γ : A → A is a weak-2-localderivation for every t ∈ Ω. The additional hypothesis on A assure that δ t ∆Γ is a linearderivation. Therefore δ t ∆Γ( X ( t ) + Y ( t )) = δ t ∆Γ( X ( t )) + δ t ∆Γ( Y ( t )) , for every X, Y in C (Ω , A ). By (2.1) δ t ∆( X + Y ) = δ t ∆Γ δ t ( X + Y ) = δ t ∆Γ δ t ( X ) + δ t ∆Γ δ t ( Y ) = δ t ∆( X ) + δ t ∆( Y ) , for every t ∈ Ω, X, Y ∈ C (Ω , A ). In particular, δ t ∆ is a linear mapping, as wedesired. (cid:3) The previous theorem can be now applied to provide new non-trivial examples ofC ∗ -algebras on which every weak-2-local derivation is a derivation. Corollary 2.5.
Let H be a complex Hilbert space. Then every weak-2-local derivation ∆ : C (Ω , B ( H )) → C (Ω , B ( H )) is a ( linear ) derivation. Furthermore, for an atomicvon Neumann algebra A , every weak-2-local derivation ∆ : C (Ω , A ) → C (Ω , A ) is a ( linear ) derivation.Proof. Theorem 2.2 proves that every derivation on C (Ω , B ( H )) is inner. It is alsoknown that every weak-2-local derivation on B ( H ) is a linear derivation (see [7, Theo-rem 3.1]). Theorem 2.4 implies that every derivation on C (Ω , B ( H )) is a linear deriva-tion.The statement for atomic von Neumann algebras follows from the same argumentsbut replacing [7, Theorem 3.1] with [7, Corollary 3.5]. (cid:3) We observe that Corollary 2.5 provides a generalization of a recent result due to Sh.Ayupov and F.N. Arzikulov (compare [4, Theorem 1]).
Corollary 2.6. [4, Theorem 1]
Let H be a complex Hilbert space. Then every 2-localderivation ∆ : C (Ω , B ( H )) → C (Ω , B ( H )) is a ( linear ) derivation. ✷ We observe that for an infinite dimensional separable complex Hilbert space H , wecan always find a derivation on the C ∗ -algebra K ( H ) of all compact operators on H which is not inner (see [19, Example 4.1.8]). Since the a similar conclusion remains TABILITY OF DERIVATIONS UNDER WEAK-2-LOCAL CONTINUOUS PERTURBATIONS 7 valid for C (Ω , K ( H )), we cannot apply Theorem 2.4 in this case. We shall see nexthow to avoid the difficulties.Throughout the paper, M n = M n ( C ) will denote the complex n × n -matrices. Foreach i, j ∈ { , . . . , n } , e ij will denote the unit matrix in M n with 1 in the ( i, j ) com-ponent and zero otherwise. Given a C ∗ -algebra A , the symbol M n ( A ) will stand forthe n × n -matrices with entries in A . It is known that M n ( A ) is a C ∗ -algebra withrespect to the product and involution defined by ( a ij ) i,j ( b ij ) i,j = n X k =1 a ik b kj ! i,j and( a ij ) ∗ i,j = ( a ∗ ji ) i,j , respectively (compare [22, § IV.3]). The space M n ( A ) also is a Banach A -bimodule for the products b ( a ij ) = ( ba ij ) , and ( a ij ) b = ( a ij b ). Given a ∈ A and i, j ∈ { , . . . , n } , the symbol a ⊗ e ij will denote the matrix in M n ( A ) with entry a inthe ( i, j )-position and zero otherwise.The following lemma might be known, it is included here due to the lack of an explicitreference. Lemma 2.7.
Let ( z λ ) and ( x λ ) be bounded nets in a von Neumann algebra M suchthat ( z λ ) → z in the weak ∗ -topology of M and ( x λ ) → x , in the norm topology of M .Then ( z λ x λ ) → z x in the weak ∗ -topology.Proof. We can assume that k z λ k , k x λ k ≤
1, for every λ . The net ( x λ − x ) → φ ∈ M ∗ , it follows from theCauchy-Schwarz inequality that | φ ( z λ ( x λ − x )) | ≤ φ ( z λ z ∗ λ ) φ (( x λ − x ) ∗ ( x λ − x )) ≤ k x λ − x k → . We deduce that ( z λ ( x λ − x )) → ∗ -topology of M . Since the product of M is separately weak ∗ -continuous, we also know that ( z λ x ) → z x in the weak ∗ -topology,and hence ( z λ x λ ) → z x in the weak ∗ -topology. (cid:3) Suppose ad x : A → A , a [ x , a ] is an inner derivation on a C ∗ -algebra. It is wellknown that the element x is not uniquely determined by ad x , for example ad x = ad y as derivations on A if and only if x − y ∈ Z ( A ).Concerning norms, it is easy to see that k [ x , . ] k ≤ k x k , where k [ x , . ] k denotesthe norm of the linear derivation in B ( A ). It is not obvious that an element in theset x + Z ( A ) can be bounded by a multiple of the norm of the inner derivation [ x , . ].In this line, R.V. Kadison, E.C. Lance and J.R. Ringrose prove, in [13, Theorem 3.1],that for each ∗ -derivation D on a C ∗ -algebra A , if e D denotes its unique extensionto a derivation on A ∗∗ , then there is a unique self-adjoint element a in A ∗∗ such that e D = ad a = [ a , . ] and, for each central projection q in A ∗∗ , we have k a q k = k e D | A ∗∗ q k .In particular k D k = 2 k a k .Let D : A → A be a derivation on a C ∗ -algebra. We can write D = D + iD , where D = ( D + D ♯ ), D = i ( D − D ♯ ) are ∗ -derivations on A with k D j k ≤ k D k , for every j ∈ { , } . Let e D, e D j : A ∗∗ → A ∗∗ denote the unique extension of D and D j to aderivation on A ∗∗ , respectively. Applying the just quoted result, we find a , b ∈ A ∗∗ sa E. JORD ´A AND A.M. PERALTA satisfying e D = ad a = [ a , . ], e D = ad b = [ b , . ] and, k a k , k b k ≤ k D k . Then e D ( x ) = [ a + ib , x ], for every x ∈ A ∗∗ , with k a + ib k ≤ k D k .In general, C (Ω , M n ) = C (Ω) ⊗ M n need not be a von Neumann algebra and itssecond dual is too big for our purposes. We have already commented that everyderivation D : C (Ω , M n ) → C (Ω , M n ) is inner, so there exists X ∈ C (Ω , M n ) satisfying D ( X ) = [ X , X ], for every X ∈ C (Ω , M n ). If D is a ∗ -derivation we can assume that X ∗ = − X . Let us assume that D is a ∗ -derivation. We know that X can be replacedwith any element in the set X + iZ ( C (Ω , M n )) sa = { X + if ⊗ I n : f ∈ C (Ω) sa } ,where I n stands for the unit in M n . The question is whether we can find an element X + if ⊗ I n satisfying k D k = k [ X + if, . ] k ≥ k X + if k . When z is a symmetric (or a skew symmetric) operator in B ( H ), J.G. Stampfliestablishes in [21, Corollary 1] that k [ z, . ] k = 2 ρ ( σ ( z )) = diam( σ ( z )) ≤ k z k , (2.2)and consequently, if 0 ∈ σ ( z ) then k z k ≤ k [ z, . ] k = diam( σ ( z )) ≤ k z k . (2.3)We shall require a variant of the previous estimations. Proposition 2.8.
Let D : C (Ω , M n ) → C (Ω , M n ) be a ∗ -derivation, where Ω is acompact Hausdorff space. Then there exists a unique Z ∈ C (Ω , M n ) satisfying Z ∗ = − Z , D ( . ) = [ Z , . ] , σ ( − iZ ) ⊆ R +0 , and k Z k = k D k . Proof.
We have already observed that there exists Z ∈ C (Ω , M n ) satisfying Z ∗ = − Z and D ( . ) = [ Z , . ]. For Z ∈ C (Ω , M n ) with Z ∗ = − Z we setdiam( σ ( Z )) := sup t ∈ Ω diam( σ ( Z ( t ))) . We claim that k D k = k [ Z , . ] k = diam( σ ( Z )) . (2.4)Indeed, for each X ∈ C (Ω , M n ) with k X k ≤
1, it follows from (2.2) that k [ Z , X ] k = sup t ∈ Ω k [ Z ( t ) , X ( t )] k ≤ sup t ∈ Ω diam( σ ( Z ( t ))) = diam( σ ( Z )) . To see the reciprocal inequality, given ε >
0, there exists t ∈ Ω such that diam( σ ( Z )) − ε < diam( σ ( Z ( t ))) . Since [ Z ( t ) , . ] : M n → M n is a bounded linear operatoron a finite dimensional space, we can find a norm-one element b ∈ M n such that k [ Z ( t ) , b ] k = k [ Z ( t ) , . ] k = diam( σ ( Z ( t ))). Clearly, Γ( b ) ∈ C (Ω , M n ) , k Γ( b ) k ≤ k [ Z , . ] k ≥ k [ Z , Γ( b )] k ≥ k [ Z ( t ) , Γ( b )( t )] k = k [ Z ( t ) , b ] k > diam( σ ( Z )) − ε, which proves the claim.We define a function σ min : Ω → C , σ min ( t ) := λ ∈ σ ( Z ( t )), where λ is the uniqueelement in σ ( Z ( t )) ⊆ i R satisfying | λ | = min {| µ | : µ ∈ σ ( Z ( t )) } . TABILITY OF DERIVATIONS UNDER WEAK-2-LOCAL CONTINUOUS PERTURBATIONS 9
Let us recall some notation. Suppose K and K are non-empty compact subsets of C . The Hausdorff distance between K and K is defined by d H ( K , K ) = max { sup t ∈ K dist( t, K ) , sup s ∈ K dist( s, K ) } . By [3, Theorem 6.2.1( v )] the inequality d H ( σ ( a ) , σ ( b )) ≤ k a − b k , holds for all normal elements a, b in a C ∗ -algebra A . Applying the above inequality,and the continuity of − Z ∗ = Z ( . ) : Ω → M n , we shall see that σ min ∈ C (Ω). Namely,fix t ∈ Ω, ε > t ∈ U such that d H ( σ ( X ( t )) , σ ( X ( t ))) ≤ k X ( t ) − X ( t ) k < ε, for every t ∈ U . Let us write σ ( X ( t )) = { σ min ( t ) = λ ( t ) , λ ( t ) , . . . , λ n ( t ) } and σ ( X ( t )) = { σ min ( t ) = λ ( t ) , λ ( t ) , . . . , λ n ( t ) } with − iσ min ( t ) ≤ − iλ ( t ) ≤ . . . ≤− iλ n ( t ) and − iσ min ( t ) ≤ − iλ ( t ) ≤ . . . ≤ − iλ n ( t ). In this case, for every t ∈ U ,there exist λ j ( t ) and λ k ( t ) such that | σ min ( t ) − λ j ( t ) | < ε and | σ min ( t ) − λ k ( t ) | < ε. If − iσ min ( t ) ≤ − iσ min ( t ) ≤ − iλ j ( t ) we have | σ min ( t ) − σ min ( t ) | ≤ | σ min ( t ) − λ j ( t ) | < ε. If − iσ min ( t ) < − iσ min ( t ) ≤ − iλ k ( t ) we have | σ min ( t ) − σ min ( t ) | ≤ | σ min ( t ) − λ k ( t ) | < ε. Therefore | σ min ( t ) − σ min ( t ) | < ε , for every t ∈ U .Clearly, σ min ⊗ I n ∈ Z ( C (Ω , M n )) and 0 ∈ σ ( Z − σ min ⊗ I n ). Applying (2.4) weconclude that k D k = k [ Z − σ min ⊗ I n , . ] k = diam( σ ( Z − σ min ⊗ I n )) = k Z − σ min ⊗ I n k , which proves the desired statement for Z = Z − σ min ⊗ I n . (cid:3) Remark 2.9.
Let N = ℓ ∞ M ≤ j ≤ m M n j be an arbitrary finite dimensional C ∗ -algebra (com-pare [22, Theorem I.11.2]). Let D : C (Ω , N ) → C (Ω , N ) be a ∗ -derivation, where Ωis a compact Hausdorff space. Then there exists a unique Z ∈ C (Ω , N ) satisfying Z ∗ = − Z , D ( . ) = [ Z , . ], σ ( − iZ ) ⊆ R +0 , and k Z k = k D k . Indeed, we can iden-tify C (Ω , N ) with the ℓ ∞ -sum ℓ ∞ M ≤ j ≤ m C (Ω , M n j ) . It is known that D ( C (Ω , M n j )) ⊆ C (Ω , M n j ) for every j (compare [7, Lemma 3.3 and its proof]). Therefore D j = D | C (Ω ,M nj ) : C (Ω , M n j ) → C (Ω , M n j ) is a derivation for every j , and we identify D with the direct sum of all D j . The desired conclusion follows by applying the aboveProposition 2.8 to each D j . In accordance with the notation in [7], henceforth, the set of all finite dimensionalsubspaces of H will be denoted by F ( H ) . This set is equipped with the natural ordergiven by inclusion, and for each F ∈ F ( H ), p F will denote the orthogonal projectionof H onto F . Given a compact Hausdorff space Ω. We set b p F = Γ( p F ), where Γ : K ( H ) → C (Ω) ⊗ K ( H ) is the mapping defined before Theorem 2.4.An approximate unit or identity in a C ∗ -algebra A is a net ( u λ ) ⊆ A satisfying0 ≤ u λ ≤ λ , u λ ≤ u µ for every λ ≤ µ and lim λ || x − u λ x k = 0 for each x in A . In these conditions, lim λ k x − xu λ k = 0 as well.The following proposition, which has not been explicitly treated in the literature, isall we shall require to deal with the case of weak-2-local derivations on C (Ω , K ( H )). Proposition 2.10.
Let D : C (Ω , K ( H )) → C (Ω , K ( H )) be a derivation, where H isa complex Hilbert space. Let τ denote the topology of Ω . Then there exists a τ -weak ∗ -continuous, bounded mapping Z : Ω → B ( H ) satisfying D ( X )( t ) = [ Z ( t ) , X ( t )] , forevery X ∈ C (Ω , K ( H )) . In particular, for each t in Ω we have δ t D Γ δ t ( A ) = δ t D ( A ) , for every A ∈ C (Ω , K ( H )) .Proof. Let us first assume that D is a ∗ -derivation.To simplify the notation we write C = C (Ω , K ( H )). Pick an arbitrary F ∈ F ( H ).Applying [14, Proposition 2.7] we deduce that the mapping b p F D b p F | b p F C b p F : b p F C b p F → b p F C b p F , b p F A b p F b p F D ( b p F A b p F ) b p F is a derivation on b p F C b p F ∼ = C (Ω , M n ), with n = dim( F ). Applying Theorem 2.2 andProposition 2.8 we find a unique Z F ∈ b p F C b p F satisfying Z ∗ F = − Z F , b p F D ( b p F A b p F ) b p F =[ Z F , b p F A b p F ], for every A ∈ C , σ ( − iZ F ) ⊆ R +0 , and k Z F k = k b p F D b p F | b p F C b p F k ≤ k D k . Fix t in Ω. The net ( Z F ( t )) F ∈ F ( H ) ⊆ K ( H ) ⊆ B ( H ) is bounded, so there exists asubnet ( Z F ( t )) F ∈ F ′ and Z ( t ) ∈ B ( H ) such that k Z ( t ) k ≤ k D k , and ( Z F ( t )) F ∈ F ′ → Z ( t ) in the weak ∗ -topology of B ( H ).It is not hard to see that the net ( b p F ) F ∈ F ( H ) is an approximate unit in C (Ω , K ( H )).Indeed, let us fix A in C (Ω , K ( H )) and ε >
0. Applying the continuity of A and thecompactness of Ω, we can find a finite open cover U , . . . , U m , and points t , . . . , t m suchthat t j ∈ U j and k A ( s ) − A ( t j ) k < ε , for each s ∈ U j . Since A ( t ) , . . . , A ( t m ) ∈ K ( H ),we can find F ∈ F ( H ) satisfying k A ( t j ) − p F A ( t j ) p F k < ε , for every j and every F ∈ F ( H ) with F ⊇ F . For each s in Ω, there exists j such that s ∈ U j , and hence k A ( s ) − p F A ( s ) p F k ≤ k A ( s ) − A ( t j ) k + k A ( t j ) − p F A ( t j ) p F k + k p F A ( t j ) p F − p F A ( s ) p F k < ε, which proves that k A − b p F A ( s ) b p F k < ε, every F ∈ F ( H ) with F ⊇ F .By the continuity of D , given A in C (Ω , K ( H )), the nets ( b p F A b p F ) F ∈ F ( H ) , and( b p F D ( b p F A b p F ) b p F ) F ∈ F ( H ) converge in norm to A and D ( A ), respectively. Consequently,for each t ∈ Ω, ( b p F A b p F ( t )) F ∈ F ′ → A ( t ), and ( b p F D ( b p F A b p F ) b p F ( t )) F ∈ F ′ → D ( A )( t ) in thenorm topology of K ( H ). Taking weak ∗ -limit in the identity b p F D ( b p F A b p F ) b p F ( t ) = [ Z F , b p F A b p F ]( t ) = [ Z F ( t ) , b p F A b p F ( t )] , TABILITY OF DERIVATIONS UNDER WEAK-2-LOCAL CONTINUOUS PERTURBATIONS 11 we deduce, via Lemma 2.7, that D ( A )( t ) = [ Z ( t ) , A ( t )].When D is a general derivation, we write D = D + iD with D and D ∗ -derivationson C (Ω , K ( H )). By the arguments above, there exist bounded maps Z , Z : Ω → B ( H ) satisfying D j ( A )( t ) = [ Z j ( t ) , A ( t )] , for every t ∈ Ω, A ∈ C (Ω , K ( H )). Therefore, D ( A )( t ) = [ Z ( t ) + iZ ( t ) , A ( t )] , for all t ∈ Ω, A ∈ C (Ω , K ( H )).We shall finally show that Z : Ω → B ( H ) is τ -weak ∗ -continuous. We have alreadyshown that for each F ∈ F ( H ) we have b p F D b p F | b p F C b p F ( . ) = b p F [ Z , b p F . b p F ] b p F = b p F [ b p F Z b p F , . ] b p F , and hence, by Theorem 2.2, there exists Z F ∈ C (Ω , p F B ( H ) p F ) such that b p F Z b p F − Z F ∈ Z ( C (Ω , p F B ( H ) p F ) = C (Ω) ⊗ I F . Therefore, b p F Z b p F ∈ C (Ω , p F B ( H ) p F ) ⊆ C (Ω , B ( H )) . (2.5)Pick a normal functional φ ∈ B ( H ) ∗ . Given ε >
0. If we identify B ( H ) ∗ with thetrace-class operators, we can easily find a finite projection p F with F ∈ F ( H ) suchthat k φ − φ ( p F .p F ) k < ε . Combining this fact with (2.5) we can easily deduce that φ ◦ Z : Ω → C is continuous, which finishes the proof. (cid:3) Remark 2.11.
We cannot assure, in general that the mapping Z : Ω → B ( H ) is τ -norm continuous. Let us take an infinite dimensional Hilbert space H . Let p n bea sequence of mutually orthogonal rank one projections in B ( H ). The von Neumannsubalgebra C of B ( H ) generated by the p n ’s is C ∗ -isomorphic to ℓ ∞ .The set Ω := { a ∈ C ∼ = ℓ ∞ : k a k ≤ } is weak ∗ -closed and hence (Ω , τ = weak ∗ ) is acompact Hausdorff space. Let Z : Ω → C ⊂ B ( H ) be the identity mapping, which is τ -weak ∗ -continuous and bounded. Clearly, Z is not weak ∗ -norm continuous.It is known that a bounded net ( a λ ) in ℓ ∞ converges in the weak ∗ -topology to anelement a if and only if for each natural n , ( | a λ ( n ) − a ( n ) | ) →
0. Suppose ( a λ ) ⊂ Ω,( a λ ) → a in Ω. Since a λ → a in the weak ∗ -topology of C , it is not hard to see that( a λ − a )( a λ − a ) ∗ → ∗ -topology of C , and hence ( a λ − a )( a λ − a ) ∗ → ∗ -topology of B ( H ).We claim that, for each finite rank projection p ∈ B ( H ), the mappings Z p, and pZ both are τ -norm continuous. Indeed, let ( a λ ) → a in Ω (i.e. in the weak ∗ -topology of B ( H )). The arguments given in the above paragraph show that ( a λ − a )( a λ − a ) ∗ → ∗ -topology of B ( H ) . Since the operator U p : B ( H ) → B ( H ), x pxp hasfinite rank, we deduce that k p ( a λ − a ) k = k ( a λ − a ) p k = k p ( a λ − a )( a λ − a ) ∗ p k λ → X ∈ C (Ω , K ( H )), Z X and XZ both lie in C (Ω , K ( H )). We may assume, without loss of generality, that k X k ≤
1. Let ( a λ ) → a in Ω, and let ε >
0. By the continuity of X , there exists λ ∈ Λ such that k X ( a λ ) − X ( a ) k < ε for every λ > λ . Since X ( a ) ∈ K ( H ), we can find a finite rankprojection p ∈ B ( H ) such that k X ( a ) − pX ( a ) k < ε . By the τ -norm continuity of Z p , there exists λ ≥ λ such that k ( Z ( a λ ) − Z ( a )) p k < ε , for all λ ≥ λ . Thereforefor all λ ≥ λ we have k ( Z X )( a λ ) − ( Z X )( a ) k ≤ k Z ( a λ )( X ( a λ ) − X ( a )) k + k ( Z ( a λ ) − Z ( a )) X ( a ) k≤ ε k ( Z ( a λ ) − Z ( a ))( X ( a ) − pX ( a )) k + k ( Z ( a λ ) − Z ( a )) pX ( a ) k≤ ε ε k ( Z ( a λ ) − Z ( a )) p k ≤ ε. This shows that Z X ∈ C (Ω , K ( H )) . The statement for XZ follows similarly. Then D : C (Ω , K ( H )) → C (Ω , K ( H )), D ( X ) = [ Z , X ] = Z X − XZ is a derivation on C (Ω , K ( H )).Though it is not true that every derivation on C (Ω , K ( H )) is inner, and henceTheorem 2.4 cannot be applied, we can extend our study to weak-2-local derivationson C (Ω , K ( H )). Theorem 2.12.
Let H be a complex Hilbert space. Then every weak-2-local derivation ∆ : C (Ω , K ( H )) → C (Ω , K ( H )) is a ( linear ) derivation.Proof. Let us fix t in Ω and φ ∈ K ( H ) ∗ . For the functional φ ⊗ δ t ∈ C (Ω , K ( H )) ∗ , A , Γ δ t ( A ) in C (Ω , K ( H )), there exists a derivation D = D A, Γ δ t ( A ) ,φ ⊗ δ t on C (Ω , K ( H ))such that ( φ ⊗ δ t )∆( A ) = φ ⊗ δ t D ( A ) and ( φ ⊗ δ t )∆Γ δ ( A ) = ( φ ⊗ δ t ) D Γ δ t ( A ). Since,by Proposition 2.10, δ t D Γ δ t ( A ) = δ t D ( A ) , we deduce that δ t ∆Γ δ t = δ t ∆ . (2.6)Arguing as in the proof of Theorem 2.4, we deduce that δ t ∆Γ : K ( H ) → K ( H ) isa weak-2-local derivation. Theorem 3.2 in [7] assures that δ t ∆Γ is a linear derivation.Applying the identity in (2.6), and following the same arguments given at the end ofthe proof of Theorem 2.4, we obtain δ t ∆( X + Y ) = δ t ∆( X ) + δ t ∆( Y ), for every X, Y in C (Ω , K ( H )) , which proves the first statement. (cid:3) The machinery developed in previous results reveals a pattern which is stated in thenext result, whose proof has been outlined in Theorems 2.4 and 2.12.
Theorem 2.13.
Let Ω be a compact Hausdorff space. Suppose A is a C ∗ -algebrasatisfying the following hypothesis: ( a ) Every weak-2-local derivation on A is a ( linear ) derivation; ( b ) For each derivation D : C (Ω , A ) → C (Ω , A ) the identity δ t D Γ δ t ( A ) = δ t D ( A ) , holds for every t ∈ Ω , A ∈ C (Ω , A ) .Then every weak-2-local derivation on C (Ω , A ) is a ( linear ) derivation. ✷ Proposition 2.10 above shows that K ( H ) satisfies the hypothesis ( b ) in the previoustheorem. We recall that every compact C ∗ -algebra B is C ∗ -isomorphic to the c -sum( L i ∈ I K ( H i )) c , where each H i is a complex Hilbert space (see [2]). Let us define aparticular approximate unit in B . Let F ( I ) denote the finite subsets of I . Let Λ bethe set of all finite tuples of the form ( F i ) i ∈ J = ( F i , . . . , F i k ), where J = { i , . . . , i k } ∈F ( I ), and F i j ∈ F ( H i j ). We shall say that ( F i ) i ∈ J ≤ ( G i ) i ∈ J if J ⊆ J and F i ⊆ G i for TABILITY OF DERIVATIONS UNDER WEAK-2-LOCAL CONTINUOUS PERTURBATIONS 13 every i ∈ J . We set p ( F i ) i ∈ J := X i ∈ J p Fi ∈ B . The net ( p ( F i ) i ∈ J ) ( Fi ) i ∈ J ∈ Λ is an approximateunit in B . When in the proof of Proposition 2.10 the approximate unit ( b p F ) F ∈ F ( H ) isreplaced with ( p ( F i ) i ∈ J ) ( Fi ) i ∈ J ∈ Λ , and Proposition 2.8 is substituted for Remark 2.9, thearguments remain valid to prove the following: Proposition 2.14.
Let D : C (Ω , B ) → C (Ω , B ) be a derivation, where Ω is a compactHausdorff space and B is a compact C ∗ -algebra. Then there exists a bounded mapping Z : Ω → B ∗∗ satisfying D ( X )( t ) = [ Z ( t ) , X ( t )] , for every X ∈ C (Ω , B ) . In particular,for each t in Ω we have δ t D Γ δ t ( A ) = δ t D ( A ) , for every A ∈ C (Ω , B ) . ✷ Finally, combining Proposition 2.14 with Proposition 3.4 in [7] and Theorem 2.13we culminate the study of weak-2-local derivation on the C ∗ -algebra of all continuousfunctions on a compact Hausdorff space with values on a compact C ∗ -algebra. Theorem 2.15.
Let B be a compact C ∗ -algebra, and let Ω be a compact Hausdorffspace. Then every weak-2-local derivation ∆ : C (Ω , B ) → C (Ω , B ) is a ( linear ) deriva-tion. In particular, every 2-local derivation on C (Ω , B ) is a ( linear ) derivation. ✷ We have already commented that, by a result of Sh. Ayupov and K. Kudaybergenovevery 2-local derivation on a von Neumann algebra M is a derivation [5], and theproblem whether the same statement remains true for general C ∗ -algebras remainsopen. We can throw new light onto the study of 2-local derivations on C ∗ -algebras.Theorem 2.13 above admits the following corollary. Corollary 2.16.
Let Ω be a compact Hausdorff space. Suppose B is a C ∗ -algebrasatisfying the following hypothesis: ( a ) Every 2-local derivation on B is a ( linear ) derivation; ( b ) For each derivation D : C (Ω , B ) → C (Ω , B ) the identity δ t D Γ δ t ( A ) = δ t D ( A ) , holds for every t ∈ Ω , A ∈ C (Ω , B ) .Then every 2-local derivation on C (Ω , B ) is a ( linear ) derivation. ✷ Let M be a von Neumann algebra and let Ω be a compact Hausdorff space. Thepreviously mentioned theorem of Ayupov and Kudaybergenov assures that M satisfieshypothesis ( a ) in the above corollary. Theorem 2.2 implies that C (Ω , M ) also satisfieshypothesis ( b ), we therefore obtain the following strengthened version of Corollary 2.6. Corollary 2.17.
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Escuela Polit´ecnica Superior de Alcoy, IUMPA, Universitat Polit´ecnica de Valen-cia, Plaza Ferr´andiz y Carbonell 1, 03801 Alcoy, Spain
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