Closed ideals and Lie ideals of minimal tensor product of certain C*-algebras
aa r X i v : . [ m a t h . OA ] M a r CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OFCERTAIN C ∗ -ALGEBRAS BHARAT TALWAR AND RANJANA JAIN
Abstract.
For a locally compact Hausdorff space X and a C ∗ -algebra A with only finitely manyclosed ideals, we discuss a characterization of closed ideals of C ( X, A ) in terms of closed idealsof A and certain (compatible) closed subspaces of X . We further use this result to prove that aclosed ideal of C ( X ) ⊗ min A is a finite sum of product ideals. We also establish that for a unital C ∗ -algebra A , C ( X, A ) has centre-quotient property if and only if A has centre-quotient property.As an application, we characterize the closed Lie ideals of C ( X, A ) and identify all closed Lieideals of C ( X ) ⊗ min B ( H ), H being a separable Hilbert space. Introduction
Let A ⊗ α B denote the completion of the algebraic tensor product of two C ∗ -algebras A and B under an algebra cross norm k · k α . A natural question arises whether the closed ideals of A ⊗ α B can be identified in terms of the closed ideals of A and B or not. It is known that the closed idealsof the Banach algebras A ⊗ h B , A ⊗ γ B and A b ⊗ B are directly related to the closed ideals of A and B , where ⊗ h , ⊗ γ and b ⊗ are the Haagerup tensor product, Banach space projective tensor productand operator space projective tensor product, respectively. Interestingly, if either A or B possessesfinitely many closed ideals then every closed ideal of these spaces is a finite sum of product ideals(see, for instance, [1, 5, 7]). However, in 1978, Wassermann [16] established an astonishing result thatnot every closed ideal of B ( H ) ⊗ min B ( H ) is a finite sum of product ideals, ⊗ min being the minimal C ∗ -tensor norm. Recall that for every x ∈ A ⊗ B , k x k min = sup {k ( π ⊗ π )( x ) k} , where π and π run over all representations of A and B respectively (see [14] for details). In the present article,we prove that this anamoly can be removed by assuming one of the C ∗ -algebras to be commutativewith the help of a totally different technique than those used for ⊗ h , ⊗ γ and b ⊗ .A Banach algebra B naturally imbibes a Lie algebra structure with the Lie bracket given by[ a, b ] = ab − ba for every a, b ∈ B . A closed subspace L of B is said to be a Lie ideal if [
B, L ] ⊆ L where [ B, L ] = span { [ b, l ] : b ∈ B, l ∈ L } . The closed Lie ideals for C ∗ -algebras are extensivelystudied, one may refer to the expository article [11] for details. Recently some research has beendone to identify the closed Lie ideals for the various tensor products of C ∗ -algebras. In [4, Section5], [6, Section 4]) the closed Lie ideals of C ( X ) ⊗ min A have been characterized in terms of closedsubspaces of X , X being a locally compact Hausdorff space and A being a simple C ∗ -algebra withat most one tracial state. However, if A is not simple nothing is known about the closed Lie idealsof such spaces. In this article, we discuss a characterization of closed Lie ideals of C ( X, A ), for any C ∗ -algebra A .We present a brief summary of the main results of the article. In Section 2, we first establishan appropriate (surjective) correspondence between a class of closed subspaces of X and the closedideals of C ( X, A ), for any C ∗ -algebra A . Interestingly, this correspondence turns out to be bijectiveif A has finitely many closed ideals. We use this correspondence to identify the image of a closedideal of C ( X, A ) in C ( X ) ⊗ min A under the canonical isomorphism. This will pave our way toestablish that every closed ideal of C ( X ) ⊗ min A is a finite sum of product ideals, where X is a Mathematics Subject Classification.
Key words and phrases.
Closed Lie ideal, C ∗ -algebra, minimal C ∗ -norm, tensor product.The first named author was supported by a Junior Research Fellowship of CSIR with file number09/045(1442)/2016-EMR-I. locally compact Hausdorff space and A is a C ∗ -algebra with finitely many closed ideals. In Section3, we characterize the closed Lie ideals of C ( X, A ) in terms of some closed subspaces of X . Toobtain a better picture of closed Lie ideals of C ( X ) ⊗ min A , we prove that if A is unital then thecentre-quotient property of A passes to C ( X, A ) and vice-versa. As an application we establish aninteresting result that a closed subspace L of C ( X ) ⊗ min B ( H ) is a Lie ideal if and only if thereexist closed subspaces S , S of X with S ⊆ S and a closed subspace K of C ( X ) ⊗ C L = J ( S ) ⊗ K ( H ) + J ( S ) ⊗ B ( H ) + K , where H is a separable Hilbert space and for F ⊆ X , J ( F ) := { f ∈ C ( X ) : f ( F ) ⊆ { }} .2. Closed ideals of C ( X ) ⊗ min A Let X be a locally compact Hausdorff space and A be any C ∗ -algebra. It is a well known factthat there is a bijective correspondence between the closed subspaces of X and the closed ideals of C ( X ) given by F ↔ J ( F ). However, if we move from complex valued functions to the vector valuedfunctions, such a correspondence is not known. Although, in the literature, it is established thatevery closed ideal of C ( X, A ) is of the form { f ∈ C ( X, A ) : f ( x ) ∈ I x , ∀ x ∈ X } where for every x ∈ X , I x is a closed ideal of A [12, V.26.2.1], but this description fails to be fruitful while movingfrom C ( X, A ) to C ( X ) ⊗ min A in order to determine the closed ideals.We first generalize the former notion to the continuous vector valued functions by establishing acorrespondence between the closed subspaces of X and closed ideals of C ( X, A ). This correspondencewill further enable us to characterize closed ideals of C ( X ) ⊗ min A in terms of closed ideals of A andsubspaces of X , when A has finitely many closed ideals. This is due to the fact that there exists anisometric ∗ -isomorphism ˜ ϕ : C ( X ) ⊗ min A → C ( X, A ), which takes f ⊗ a to af for every f ∈ C ( X )and a ∈ A , where ( af )( x ) = f ( x ) a (see, [14, Theorem 4.14 (iii)], [9, Proposition 1.5.6]).Let us first fix some notations for further use. For any t ∈ N , the set { , , , . . . , t } is be denoted by N t . The spaces C b ( X, A ) and C c ( X, A ), as usual, denote the C ∗ -algebras of all bounded continuousfunctions and compactly supported continuous functions, respectively, from X to A endowed withsup norm. For a non-unital C ∗ -algebra A , ˜ A will denote its unitization. For a locally compactHausdorff space X and any function g ∈ C ( X ), we define ˆ g ∈ C ( X, ˜ A ) (resp., ˆ g ∈ C ( X, A )) byˆ g ( x ) = g ( x )1, where 1 is the unit of ˜ A (resp., of A ) if A is non unital (resp., if A is unital). For any a ∈ A , we denote by a ′ the constant function in C b ( X, A ) such that a ′ ( x ) = a for every x ∈ X .For an indexing set ∆, let S = { S i } i ∈ ∆ and T = { T i } i ∈ ∆ be collections of subspaces of some sets Y and Z . We define S to be compatible with T if whenever for some subset γ of ∆, ∩ j ∈ γ T j = T i for some i ∈ ∆, then ∩ j ∈ γ S j = S i . For a locally compact Hausdorff space X , α ∈ ∆ and T as above, define a collection T αT := { S = { S i } i ∈ ∆ : S i is a closed subspace of X for every i ∈ ∆ , S is compatible with T and S α = X } . Theorem 2.1.
Let X be a locally compact Hausdorff space, A be a C ∗ -algebra and I = { I i } i ∈ ∆ bethe collection of all closed ideals of A with I β = A . Then there exists a surjection θ from T β I into K ,the set of all closed ideals of C ( X, A ) .Proof. Define θ : T β I → K by θ ( S ) = J ( S ) where J ( S ) = { f ∈ C ( X, A ) : f ( S i ) ⊆ I i , ∀ i ∈ ∆ } .Clearly θ is well defined. To see that θ is onto, consider J ∈ K . For each i ∈ ∆, set S i = ∩ f ∈ J f − ( I i ).If I i = ∩ j ∈ γ I j for some subset γ of ∆, then S i = ∩ f ∈ J f − ( I i ) = ∩ f ∈ J f − ( ∩ j ∈ γ I j ) = ∩ f ∈ J ∩ j ∈ γ f − ( I j ) = ∩ j ∈ γ ∩ f ∈ J f − ( I j ) = ∩ j ∈ γ S j , so that S = { S i } i ∈ ∆ is compatible with I . Since, S β = ∩ f ∈ J f − ( I β ) = ∩ f ∈ J f − ( A ) = X , we obtain S ∈ T β I . Clearly J ⊆ J ( S ), so it is sufficient to prove that J is dense in J ( S ). For this, let f ∈ J ( S )be non-zero and ǫ > x ∈ X , there exists an element h x ∈ J such that k h x ( x ) − f ( x ) k ≤ ǫ .Indeed, if γ ′ = { i ∈ ∆ : x ∈ S i } , then x ∈ ∩ i ∈ γ ′ S i and f ( x ) ∈ ∩ i ∈ γ ′ I i = I j for some j ∈ ∆. Let I r denote the closed ideal of A generated by the set { g ( x ) : g ∈ J } . Then x ∈ S r which implies that f ( x ) ∈ I r . Hence k P ki =1 a i g i ( x ) b i − f ( x ) k ≤ ǫ for some a , a , . . . a k , b , b , . . . b k in A (resp., in ˜ A ) LOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT 3 if A is unital (resp., if A is non unital). From [8, Corollary 4.2.10], we know that J is a closed idealof C ( X, ˜ A ), thus we have a function h x = P ni =1 a ′ i g i b ′ i in J which satisfies the required condition.Denote by ˜ X = X ∪ {∞} , the one point compactification of X . For each x ∈ X , let ˜ h x and ˜ f be the continuous extensions of h x and f to ˜ X which take ∞ to 0, and let ˜ h x be the continuousextension of h x , the zero function, to ˜ X such that ˜ h x ( ∞ ) = 0. As ˜ h x ( ∞ ) = ˜ f ( ∞ ) = 0, there existsa neighbourhood V x of ∞ in ˜ X such that k ˜ h x ( y ) − f ( y ) k ≤ ǫ for every y ∈ V x . Further ˜ f and˜ h x are continuous at x ∈ X , so there exists a neighbourhood V x of x such that k h x ( y ) − f ( y ) k ≤ ǫ for every y ∈ V x .The collection { V x , V x } x ∈ ˜ X is an open cover for ˜ X . As ˜ X is compact, there is a finite subcover { V x , V x , V x , . . . , V x k } of ˜ X . Then there exists a partition of unity { g , g , g , . . . g k } subordinateto this subcover such that for every i ∈ { , , , . . . , k } , g i ∈ C ( ˜ X ), 0 ≤ g i ≤
1, supp( g i ) ⊆ V x i andfor each x ∈ ˜ X , P ki =0 g i ( x ) = 1. Now for ˜ h = P ki =0 ˆ g i ˜ h x i ∈ C ( ˜ X, A ), its restriction to X , given by h = P ki =0 ˆ g i | X h x i , is an element in J . Hence for any y ∈ X , we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =0 ˆ g i | X ( y ) h x i ( y ) − f ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X i =0 ˆ g i ( y ) h x i ( y ) − k X i =0 ˆ g i ( y ) f ( y ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k X i =0 k ˆ g i ( y ) kk f ( y ) − h x i ( y ) k≤ k X i =0 g i ( y ) ǫ ≤ ǫ, thus k h − f k ≤ ǫ . (cid:3) Remark 2.2.
Note that the notation J ( S ) given in the above theorem coincides with the previousnotation of J ( F ) by taking S = { F, X } and I = {{ } , A = C } . It is interesting to note that if A possesses only finitely many closed ideals, then the abovedefined mapping θ turns out to be injective, hence providing a nice characterization of the closedideals of C ( X, A ). We would like to mention that the case when A is a simple C ∗ -algebra isdiscussed in [6, Proposition 4.1], wherein we proved that every closed ideal of C ( X, A ) is of the form { f ∈ C ( X, A ) : f ( x ) = 0 , ∀ x ∈ F } , for some closed subspace F of X. Theorem 2.3.
Let X be a locally compact Hausdorff space, A be a C ∗ -algebra and I = { I , I , . . . , I n = A } , n ∈ N , be the set of all closed ideals of A . Then the map θ from T n I to the set of all closedideals of C ( X, A ) is a bijection. In particular, every closed ideal of C ( X, A ) is precisely of the form J ( S ) = { f ∈ C ( X, A ) : f ( S i ) ⊆ I i , ∀ i ∈ N n } , for a unique S = { S i } compatible with I .Proof. Let S = { S i } i ∈ N n and S ′ = { S ′ i } i ∈ N n be two distinct elements of T n I so that S i = S ′ i forsome i ∈ N n \ { n } . Without loss of generality, we may assume that there exists x ∈ S i \ S ′ i .Define a non-empty subset γ = { j ∈ N n : x / ∈ S ′ j } of N n . By Urysohn’s Lemma [13, Theorem2.12] , as V x = ( ∪ i ∈ γ S ′ i ) c is an open set containing x , there exists g ∈ C c ( X ) such that g ( x ) = 1, g ( X ) ⊆ [0 ,
1] and supp ( g ) ⊆ V x . It is easy to see that ( ∩ j ∈ γ c I j ) \ I i = ∅ , because if ∩ j ∈ γ c I j ⊆ I i then ∩ j ∈ γ c S ′ j ⊆ S ′ i , which is not true as x ∈ ∩ j ∈ γ c S ′ j but x / ∈ S ′ i . Let a ∈ ( ∩ j ∈ γ c I j ) \ I i . Consider thefunction h = a ′ ˆ g ∈ C ( X, A ). Observe that h / ∈ J ( S ), since x ∈ S i but h ( x ) = a ′ ( x )ˆ g ( x ) = a / ∈ I i .However, we assert that h ∈ J ( S ′ ) which proves J ( S ) = J ( S ′ ). For k ∈ γ , y ∈ S ′ k implies y ∈ V cx ,so that h ( y ) = a ˆ g ( y ) = 0 ∈ I k . Also, if k ∈ γ c , then h ( y ) = a ˆ g ( y ) ∈ ( ∩ j ∈ γ c I j ) ⊆ I k for every y ∈ S ′ k . (cid:3) In the quest of proving the main result regarding the characterization of closed ideals of C ( X ) ⊗ min A , we require few more ingredients. B. TALWAR AND R. JAIN
Lemma 2.4.
Let X be a locally compact Hausdorff space and A be a C ∗ -algebra. Then for a closedsubspace C of A and a closed ideal J ( Y ) of C ( X ) , Y ⊆ X being closed, there is an isometricisomorphism of Banach spaces J ( Y ) ⊗ C min ∼ = { f ∈ C ( X, A ) : f ( Y ) = { } , f ( X ) ⊆ C } . Proof.
Denote by J the closed subspace { f ∈ C ( X, A ) : f ( Y ) = { } , f ( X ) ⊆ C } of C ( X, A ), and I = J ( Y ). Let ϕ denote the restriction of ˜ ϕ to I ⊗ C min , where ˜ φ : C ( X ) ⊗ min A → C ( X, A )is the isometric ∗ -isomorphism as discussed earlier. Then for P nj =1 f j ⊗ c j ∈ I ⊗ C , we have ϕ ( P j ∈ N n f j ⊗ c j )( Y ) = { } , so that ϕ ( I ⊗ C ) ⊆ J . Since ϕ is an isometry, it is sufficient to provethat ϕ ( I ⊗ C ) is dense in J .Let g ∈ J and ǫ > J is also a closed subspace of C ( X, C ) and C c ( X, C ) isdense in C ( X, C ), there exists a function h ∈ C c ( X, C ) such that k g − h k < ǫ/
2. Let K := supp( h ), B r ( b ) := { c ∈ C : k c − b k < r } and B × r ( b ) := B r ( b ) \ { } , where b ∈ C and r >
0. Since k h ( y ) k = k g ( y ) − h ( y ) k < ǫ/ y ∈ Y , the collection { h − ( B × ǫ ( h ( x )) \ h ( Y )) : x ∈ K \ Y }∪ h − ( B ǫ (0)) forms an open cover of the compact set K . Fix a finite subcover, say, h − ( B ǫ (0)) ∪{ h − ( B × ǫ ( h ( x i )) \ h ( Y )) : 1 ≤ i ≤ n } . Since K is a compact subspace of a locally compact Hausdorffspace X , there exists a partition of unity subordinate to this finite subcover, i.e. there exist functions f , f , . . . , , f n in C c ( X ) such that 0 ≤ f i ≤ ≤ i ≤ n , supp( f ) ⊆ U := h − ( B ǫ (0)),supp( f i ) ⊆ U i := h − ( B × ǫ ( h ( x i )) \ h ( Y )) for all 1 ≤ i ≤ n and P ni =0 f i ( x ) = 1 for x ∈ K (see [13,Theorem 2.13]).Let V = ( P ni =0 f i ) − (0 , / V ∩ ( ∪ ni =0 U i ) is an open set containing K . Pick ˜ f ∈ C c ( X )such that ˜ f is 1 on K , supp( ˜ f ) ⊆ V ∩ ( ∪ ni =0 U i ) and 0 ≤ ˜ f ≤
1. Then for ˜ f i = ˜ f f i , supp( ˜ f i ) ⊆ V ∩ U i because supp( f i ) ⊆ U i and supp( ˜ f ) ⊆ V . Now for x ∈ K , we have n X i =0 ˜ f i ( x ) = n X i =0 ˜ f ( x ) f i ( x ) = n X i =0 f i ( x ) = 1 . Also notice that 0 ≤ P ni =0 ˜ f i ≤ / x ∈ V ∩ ( ∪ ni =0 U i ), P ni =0 ˜ f i ( x ) = P ni =0 ˜ f ( x ) f i ( x ) ≤ P ni =0 f i ( x ) = 3 / x ∈ ( V ∩ ( ∪ ni =0 U i )) c , we have P ni =0 ˜ f i ( x ) = 0 .Now for 1 ≤ i ≤ n , the open set U i , and thus V ∩ U i is disjoint from Y so that P ni =1 ˜ f i ⊗ h ( x i ) ∈ I ⊗ C . Fix x ∈ K c , then for each x ∈ X k h ( x ) − n X i =1 ˜ f i ( x ) h ( x i ) k = k h ( x ) n X i =0 ˜ f i ( x ) − n X i =0 ˜ f i ( x ) h ( x i ) k≤ n X i =0 k h ( x ) − h ( x i ) k ˜ f i ( x )= X i : x ∈ U i ∩ V k h ( x ) − h ( x i ) k ˜ f i ( x ) (since supp( ˜ f i ) ⊆ U i ∩ V ) < ǫ. Hence we obtain k g − ϕ ( P ni =1 ˜ f i ⊗ h ( x i )) k < ǫ , proving that ϕ ( I ⊗ C ) is dense in J . (cid:3) As a consequence of the above result, we have an interesting observation which identifies certainclosed ideals of C ( X, A ) with some closed ideals of C ( X ) ⊗ min A . Corollary 2.5.
Let Y be a closed subspace of a locally compact Hausdorff space X . For any closedideal I of a C ∗ -algebra A , we have C ( X ) ⊗ min I + J ( Y ) ⊗ min A = { f ∈ C ( X, A ) : f ( Y ) ⊆ I } Proof.
Let I = { I i } i ∈ ∆ be the set of all closed ideals of A with I β = A , for some β ∈ ∆, and set I = I t . If t = β then the result is trivial. Otherwise, let J = C ( X ) ⊗ min I t + J ( Y ) ⊗ min A and LOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT 5 J = { f ∈ C ( X, A ) : f ( Y ) ⊆ I t } . Then, by Theorem 2.1, there exist elements S = { S i } i ∈ ∆ and S ′ = { S ′ i } i ∈ ∆ in T β I such that J = J ( S ) and J = J ( S ′ ). It is sufficient to prove that S = S ′ .We first mention a common trick used in the proof. For any x ∈ X and a closed subspace F of X with x / ∈ F , Urysohn’s Lemma implies that there exists f ∈ C c ( X ) such that f ( x ) = 1 and f ( F ) = 0.Then for any fixed a ∈ A and any y ∈ X , there exists a function g ( y ) := f ( y ) a in C ( X, A ) suchthat g ( x ) = a and g vanishes on F .We now claim that S t = ∩ f ∈ J f − ( I t ) = Y . For f ∈ J , f = f + f for some f ∈ C ( X ) ⊗ min I t and f ∈ J ( Y ) ⊗ min A . Thus, for any y ∈ Y , f ( y ) = f ( y ) + f ( y ) ∈ I t as f ( y ) = 0 by Lemma 2.4, sothat Y ⊆ S t . For the reverse containment assume that Y ( S t . Pick a ∈ A \ I t and x ∈ S t \ Y , thenthere exists a function in C ( X, A ) which vanishes on Y and maps x to a which is a contradictionto the definition of S t . On the similar lines, using the fact that S ′ t = ∩ g ∈ J g − ( I t ), one can easilydeduce that S t = Y = S ′ t .Now fix i ∈ ∆ with i = β, i = t . Note that J = ∩ i ∈ ∆ { f ∈ C ( X, A ) : f ( S ′ i ) ⊆ I i } , so that G t := { f ∈ C ( X, A ) : f ( S ′ t ) ⊆ I t } ⊆ { f ∈ C ( X, A ) : f ( S ′ i ) ⊆ I i } = G i (say), for every i ∈ ∆.Case(i): I i ( I t , then S ′ i = ∅ = S i . Because for y ∈ S ′ i ⊆ S ′ t and a ∈ I t \ I i , there exists a functionin C ( X, A ) which takes y to a . Then such a function is in G t but not in G i . Also, if there exists an x ∈ S i ⊆ S t = Y , then a ′ g as defined above will be a function in C ( X ) ⊗ min I t ⊂ J which takes anelement x of S i to a which does not belong to I i , which is a contradiction to the definition of S i .Case(ii): I t ( I i , then S i = S t = S ′ t = S ′ i . To see this, if S ′ t is properly contained in S ′ i , thenfor x ∈ S ′ i \ S ′ t and a / ∈ I i , there exists a function in C ( X, A ) which takes x to a and S ′ t to 0.This function belongs to G t but does not belong to G i , which is a contradiction. Similarly, if S t isproperly contained in S i , then for x ∈ S i \ S t and a / ∈ I i , there is a function in J ( Y ) ⊗ min A ⊂ J which takes x outside I i which contradicts the definition of S i .Case(iii): I i is neither a subset nor a superset of I t , then we claim that S i = S ′ i = ∅ . If I j = I t ∩ I i ,then I j ( I t so that by Case(i), S t ∩ S i = S j = ∅ = S ′ j = S ′ t ∩ S ′ i . Now, for x ∈ S ′ i , x / ∈ S ′ t , as argued inCase (ii), we obtain that G t is not contained in G i , which is a contradiction, thus S ′ i = ∅ . Similarly,for x ∈ S i , x is not a member of Y = S t . So for any a ∈ I t \ I i , applying the technique mentioned inthe beginning, we get a function g in J ( Y ) ⊗ min A ⊆ J such that g ( x ) / ∈ I i , a contradiction.This proves that S = S ′ and hence J = J . (cid:3) We are now ready to prove the main result of this section. Note that a product ideal is a closedideal of the form I ⊗ min J , where I and J are closed ideals of A and B , respectively. Theorem 2.6.
Let X be a locally compact Hausdorff space and A be a C ∗ -algebra with finitelymany closed ideals, say, I , I , . . . , I n with I = { } and I n = A . Then for any closed ideal K of C ( X ) ⊗ min A , there exists S = { S i } i ∈ N n ∈ T n I , where I = { I i } i ∈ N n , such that K = n X j =2 J ( ∪ k ∈ γ j S k ) ⊗ min I j , where γ j = { i ∈ N n : I j * I i } , for every j ∈ { , , . . . , n } .In particular, every closed ideal of C ( X ) ⊗ min A is a finite sum of product ideals.Proof. By Theorem 2.3, there exists S = { S i } i ∈ N n ∈ T n I such that K = J ( S ) = { f ∈ C ( X, A ) : f ( S i ) ⊆ I i , i ∈ N n } . Set K ′ = P nj =2 J ( ∪ k ∈ γ j S k ) ⊗ min I j , then by Lemma 2.4, K ′ can be considered asa closed ideal of C ( X, A ). By virtue of Theorem 2.1, it is sufficient to prove that S i = ∩ f ∈ K ′ f − ( I i )for every i ∈ N n .It is clear that S n = X = ∩ f ∈ K ′ f − ( A ). Fix i ∈ N n − and consider any x ∈ S i . For f ∈ K ′ , f = f + f + · · · + f n , where f r ∈ J ( ∪ k ∈ γ r S k ) ⊗ min I r for every r ∈ { , , . . . , n } . Then for any such r , either i ∈ γ r or i ∈ γ cr . If i ∈ γ r then f r ( x ) = 0 ∈ I i . If i ∈ γ cr then f r ( x ) ∈ I r ⊆ I i . These twoconclusions together imply that S i ⊆ ∩ f ∈ K ′ f − ( I i ).Next, pick x / ∈ S i and define α i = { j ∈ N n : I j * I i } . Note that α i is non empty as n ∈ α i .It is sufficient to prove the existence of a function f ∈ K ′ such that f ( x ) / ∈ I i . We shall actually B. TALWAR AND R. JAIN prove that such a function exists in the subset P r ∈ α i J ( ∪ k ∈ γ r S k ) ⊗ min I r of K ′ . It is further enoughto prove that there exists an r ∈ α i such that x / ∈ ∪ k ∈ γ r S k , so that the required function f existsin J ( ∪ k ∈ γ r S k ) ⊗ min I r . In fact, by Urysohn’s Lemma there exists a function g ∈ C c ( X ) such that0 ≤ g ≤ g ( x ) = 1 and g ( ∪ k ∈ γ r S k ) = { } . Then by Lemma 2.4, for a ∈ I r \ I i (since I r * I i ), thefunction a ′ ˆ g serves the purpose. We claim that ∩ r ∈ α i ( ∪ k ∈ γ r S k ) = S i , which will ensure the existenceof such an r .When r ∈ α i , we have i ∈ γ r and hence S i ⊆ ∩ r ∈ α i ( ∪ k ∈ γ r S k ). We now prove the reverseinclusion. Set α i = { r , r , . . . , r q } and for each r j ∈ α i , let there be p r j number of elements in γ r j ,say γ r j = { r j,t rj : 1 ≤ t r j ≤ p r j } . So \ r ∈ α i ( ∪ k ∈ γ r S k ) = [ ≤ t rj ≤ p rj ( S r ,tr ∩ S r ,tr ∩ · · · ∩ S r q,trq ) . We have obtained that ∩ r ∈ α j ( ∪ k ∈ γ r S k ) is a union of Π qj =1 p r j objects, each of which is anintersection of q objects which looks like S r ,tr ∩ S r ,tr ∩ · · · ∩ S r q,trq . Pick an ideal I j r ,tr ∩ I j r ,tr ∩ · · · ∩ I j rq,trq . Then there exists an m ∈ N n such that I m = I r ,tr ∩ I r ,tr ∩ · · · ∩ I r q,trq , andhence S m = S r ,tr ∩ S r ,tr ∩ · · · ∩ S r q,trq . If S m ⊆ S i , we are done. Otherwise, S m * S i will imply I m * I i and hence m ∈ α i . Thus m = r l for some l ∈ { , , . . . k } . Then r l,t rl ∈ γ r l which implies( I m =) I r l * I r l,trl , which is a contradiction to the fact that I m = I r ,tr ∩ I r ,tr ∩ · · · ∩ I r q,trq . (cid:3) Let us demonstrate the above theorem with the help of few examples.
Example 2.7.
Let H be a separable Hilbert space and A = B ( H ) ⊕ B ( H ) . For I = { I , . . . , I = A } ,the lattice of closed ideals of A is given in the diagram below. B ( H ) ⊕ B ( H )(= I ) B ( H ) ⊕ K ( H )(= I ) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ K ( H ) ⊕ B ( H )(= I ) j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ B ( H ) ⊕ { } (= I ) O O K ( H ) ⊕ K ( H )(= I ) j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ { } ⊕ B ( H )(= I ) O O K ( H ) ⊕ { } (= I ) O O ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ { } ⊕ K ( H )(= I ) O O j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ { } ⊕ { } (= I ) j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Looking at the diagram, we obtain that γ = { , , } , γ = { , , } , γ = { , , , , , } , γ = { , , , , } , γ = { , , , , , } , γ = { , , , , , , } , γ = { , , , , , , } and γ = { , , , , , , , } . Hence, every closed ideal of C ( X ) ⊗ min A is of the form: J ( S ) ⊗ min I + J ( S ) ⊗ min I + J ( S ) ⊗ min I + J ( S ∪ S ) ⊗ min I + J ( S ) ⊗ min I + J ( S ) ⊗ min I + J ( S ) ⊗ min I + J ( S ∪ S ) ⊗ min I , for a unique S = ( S , S , . . . , S ) ∈ T I , where a pictorial LOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT 7 representation of S is the following (here an arrow from S j to S k means that S j ⊆ S k , for j, k ∈ N ): S S > > ⑥⑥⑥⑥⑥⑥⑥ S ` ` ❆❆❆❆❆❆❆ S O O S ` ` ❆❆❆❆❆❆❆ > > ⑥⑥⑥⑥⑥⑥⑥ S O O S O O > > ⑥⑥⑥⑥⑥⑥⑥ S O O ` ` ❆❆❆❆❆❆❆ S ` ` ❆❆❆❆❆❆❆ > > ⑥⑥⑥⑥⑥⑥⑥ We next discuss the precise form of closed ideals of C ( X ) ⊗ min B ( H ), in terms of product ideals.Note that for a Hilbert space H , the set of all closed ideals forms a chain (see, [10, Corollary 6.2]).In the following, w denotes the cardinality of the set of all natural numbers and for every i ∈ N , let w i = 2 w i − so that w is continuum. Example 2.8.
Let X be a locally compact Hausdorff space and H be a Hilbert space with w n ( n ∈ N ) as its Hilbert dimension. Then the closed ideals of C ( X ) ⊗ min B ( H ) are of the form P n +3 j =2 J ( S j − ) ⊗ min I j , where I j ’s are closed ideals of B ( H ) and S j ’s are some closed subspaces of X .To see this, let { } = I ( I ( I ( · · · ( I n +3 = B ( H ) be the chain of closed ideals of B ( H ) and J be a closed ideal of C ( X ) ⊗ min B ( H ) . By Theorem 2.1, there exists n + 3 closed subspaces S ⊆ S ⊆ S ⊆ . . . ⊆ S n +3 = X such that J = J ( S ) where S = { S i } n +3 i =1 . As in Theorem 2.6, for j ∈ N n +3 , ∪ k ∈ γ j S k = S j − and hence J = P n +3 j =2 J ( S j − ) ⊗ min I j . Closed Lie ideals of C ( X, A )The Lie normalizer of a subspace S of a Lie algebra A is defined by N ( S ) := { a ∈ A : [ a, A ] ⊆ S } .It can be easily verified that N ( I ) is a closed subalgebra of A for a closed ideal I in A . The Lienormalizer plays an important role in determining the Lie ideals of A (for instance, see [4],[6]). Weidentify the Lie normalizer of ideals of C ( X, A ) and use this identification to characterize its closedLie ideals.
Theorem 3.1.
Let X be a locally compact Hausdorff space and A be a C ∗ -algebra with I = { I i } i ∈ ∆ as the collection of all closed ideals such that I β = A . Then a closed subspace L of C ( X, A ) is aclosed Lie ideal if and only if there is an element S = { S i } i ∈ ∆ ∈ T β I such that { f ∈ C ( X, A ) : f ( S i ) ⊆ [ I i , A ] , ∀ i ∈ ∆ } ⊆ L ⊆ { f ∈ C ( X, A ) : f ( S i ) ⊆ N ( I i ) , ∀ i ∈ ∆ } . Proof.
We know that a closed subspace L of the C ∗ -algebra C ( X, A ) is a Lie ideal if and only ifthere exists a closed ideal J ⊆ C ( X, A ) such that [
J, C ( X, A )] ⊆ L ⊆ N ( J ) ([3, Proposition 5.25,Theorem 5.27]). By Theorem 2.1, J = J ( S ) for some S = { S i } i ∈ ∆ ∈ T β I . Since for any fixed a ∈ A and x ∈ X , there is an element in C ( X, A ) which takes x to a , we have N ( J ) = { f ∈ C ( X, A ) : [ f, g ] ∈ J ( S ) , ∀ g ∈ C ( X, A ) } = { f ∈ C ( X, A ) : [ f, g ]( x ) ∈ I i , ∀ x ∈ S i , g ∈ C ( X, A ) , ∀ i ∈ ∆ } = { f ∈ C ( X, A ) : f ( x ) ∈ N ( I i ) , ∀ x ∈ S i , i ∈ ∆ } = { f ∈ C ( X, A ) : f ( S i ) ⊆ N ( I i ) , ∀ i ∈ ∆ } . B. TALWAR AND R. JAIN
We know from [3, Proposition 5.25] that [
I, B ] = I ∩ [ B, B ] for closed ideal I of a C ∗ -algebra B .This fact, along with Lemma 2.4 gives[ J, C ( X, A )] = [ C ( X, A ) , C ( X, A )] ∩ J = [ C ( X ) ⊗ min A, C ( X ) ⊗ min A ] ∩ J = C ( X ) ⊗ [ A, A ] ∩ J = C ( X ) ⊗ [ A, A ] ∩ J = { f ∈ C ( X, A ) : f ( X ) ⊆ [ A, A ] } ∩ { f ∈ C ( X, A ) : f ( S i ) ⊆ I i , ∀ i ∈ ∆ } = { f ∈ C ( X, A ) : f ( S i ) ⊆ [ A, A ] ∩ I i , ∀ i ∈ ∆ } = { f ∈ C ( X, A ) : f ( S i ) ⊆ [ I i , A ] , ∀ i ∈ ∆ } . Hence the result. (cid:3)
From the last result one can observe that if N ( I ) = I + Z ( A ), Z ( A ) being the centre of A , forevery closed ideal I of A , then for a closed ideal J of C ( X, A ), N ( J ) = { f ∈ C ( X, A ) : f ( S i ) ⊆ I i + Z ( A ) , ∀ i ∈ ∆ } for some { S i } i ∈ ∆ ∈ T β I . The question that comes next is the following: Can wewrite every element of N ( J ) as g + h such that g ( S i ) ⊆ I i for every i ∈ ∆ and h is Z ( A )-valued? Weshall observe in Corollary 3.6 that a positive answer of this question will help us to obtain a betterrepresentation of all closed Lie ideals of C ( X, A ).Recall that a C ∗ -algebra A is said to have the centre-quotient property if Z ( A/I ) = ( Z ( A ) + I ) /I for every closed ideal I of A (see [2] for details). Using the fact that for the natural quotientmap π : A → A/I , N ( I ) = π − ( Z ( A/I )), we have a nice relation between Lie normalizer and thecentre-quotient property.
Lemma 3.2. A C ∗ -algebra A has centre-quotient property if and only if N ( I ) = I + Z ( A ) for everyclosed ideal I in A . A unital C ∗ -algebra A is called weakly central if the continuous surjection ψ : Max( A ) → Max( Z ( A )) given by ψ ( I ) = I ∩ Z ( A ) is an injection, where Max( B ) denotes the space of allmaximal ideals of a C ∗ -algebra B endowed with the hull-kernel topology. It is well known thata unital C ∗ -algebra with unique maximal ideal must have one dimensional centre [2, Lemma 2.1].Since weak centrality and centre-quotient property are equivalent in unital C ∗ -algebras (see, [15,Theorem 1 and 2]), presence of unique maximal ideal in a unital C ∗ -algebra A implies that A hascentre-quotient property. In [6, Lemma 4.6] it was observed that for a simple unital C ∗ -algebra A and a closed ideal I ⊆ A , N ( I ) = I + C ( X, C Theorem 3.3.
Let X be a locally compact Hausdorff space and A be a unital C ∗ -algebra with uniquemaximal ideal. Then N ( J ) = J + C ( X, C for any closed ideal J of C ( X, A ) .Proof. Note that Z ( C ( X, A )) = C ( X, Z ( A )) = C ( X, C
1) and hence J + C ( X, C ⊆ N ( J ). Let I = { I i } i ∈ ∆ be the collection of all closed ideals of A with A = I β and let I β ′ be the unique maximalideal of A , β, β ′ ∈ ∆. Then, Theorem 2.1, there exists an element S = { S i } i ∈ ∆ ∈ T β I such that J = J ( S ). Since A has centre-quotient property, by Lemma 3.2, N ( I i ) = I i + C i ∈ ∆.Let f ∈ N ( J ) = { g ∈ C ( X, A ) : g ( S i ) ⊆ I i + C , ∀ i ∈ ∆ } as noted in Theorem 3.1. Since I β ′ is theunique maximal ideal of A and S ∈ T β I , we have S α ⊆ S β ′ for every α ∈ ∆ \ { β } .On S β ′ , write f = g + h which satisfy h ( S β ′ ) ⊆ C g ( S i ) ⊆ I i for every i ∈ ∆ \ { β } . This ispossible as no proper ideal of A can intersect C
1. Since I β ′ ∩ C { } , by Hahn-Banach Theorem,there exists T ∈ A ∗ such that k T k = 1, T ( I β ′ ) = { } and T ( λ
1) = λ for λ ∈ C . Then T f = T h on S β ′ . Also f vanishes at infinity and k T k = 1, so we obtain that T f vanishes at infinity because k T f k ≤ k f k . Since k T h k = k h k , T h and hence h is a continuous function vanishing at infinity. So g = f − h is continuous on S β ′ and is vanishing at infinity. By [6, Theorem 4.5], there exists an h ′ ∈ C ( X ) such that h ′| Sβ ′ = h . For x ∈ S cβ ′ , define g ′ ( x ) = f ( x ) − h ′ ( x ). Then f = g ′ + h ′ with g ′ ∈ J ( S ) and h ′ ∈ C ( X, C
1) and we are done. (cid:3)
LOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT 9
It is observed in the previous result that if A is a unital C ∗ -algebra with a unique maximal idealthen C ( X, A ) has the centre-quotient property. We now generalize this result and prove that thecentre-quotient property of A passes to C ( X, A ). We would like to point out that the proof givenabove does not work when A has more than one maximal ideal because in this case a closed idealmay intersect the centre non-trivially, that is, I ∩ Z ( A ) = { } , and this is where the proof will fail.As an intermediate step towards this generalization, we provide the following result. Proposition 3.4.
Let X be a compact Hausdorff space and A be a unital C ∗ -algebra having centre-quotient property. Then C ( X, A ) has centre-quotient property.Proof. It is sufficient to prove that C ( X, A ) is weakly central. Consider maximal ideals J and J of C ( X, A ) such that J ∩ Z ( C ( X, A )) = J ∩ Z ( C ( X, A )). For i = 1 ,
2, there exist maximal ideals I i of A and x i ∈ X such that J i = { f ∈ C ( X, A ) : f ( x i ) ∈ I i } ([12, Corollary V.26.2.2]). With the help ofUrysohn’s Lemma, one can easily verify that I i and x i are unique. Since Z ( C ( X, A )) = C ( X, Z ( A )),we have J i ∩ Z ( C ( X, A )) = { f ∈ C ( X, Z ( A )) : f ( x i ) ∈ I i ∩ Z ( A ) } . By the uniqueness, x = x and I ∩ Z ( A ) = I ∩ Z ( A ). Since A is weakly central, we obtain I = I and hence J = J . (cid:3) Theorem 3.5. If X is a locally compact Hausdorff topological space and A is a C ∗ -algebra. If C ( X, A ) has centre-quotient property then A has centre-quotient property. Converse is true if A isunital.Proof. In order to prove that A has centre-quotient property, it is sufficient to prove that for aclosed ideal I of A , N ( I ) ⊆ I + Z ( A ). For a fixed x ∈ X , consider a closed ideal J of C ( X, A )given by { f ∈ C ( X, A ) : f ( x ) ∈ I } . Now, for a ∈ N ( I ), let f ∈ C ( X, A ) such that f ( x ) = a .For any g ∈ C ( X, A ), ( f g − gf )( x ) = ag ( x ) − g ( x ) a ∈ I , which implies that f ∈ N ( J ). Since C ( X, A ) has centre-quotient property, f = g + h , for some g ∈ J and h ∈ C ( X, Z ( A )). Thus a = f ( x ) = g ( x ) + h ( x ) ∈ I + Z ( A ).Conversely, suppose that A is unital and has centre-quotient property. If ˜ X denotes the one pointcompactification of X , then we have a natural inclusion C ( X, A ) ⊆ C ( ˜ X, A ). For a closed ideal J of C ( X, A ), we claim that N ( J ) C ( X,A ) ⊆ N ( J ) C ( ˜ X,A ) , where N ( J ) B represents the Lie normalizerin B . Let f ∈ N ( J ) C ( X,A ) and { f µ } be a quasi central approximate identity of the closed ideal C ( X, A ) of C ( ˜ X, A ). Then for any g ∈ C ( ˜ X, A ), f g − gf = lim f f µ g − lim gf f µ = lim( f f µ g − gf f µ + f µ gf − f µ gf )= lim( f f µ g − f µ gf + f µ gf − gf f µ )= lim( f f µ g − f µ gf ) , since { f µ } is quasi central approximate identity and gf ∈ C ( X, A ). Note that f µ g ∈ C ( X, A ) and f ∈ N ( J ) C ( X,A ) together imply that f f µ g − f µ gf ∈ J for every µ . Thus f g − gf ∈ J which gives f ∈ N ( J ) C ( ˜ X,A ) .Since C ( ˜ X, A ) has centre-quotient property (Theorem 3.5) and J is also a closed ideal of C ( ˜ X, A ),we have N ( J ) C ( ˜ X,A ) = J + C ( ˜ X, Z ( A )). Now, for f ∈ N ( J ) C ( X,A ) , there exist g ∈ J and h ∈ C ( ˜ X, Z ( A )) such that f = g + h . If ˜ X = X ∪ {∞} , then f ( ∞ ) = 0 = g ( ∞ ) which gives h ( ∞ ) = f ( ∞ ) − g ( ∞ ) = 0 so that h ∈ C ( X, Z ( A )). Thus N ( J ) C ( X,A ) ⊆ J + C ( X, Z ( A )) and hence theresult holds. (cid:3) We now characterize the Lie ideals of a class of C ∗ -algebras. Recall that a bounded linearfunctional f on a C ∗ -algebra A is said to be a tracial state if f is positive of norm 1 and f ([ a, b ]) = 0for every a, b ∈ A . Corollary 3.6.
Let X be a locally compact Hausdorff space and A be a unital C ∗ -algebra withcentre-quotient property and no tracial states. Then a closed subspace L of C ( X, A ) is a Lie ideal if and only if it is of the form J + K for some closed ideal J of C ( X, A ) and a closed subspace K of C ( X, Z ( A )) .Proof. Since A has no tracial states, from [6, Lemma 2.4], C ( X, A ) has no tracial states. Thus, by[3, Proposition 5.25], [
J, A ] = J for every closed ideal J of C ( X, A ). From [3, Theorem 5.27] andTheorem 3.5, a closed subspace L of C ( X, A ) is a Lie ideal if and only if there exists a closed ideal J of C ( X, A ) such that J ⊆ L ⊆ J + C ( X, Z ( A )). Hence L must be of the form J + K for someclosed subspace K of C ( X, Z ( A )). (cid:3) As a consequence, we can now characterize all closed ideals of C ( X ) ⊗ min B ( H ). Corollary 3.7.
For a separable Hilbert space H and a locally compact Hausdorff space X , a closedsubspace L of C ( X ) ⊗ min B ( H ) is a Lie ideal if and only if there exist two closed subspaces S ⊆ S of X and a closed subspace K of C ( X ) ⊗ C such that L = J ( S ) ⊗ K ( H ) + J ( S ) ⊗ B ( H ) + K. Proof.
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