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Grothendieck's Tensor Norms Grothendieck's tensor norms Vandana Shivaji College, University of Delhi, India April, 2015 Vandana Grothendieck's tensor norms Overview Grothendieck's tensor norms Operator spaces Tensor products of Operator spaces Schur tensor product Vandana Grothendieck's tensor norms Grothendieck's tensor norms A Banach space is a complete normed space. For Banach spaces X and Y , X ⊗ Y = spanfx ⊗ y : x 2 X; y 2 Y g, where x ⊗ y is the functional on B(X∗ × Y ∗; C) given by x ⊗ y(f; g) = f(x)g(y) for f 2 X∗ and g 2 Y ∗. For a pair of arbitrary Banach spaces X and Y , the norm on X ⊗ Y induced by the embedding X ⊗ Y ! B(X∗ × Y ∗; C) is known as Banach space injective tensor norm. That is , for u 2 X ⊗ Y , the Banach space injective tensor norm is defined to be n n o X ∗ ∗ kukλ = sup f(xi)g(yi) : f 2 X1 ; g 2 Y1 : i=1 Vandana Grothendieck's tensor norms Grothendieck's tensor norms Question is How can we norm on X ⊗ Y ? n X kx ⊗ ykα ≤ kxkkyk, then, for u = xi ⊗ yi, by triangle's i=1 n X inequality it follows that kukα ≤ kxikkyik. Since this holds for i=1 n X every representation of u, so we have kukα ≤ inff kxikkyikg. i=1 For a pair of arbitrary Banach spaces X and Y and u an element in the algebraic tensor product X ⊗ Y , the Banach space projective tensor norm is defined to be n n X X kukγ = inff kxikkyik : u = xi ⊗ yi; n 2 Ng: i=1 i=1 X ⊗γ Y will denote the completion of X ⊗ Y with respect to this norm. Vandana Grothendieck's tensor norms Tensor Products: Properties For Banach spaces X1, X2, X~1, X~2 and X3, projective, For the quotient maps Q : X1 ! X~1 and R : X2 ! X~2, the corresponding map Q ⊗ R : X1 ⊗ X2 ! X~1 ⊗ X~2 extends to a γ γ γ quotient map Q ⊗ R : X1 ⊗ X2 ! X~1 ⊗ X~2. γ γ commutative, X1 ⊗ X2 is isometrically isomorphic to X2 ⊗ X1; γ γ associative, X1 ⊗ (X2 ⊗ X3) is isometrically isomorphic to γ γ (X1 ⊗ X2) ⊗ X3; If φi : Xi ! Yi; i = 1; 2, are bounded maps between the Banach spaces, then φ1 ⊗ φ2 : X1 ⊗ X2 ! Y1 ⊗ Y2 extends to a bounded γ γ γ map φ1 ⊗ φ2 : X1 ⊗ X2 ! Y1 ⊗ Y2 with γ kφ1 ⊗ φ2k = kφ1kkφ2k; In general, Banach space projective tensor product does not respect γ subspaces, that is, if Y1 is a subspace of X1, then Y1 ⊗ X2 is not, γ in general, a subspace of X1 ⊗ X2. However, it does for closed ∗-subalgebras of C∗-algebras. Also, for Banach spaces X and Y , X ⊗γ Y is a closed subspace of X∗∗ ⊗γ Y ∗∗. Vandana Grothendieck's tensor norms Tensor Products: Properties The key property of this tensor product is that it linearizes bounded bilinear maps just as the algebraic tensor product linearizes bilinear mappings. Let X, Y and Z be Banach spaces. Then there exists a natural isometric isomorphism B(X × Y; Z) =∼ B(X ⊗γ Y; Z) under which any element B 2 B(X × Y; Z) is mapped to the element B~ of B(X ⊗γ Y; Z) given by B~(x ⊗ y) = B(x; y)(x 2 X; y 2 Y ). In particular, (X ⊗γ Y )∗ = B(X × Y; C). Vandana Grothendieck's tensor norms Tensor Products: Properties For Banach algebras A and B, A ⊗γ B is a Banach algebra, It is commutative if and only if both A and B are. If A and B are Banach ∗-algebras, then A ⊗γ B is a Banach ∗-algebra under the natural involution (a ⊗ b)∗ = a∗ ⊗ b∗ for a 2 A and b 2 B. Vandana Grothendieck's tensor norms Tensor Products: Properties For Banach spaces X1, X2, Y1, Y2 and X3, If ui : Xi ! Yi are isometries for i = 1; 2 then the corresponding λ λ map u1 ⊗ u2 : X1 ⊗ X2 ! Y1 ⊗ Y2 is an isometry too. λ λ X1 ⊗ X2 is isometrically isomorphic to X2 ⊗ X1; λ λ λ λ X1 ⊗ (X2 ⊗ X3) is isometrically isomorphic to (X1 ⊗ X2) ⊗ X3; If φi : Xi ! Yi; i = 1; 2, are bounded maps between the Banach spaces, then φ1 ⊗ φ2 : X1 ⊗ X2 ! Y1 ⊗ Y2 extends to a bounded λ λ λ map φ1 ⊗ φ2 : X1 ⊗ X2 ! Y1 ⊗ Y2 with λ kφ1 ⊗ φ2k = kφ1kkφ2k; In general, Banach space injective tensor product does not respect quotient. Vandana Grothendieck's tensor norms Tensor Products: Properties For Banach algebras A and B, A ⊗λ B is not a Banach algebra in general. ∗ Blecher showed that if A and B are unital C -algebras then k · kλ is submultiplicative on A ⊗ B if and only if A or B is commutative. Vandana Grothendieck's tensor norms Introduction to Operator Spaces An (concrete) operator space V is a closed subspace of B(H) together with the natural norms on Mn(V ) inherited from n Mn(B(H)) = B(H ). A normed space V with a sequence of norms k · kn : Mn(V ) ! [0; 1); n 2 N is said to be an (abstract) operator space if: (i) kv ⊕ wkn+m = maxfkvkn; kwkmg for all v 2 Mn(V ); w 2 Mm(V ), v 0 where v ⊕ w denotes the matrix 0 w 2 Mn+m(V ). (ii) kαvβkm ≤ kαkkvknkβk, for all α 2 Mm;n; β 2 Mn;m; v 2 Mn(V ). Vandana Grothendieck's tensor norms Operator Spaces: Morphisms An operator φ : V ! W between operator spaces V and W is said to be completely bounded (abbreviated as c.b.) if kφkcb := supfkφnk : n 2 Ng < 1; where φn : Mn(V ) ! Mn(W ) is defined by φn((xij)) = (φ(xij)) for all (xij) 2 Mn(V ): The set of all completely bounded maps from V into W is denoted by CB(V; W ). Two operator spaces V and W are said to be completely isometrically isomorphic if there exists a completely bounded isometry φ : V ! W whose inverse is also completely bounded. Vandana Grothendieck's tensor norms Operator Spaces Ruan(1988) If V is an abstract operator space, then V is completely isometrically isomorphic to a closed linear subspace of B(H) for some Hilbert space H. Vandana Grothendieck's tensor norms Introduction to Operator Space: Examples By Gelfand-Naimark theorem, every C∗-algebra is an operator space. For Hilbert spaces H and K, B(H; K) is an operator space by the embedding B(H; K) ,!B(H ⊕ K). The matrix norms are given by ∼ n n Mn(B(H; K)) = B(H ;K ). Every Banach space X possess a structure of operator space. To see ∗ ∗ this, consider Γ = X1 , which is compact in the w -topology by ∗ Alogalu Theorem, so that C(Γ) ⊆ B(`2(Γ)) is a C -algebra. The isometric embedding X,! C(Γ) via x ! fx, where fx(g) := g(x) for g 2 Γ, equips X with an operator space structure. Vandana Grothendieck's tensor norms Quantum tensor product-Tensor Products of operator spaces For operator spaces V and W and V ⊗ W their algebraic tensor product. Assumptions: k · kn on Mn(V ⊗ W ) satisfying Ruan's Theorem, k · kn on Mn(V ⊗ W ) are subcross matrix norms, where an operator space matrix norm k · kµ on V ⊗ W is called a subcross matrix norm if kv ⊗ wkµ ≤ kvkkwk for all v 2 Mp(V ) and w 2 Mq(W ), p; q 2 N; if in addition, kv ⊗ wkµ = kvkkwk, then k · kµ is called a cross matrix norm on V ⊗ W . For operator spaces V and W , the operator space projective tensor product denoted by V ⊗bW , is the completion of the algebraic tensor product V ⊗ W under the norm kuk^ = inffkαkkvkkwkkβk : u = α(v ⊗ w)βg; u 2 Mn(V ⊗ W ); where infimum runs over arbitrary decompositions with v 2 Mp(V ), w 2 Mq(W ); α 2 Mn;pq; β 2 Mpq;n and p; q 2 N arbitrary. Vandana Grothendieck's tensor norms Quantum tensor product-Tensor Products of operator spaces γ The analogy of ⊗b and ⊗ is not completely transparent from the definition; the following universal properties of ⊗b confirm the parallelism. Theorem [Effros-Ruan and Blecher-Paulsen] If V , W and Z are operator spaces, then there are natural completely isometric identifications : ∼cb CB(V ⊗bW; Z) = JCB(V × W; Z): cb ∗ ∼ In particular, if Z = C, (V ⊗bW ) = JCB(V × W; C). where a bilinear map u : V × W ! Z is said to be jointly completely bounded (in short, j.c.b.) if the associated maps un : Mn(V ) × Mn(W ) ! Mn2 (Z) given by un (vij); (wkl) = u(vij; wkl) ; n 2 N are uniformly bounded, and in this case we denote kukjcb = sup kunk. n Vandana Grothendieck's tensor norms Quantum tensor Products Given operator spaces X ⊆ B(H) and Y ⊆ B(K), the norm induced on X ⊗ Y via the inclusion X ⊗ Y ⊆ B(H ⊗2 K) is known as the operator space injective tensor norm. Vandana Grothendieck's tensor norms Tensor Products For u 2 Mn(X ⊗ Y ), the Haagerup tensor norm is defined as kukh = inffkvkkwk : u = v w; v 2 Mn;p(X); w 2 Mp;n(Y ); p 2 Ng; Pp where v w = k=1 vik ⊗ wkj ij. For C∗-algebras A and B, the Haagerup norm of an element u 2 A ⊗ B takes a simpler and convenient form given by ∗ 1=2 ∗ 1=2 n kukh = inf kΣi aiai k kΣi bi bik : u = Σi=1ai ⊗ bi; n 2 N : Vandana Grothendieck's tensor norms Tensor Products Like the projective tensor product of Banach spaces and the operator space projective tensor product of operator spaces, the Haagerup tensor norm is naturally associated with the completely bounded bilinear maps through the following identifications: cb cb h h ∗ CB(V × W; Z) =∼ CB(V ⊗ W; Z) and CB(V × W; C) =∼ (V ⊗ W ) : By a completely bounded (in short, c.b.) bilinear map, we mean a bilinear map u : V × W ! Z for which the associated maps un : Mn(V ) × Mn(W ) ! Mn(Z) given by X un (vij); (wkl) = u(vik; wkj) ; n 2 N k are uniformly bounded, and in this case we denote kukcb = sup kunk.
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