Systems of seminorms and weak bounded approximation properties

Systems of seminorms and weak bounded approximation properties Aleksei Lissitsin University of Tartu NoLiFA Valencia 19.10.2017

Notation X, Y, Z - Banach spaces, A - an operator ideal, F, K - ideals of finite-rank and compact operators, respectively, L, X, W - ideals of all, separable and reflexive Banach spaces, respectively, τ s, τ c - topologies of pointwise convergence and uniform convergence on compact sets, respectively. Aleksei Lissitsin Seminorm systems and weak BAPs 2 / 19

Old open problem: X AP X MAP? Classical partial solution with many different proofs: Yes, if X or X has Radon Nikodým property (RNP). Aleksei Lissitsin Seminorm systems and weak BAPs 4 / 19

Old open problem: X AP X MAP? Classical partial solution with many different proofs: Yes, if X or X has Radon Nikodým property (RNP). Proof of Oja, 2006: X has AP X has MAP X has weak MAP norms RNP X has MAP norms Aleksei Lissitsin Seminorm systems and weak BAPs 4 / 19

Approximation properties A Banach space X has: approximation property (AP) if I X F(X) τc metric AP (MAP) if I X {S F(X) : S 1} τc MAP for A if I X {S F(X) : Y T A(X, Y ) T S T } τc weak MAP if MAP for K. weak MAP for A if I X {S F(X) : Y T A(X, Y ) T S T } τs Aleksei Lissitsin Seminorm systems and weak BAPs 5 / 19

Approximation properties A Banach space X has: approximation property (AP) if I X F(X) τc metric AP (MAP) if I X {S F(X) : S 1} τc MAP for A if I X {S F(X) : Y T A(X, Y ) T S T } τc weak MAP if MAP for K. weak MAP for A if I X {S F(X) : Y T A(X, Y ) T S T } τs MAP for L MAP, Aleksei Lissitsin Seminorm systems and weak BAPs 5 / 19

Approximation properties A Banach space X has: approximation property (AP) if I X F(X) τc metric AP (MAP) if I X {S F(X) : S 1} τc MAP for A if I X {S F(X) : Y T A(X, Y ) T S T } τc weak MAP if MAP for K. weak MAP for A if I X {S F(X) : Y T A(X, Y ) T S T } τs MAP for L MAP, weak MAP MAP for K(L, X W) MAP for W MAP for RN dual Aleksei Lissitsin Seminorm systems and weak BAPs 5 / 19

Approximation properties A Banach space X has: A-approximation property (A-AP) if I X A τc metric A-AP (A-MAP) if I X {S A : S 1} τc A-MAP for A if I X {S A : Y T A(X, Y ) T S T } τc weak A-MAP if A-MAP for K weak A-MAP for A if I X {S A : Y T A(X, Y ) T S T } τs where A L(X) is convex and contains 0. A-MAP for L A-MAP, weak A-MAP A-MAP for K(L, X W) Aleksei Lissitsin Seminorm systems and weak BAPs 5 / 19

Approximation properties A Banach space X has: A-approximation property (A-AP) if I X A τc metric A-AP (A-MAP) if I X {S A : S 1} τc A-MAP for A if I X {S A : Y T A(X, Y ) T S T } τc weak A-MAP if A-MAP for K weak A-MAP for A if I X {S A : Y T A(X, Y ) T S T } τs where A L(X) is convex and contains 0. A-MAP for L A-MAP, weak A-MAP A-MAP for K(L, X W) A-MAP for W A-MAP for RN dual (if A K(X)) Aleksei Lissitsin Seminorm systems and weak BAPs 5 / 19

Abstract definition Let V be a vector space, let q be a seminorm on V, and let A V. Denote A q := {x A : q(x) 1}. Aleksei Lissitsin Seminorm systems and weak BAPs 6 / 19

Abstract definition Let V be a vector space, let q be a seminorm on V, and let A V. Denote A q/λ := {x A : q(x) λ}, λ > 0. Aleksei Lissitsin Seminorm systems and weak BAPs 6 / 19

Abstract definition Let V be a vector space, let q be a seminorm on V, and let A V. Denote A q/λ := {x A : q(x) λ}, λ > 0. If V is a TVS and (0, 1) A A, then A q/µ = A q/λ, λ > 0. µ>λ Aleksei Lissitsin Seminorm systems and weak BAPs 6 / 19

Abstract definition Let V be a vector space, let q be a seminorm on V, and let A V. Denote A q/λ := {x A : q(x) λ}, λ > 0. If V is a TVS and (0, 1) A A, then A q/µ = A q/λ, λ > 0. µ>λ Let λ > 0, let Q be a system of seminorms on V, and let τ be a linear topology on V. Denote (λ, Q, τ) -bap(a) := {x V : x τ A q/µ q Q}. µ>λq(x) Aleksei Lissitsin Seminorm systems and weak BAPs 6 / 19

Abstract definition Let V be a vector space, let q be a seminorm on V, and let A V. Denote A q/λ := {x A : q(x) λ}, λ > 0. If V is a TVS and (0, 1) A A, then A q/µ = A q/λ, λ > 0. µ>λ Let λ > 0, let Q be a system of seminorms on V, and let τ be a linear topology on V. Denote (λ, Q, τ) -bap(a) := {x V : x A q/λq(x) τ q Q}. Aleksei Lissitsin Seminorm systems and weak BAPs 6 / 19

Abstract definition Let V be a vector space, let q be a seminorm on V, and let A V. Denote A q/λ := {x A : q(x) λ}, λ > 0. If V is a TVS and (0, 1) A A, then A q/µ = A q/λ, λ > 0. µ>λ Let λ > 0, let Q be a system of seminorms on V, and let τ be a linear topology on V. Denote (λ, Q, τ) -bap(a) := {x V : x {y A : q(y) λq(x)} τ q Q}. Aleksei Lissitsin Seminorm systems and weak BAPs 6 / 19

BAP closure of A Let λ > 0, let Q be a system of seminorms on V, and let τ be a linear topology on V. Denote (λ, Q, τ) -bap(a) := {x V : x τ A q/µ q Q} µ>λq(x) and (Q, τ) -map(a) := (1, Q, τ) -bap(a). Omit Q if V is a normed space and Q consists of its norm. For example, X has MAP I X τ c -map(f(x)). Aleksei Lissitsin Seminorm systems and weak BAPs 7 / 19

Obvious case of AP = MAP Let τ(q) denote the linear topology induced by Q. Proposition 1. If τ(q) τ, then A τ = (Q, τ) -map(a). Aleksei Lissitsin Seminorm systems and weak BAPs 8 / 19

Obvious case of AP = MAP Let τ(q) denote the linear topology induced by Q. Proposition 1. If τ(q) τ, then A τ = (Q, τ) -map(a). Proof. Take x A τ and q Q. We want x τ A q/µ. µ>q(x) Aleksei Lissitsin Seminorm systems and weak BAPs 8 / 19

Obvious case of AP = MAP Let τ(q) denote the linear topology induced by Q. Proposition 1. If τ(q) τ, then A τ = (Q, τ) -map(a). Proof. Take x A τ and q Q. We want x τ A q/µ. µ>q(x) There is a net (x α ) A such that x α τ x. Then q(x α ) q(x). Hence, for any µ > q(x) there is some α such that q(x β ) µ for all β α. Aleksei Lissitsin Seminorm systems and weak BAPs 8 / 19

Obvious case of AP = MAP Let τ(q) denote the linear topology induced by Q. Proposition 1. If τ(q) τ, then A τ = (Q, τ) -map(a). Proof. Take x A τ and q Q. We want x τ A q/µ. µ>q(x) There is a net (x α ) A such that x α τ x. Then q(x α ) q(x). Hence, for any µ > q(x) there is some α such that q(x β ) µ for all β α. Actually, A (Q, τ) -map(a) is always a closure operator of some topology but it is probably almost never linear. Aleksei Lissitsin Seminorm systems and weak BAPs 8 / 19

Equivalence of seminorm systems Define Q 1 Q 2 by: Aleksei Lissitsin Seminorm systems and weak BAPs 9 / 19

Equivalence of seminorm systems Define Q 1 Q 2 by: x V and q 1 Q 1 q 2 Q 2 and c > 0 such that q 1 cq 2 and q 1 (x) = cq 2 (x). Aleksei Lissitsin Seminorm systems and weak BAPs 9 / 19

Equivalence of seminorm systems Define Q 1 Q 2 by: x V and q 1 Q 1 q 2 Q 2 and c > 0 such that q 1 cq 2 and q 1 (x) = cq 2 (x). Q 1 Q 2 Q 1 Q 2 and Q 2 Q 1. Aleksei Lissitsin Seminorm systems and weak BAPs 9 / 19

Equivalence of seminorm systems Define Q 1 Q 2 by: x V and q 1 Q 1 q 2 Q 2 and c > 0 such that q 1 cq 2 and q 1 (x) = cq 2 (x). Q 1 Q 2 Q 1 Q 2 and Q 2 Q 1. Note that if Q 1 Q 2, then (λ, Q 2, τ) -bap(a) (λ, Q 1, τ) -bap(a) Aleksei Lissitsin Seminorm systems and weak BAPs 9 / 19

Equivalence of seminorm systems Define Q 1 Q 2 by: x V and q 1 Q 1 q 2 Q 2 and c > 0 such that q 1 cq 2 and q 1 (x) = cq 2 (x). Q 1 Q 2 Q 1 Q 2 and Q 2 Q 1. Note that if Q 1 Q 2, then (λ, Q 2, τ) -bap(a) (λ, Q 1, τ) -bap(a) and, of course, τ(q 1 ) τ(q 2 ). Aleksei Lissitsin Seminorm systems and weak BAPs 9 / 19

Seminorms on operator spaces For T L(X, Y ) and a Banach space Z define seminorms r T [Z](S) := ST and l T [Z](S) := T S on L(Y, Z) and L(Z, X), respectively. Put r A := {r T [Z] : T A, Z is a Banach space }, l A := {l T [Z] : T A, Z is a Banach space }. Then r A and l A are classes of seminorms defined on all operator spaces L(X, Y ). Aleksei Lissitsin Seminorm systems and weak BAPs 10 / 19

Seminorms on operator spaces For T L(X, Y ) and a Banach space Z define seminorms r T [Z](S) := ST and l T [Z](S) := T S on L(Y, Z) and L(Z, X), respectively. Put r A := {r T [Z] : T A, Z is a Banach space }, l A := {l T [Z] : T A, Z is a Banach space }. Then r A and l A are classes of seminorms defined on all operator spaces L(X, Y ). For example, X has MAP for A I X (l A, τ c ) -map(f(x)). Aleksei Lissitsin Seminorm systems and weak BAPs 10 / 19

Seminorms on operator spaces For T L(X, Y ) and a Banach space Z define seminorms r T [Z](S) := ST and l T [Z](S) := T S on L(Y, Z) and L(Z, X), respectively. Put r A := {r T [Z] : T A, Z is a Banach space }, l A := {l T [Z] : T A, Z is a Banach space }. Then r A and l A are classes of seminorms defined on all operator spaces L(X, Y ). For example, X has MAP for A I X (l A, τ c ) -map(f(x)). Note that τ c = τ(r K ). Aleksei Lissitsin Seminorm systems and weak BAPs 10 / 19

Abuse the notation: A(X, Y ) := L(X, Y ) A, A(A, B) := {T A(V, W ) : V A, W B}, A dual := {T L : T A}, A a := {T : T A}. Proposition 2. Let A be an operator ideal, A a space ideal. If A = A Op(A), then r A r A(A,L). If A = Op(A) A, then l A l A(L,A). Reason (e.g., Heinrich, 1980): If T factors through some space Z as T = T 1 T 2, then you can pick an equivalent norm on Z such that T = T 1 T 2. Aleksei Lissitsin Seminorm systems and weak BAPs 11 / 19

Adjoint systems A seminorm q on L(Y, X ) induces a seminorm q on L(X, Y ) via the natural inclusion L(X, Y ) L(Y, X ), i.e., q (S) := q(s ) for S L(X, Y ). Put Q = {q : q Q}. Aleksei Lissitsin Seminorm systems and weak BAPs 12 / 19

Adjoint systems A seminorm q on L(Y, X ) induces a seminorm q on L(X, Y ) via the natural inclusion L(X, Y ) L(Y, X ), i.e., q (S) := q(s ) for S L(X, Y ). Put Clearly, Q 1 Q 2 if Q 1 Q 2. Q = {q : q Q}. Aleksei Lissitsin Seminorm systems and weak BAPs 12 / 19

Adjoint systems A seminorm q on L(Y, X ) induces a seminorm q on L(X, Y ) via the natural inclusion L(X, Y ) L(Y, X ), i.e., q (S) := q(s ) for S L(X, Y ). Put Clearly, Q 1 Q 2 if Q 1 Q 2. Q = {q : q Q}. Similarly, define τ for a system of topologies τ. Note that τ(q ) = τ(q). Aleksei Lissitsin Seminorm systems and weak BAPs 12 / 19

Adjoint systems A seminorm q on L(Y, X ) induces a seminorm q on L(X, Y ) via the natural inclusion L(X, Y ) L(Y, X ), i.e., q (S) := q(s ) for S L(X, Y ). Put Clearly, Q 1 Q 2 if Q 1 Q 2. Q = {q : q Q}. Similarly, define τ for a system of topologies τ. Note that τ(q ) = τ(q). Observe that by definition r A = l A a and l A = r A a. Aleksei Lissitsin Seminorm systems and weak BAPs 12 / 19

Observe that by definition r A = l A a and l A = r A a. Proposition 2. Let A be an operator ideal, and let A be a space ideal. If A = A Op(A), then r A r A(A,L). If A = Op(A) A, then l A l A(L,A). W - weakly compact operators, W - reflexive spaces, W = Op(W): Aleksei Lissitsin Seminorm systems and weak BAPs 13 / 19

Observe that by definition r A = l A a and l A = r A a. Proposition 2. Let A be an operator ideal, and let A be a space ideal. If A = A Op(A), then r A r A(A,L). If A = Op(A) A, then l A l A(L,A). W - weakly compact operators, W - reflexive spaces, W = Op(W): Proposition 3. Let A = A W. Then l A dual r A. Aleksei Lissitsin Seminorm systems and weak BAPs 13 / 19

Observe that by definition r A = l A a and l A = r A a. Proposition 2. Let A be an operator ideal, and let A be a space ideal. If A = A Op(A), then r A r A(A,L). If A = Op(A) A, then l A l A(L,A). W - weakly compact operators, W - reflexive spaces, W = Op(W): Proposition 3. Let A = A W. Then l A dual r A. In particular, l K r K and l W r W. Aleksei Lissitsin Seminorm systems and weak BAPs 13 / 19

Oja s RNP impact lemma Let A L(X) be convex, λ 1, τ be a locally convex topology on L(X), I X (λ, l K, τ) -bap(a), T L(X, Y ) be such that {T S : S A} K(X, Y ), X or Y have RNP. Then I X (λ, {l T }, τ) -bap(a). Aleksei Lissitsin Seminorm systems and weak BAPs 14 / 19

Oja s RNP impact lemma Let A L(X) be convex, was A = F(X), λ 1, τ be a locally convex topology on L(X), was τ = τ c, I X (λ, l K, τ) -bap(a), T L(X, Y ) be such that {T S : S A} K(X, Y ), X or Y have RNP. Then I X (λ, {l T }, τ) -bap(a). Aleksei Lissitsin Seminorm systems and weak BAPs 14 / 19

In the case, when A K(X) and X or X have the RNP, we can replace the seminorm systems with the norm. Theorem Let X be a Banach space such that X or X has the RNP. Let A K(X) be convex, let λ 1, and let τ be a locally convex topology on L(X) or on L(X ). 1. If I X (λ, l K, τ) -bap(a), then I X (λ, τ) -bap(a), 2. if I X (λ, r K, τ) -bap(a a ), then I X (λ, τ) -bap(a a ), 3. if I X A τ and τ τ c, then I X τ -map(a), 4. if I X A aτ and τ τ c, then I X τ -map(a a ). Aleksei Lissitsin Seminorm systems and weak BAPs 15 / 19

Proof of theorem 1. If I X (λ, l K, τ) -bap(a), then I X (λ, τ) -bap(a): Take T = I X in Oja s lemma, then l T =. Aleksei Lissitsin Seminorm systems and weak BAPs 16 / 19

Proof of theorem 1. If I X (λ, l K, τ) -bap(a), then I X (λ, τ) -bap(a): Take T = I X in Oja s lemma, then l T =. 2. if I X (λ, r K, τ) -bap(a a ), then I X (λ, τ) -bap(a a ): Since r K l K, we get that I X (λ, l K, τ ) -bap(a). Then I X (λ, τ ) -bap(a), which is equivalent to the claim. 3. if I X A τ and τ τ c, then I X τ -map(a): Note that τ c = τ(r K ) = τ(r K ) = τ(l K). So Proposition 1 yields that I X (l K, τ) -map(a), and the claim follows from 1. Aleksei Lissitsin Seminorm systems and weak BAPs 16 / 19

Proof of theorem 1. If I X (λ, l K, τ) -bap(a), then I X (λ, τ) -bap(a): Take T = I X in Oja s lemma, then l T =. 2. if I X (λ, r K, τ) -bap(a a ), then I X (λ, τ) -bap(a a ): Since r K l K, we get that I X (λ, l K, τ ) -bap(a). Then I X (λ, τ ) -bap(a), which is equivalent to the claim. 3. if I X A τ and τ τ c, then I X τ -map(a): Note that τ c = τ(r K ) = τ(r K ) = τ(l K). So Proposition 1 yields that I X (l K, τ) -map(a), and the claim follows from 1. 4. if I X A aτ and τ τ c, then I X τ -map(a a ): This is immediate from 3. Aleksei Lissitsin Seminorm systems and weak BAPs 16 / 19

Theorem... 3. if I X A τ and τ τ c, then I X τ -map(a), 4. if I X A aτ and τ τ c, then I X τ -map(a a ). Oja used the lemma for τ c to go from weak MAP of X to MAP of X. If we use the lemma for τ c, we go directly: X has AP RNP X has MAP. Aleksei Lissitsin Seminorm systems and weak BAPs 17 / 19

Another application - weakly compact AP Let X be a Banach space such that X has the RNP. Then I X W(X) aτ(r W ) I X (l V, τ(r W ) ) -map(w(x)). Aleksei Lissitsin Seminorm systems and weak BAPs 18 / 19

Papers S. Heinrich, Closed operator ideals and interpolation, J. Funct. Anal. 35 (1980) 397 411. Å. Lima and E. Oja, The weak metric approximation property, Math. Ann. 333 (2005) 471 484. E. Oja, The impact of the Radon-Nikodým property on the weak bounded approximation property, Rev. R. Acad. Cien. Serie A. Mat. 100 (2006) 325 331. A. L., Impact of the Radon Nikodým property on convex approximation properties, Archiv der Math. 105(2) (2015) 163 171. Aleksei Lissitsin Seminorm systems and weak BAPs 19 / 19