Grothendieck spaces: the landscape and perspectives
GGROTHENDIECK SPACES: THE LANDSCAPE AND PERSPECTIVES
MANUEL GONZÁLEZ AND TOMASZ KANIA
Abstract.
In 1973, Diestel published his seminal paper
Grothendieck spaces and vectormeasures that drew a connection between Grothendieck spaces (Banach spaces for whichweak- and weak*-sequential convergences in the dual space coincide) and vector measures.This connection was developed in his book with J. Uhl Jr.
Vector measures . Addition-ally, Diestel’s paper included a section with several open problems about the structuralproperties of Grothendieck spaces, and only half of them have been solved to this day.The present paper aims at synthetically presenting the state of the art at subjectivelyselected corners of the theory of Banach spaces with the Grothendieck property, describ-ing the main examples of spaces with this property, recording the solutions to Diestel’sproblems, providing generalisations/extensions or new proofs of various results concern-ing Grothendieck spaces, and adding to the list further problems that we believe are ofrelevance and may reinvigorate a better-structured development of the theory.
Date : March 5, 2021.2010
Mathematics Subject Classification.
Primary: 46A35; 46B20.
Key words and phrases.
Grothendieck space, G-space, property ( V ) , property ( V ∞ ) , Banach space, Gro-thendieck operator, C*-algebra, Banach lattice, positive Grothendieck property, quantitative Grothendieckproperty, pseudo-intersection number, locally convex Grothendieck space, twisted sum, push-out.Research of M. González was partially supported by MICINN (Spain), Grant PID2019-103961GB-C22.T. Kania acknowledges with thanks funding received from SONATA 15 No. 2019/35/D/ST1/01734. a r X i v : . [ m a t h . F A ] M a r M. GONZÁLEZ AND T. KANIA
A Banach space X is Grothendieck whenever every weakly* convergent sequence in thedual space X ∗ converges weakly. The notion of a Grothendieck space was coined afterGrothendieck proved that for any index set Γ the space of bounded functions on Γ , (cid:96) ∞ (Γ) ,and more generally, the space C ( K ) of continuous functions on a compact, extremallydisconnected ( Stonean ) space K have this property ([66, Théorème 9]).Without a doubt, reflexive spaces are Grothendieck, and from the point of view of thedual space, the Grothendieck property may be indeed viewed as a form of ‘sequential re-flexivity’. The Grothendieck property may be regarded as complementary to being weaklycompactly generated—in particular, these two properties together imply reflexivity. Gro-thendieck property is of relevance in summability theory in Banach spaces (see [63] andreferences therein) or in the theory of C -semigroups of operators, where many featuresavailable for semigroups of operators on reflexive spaces may be emulated.In [42], Diestel explained the relevance of the Grothendieck property in the study offinitely additive vector-valued measures; the paper itself includes a section of open problemsconcerning this property, most of which to this day remain open. Even though manyparticular examples (and non-examples) of Grothendieck spaces have been identified since,little progress on stability properties of Grothendieck spaces has been obtained so far.The present paper aims at synthetically presenting various corners of the theory of Ba-nach spaces with the Grothendieck property with the perspective of putting it on a moresystematic footing. For this reason, we describe the main isomorphic properties of Grothen-dieck spaces, collect the answers that have been obtained for some of Diestel’s problems,revisit the stubbornly open ones as well as raise a number of new ones. Occasionally, wegeneralise certain facts concerning Grothendieck spaces. In order to properly motivatethe problems by related results, we have stated them along the text, as well as we haveincluded a comprehensive list of open problems in the final section. Let us briefly describethe contents of the paper.Chapter 2 is devoted to introducing concepts that will be used throughout our exposition.Chapter 3 includes several characterisations and properties of Grothendieck Banach spaces.In Chapter 4 we exemplify Grothendieck spaces within certain familiar classes of Banachspaces such as C ( K ) -spaces, C ∗ -algebras, L ∞ -spaces, Banach lattices, etc. as well aswe provide sufficient conditions for spaces in these classes to be Grothendieck. Chapter5 touches upon preservation of the Grothendieck property via various constructions anddiscusses methods of building new Grothendieck spaces from the already known ones. Thisincludes twisted-sums, tensor products, ultraproducts, and more. In Chapter 6 we briefly describe some additional results on the Grothendieck property, and Chapter 7 providesa concise list of the problems that we have collected in the previous chapters. The Banach spaces we consider are either real or complex, however when we talk about C ∗ -algebras, the underlying Banach spaces are always complex. An operator is a continuouslinear operator between Banach spaces (unless we explicitly state otherwise). We denoteby B X the closed unit ball of a Banach space X . The symbol B ( X, Y ) stands for the spaceof all operators between Banach spaces X and Y ; the spaces of compact operators K ( X, Y ) and weakly compact operators W ( X, Y ) are closed subspaces thereof; and we write B ( X ) , K ( X ) , and W ( X ) when X = Y . We write X ∼ = Y , whenever two Banach spaces X and Y are isomorphic and X ≡ Y in the case where they are isometrically isomorphic.When X and Y are isomorphic, we denote the (multiplicative) Banach–Mazur distancebetween X and Y by d BM ( X, Y ) = inf (cid:8) (cid:107) T − (cid:107) : T ∈ B ( X, Y ) is a norm-one isomorphism (cid:9) . The density character of a Banach space X , denoted dens X , is the least cardinal λ forwhich X has a dense subset of cardinality λ .A Banach space X is weakly compactly generated (WCG, for short) if it contains a weaklycompact subset that spans a dense subspace. Every separable Banach space, every reflexiveBanach space, and spaces of the form c (Γ) for any set Γ are WCG.A Banach space X has weak ∗ sequentially compact dual ball (has W ∗ SC dual ball, forshort) whenever every sequence in B X ∗ has a weak ∗ -convergent subsequence; we refer to[44, Chapter XIII] for systematic study of spaces with this property. Each subspace of aWCG Banach space has W ∗ SC dual ball [44, Theorem XIII.4]. Also the spaces whose dualspaces contain no copies of (cid:96) , and the spaces isomorphic to the dual of a separable spacecontaining no copies of (cid:96) . Clearly, a Grothendieck space with weak ∗ sequentially compactdual ball is reflexive. We refer to [44, Chapter XIII] for additional information.An operator T : X → Y is weakly pre-compact if for every bounded sequence ( x n ) ∞ n =1 in X , ( T x n ) ∞ n =1 has a weakly Cauchy subsequence; equivalently, by Rosenthal’s (cid:96) -theorem[44, Chapter XI], if there exists no subspace of X isomorphic to (cid:96) on which the operator T is an isomorphism. We denote by R ( X, Y ) the space of weakly pre-compact operatorsfrom X into Y .A series (cid:80) ∞ n =1 x n in X is weakly unconditionally converging if for every f ∈ X ∗ onehas (cid:80) ∞ n =1 |(cid:104) f, x n (cid:105)| < ∞ . An operator T : X → Y is unconditionally converging if for eachweakly unconditionally converging series (cid:80) ∞ n =1 x n in X , the series (cid:80) ∞ n =1 T x n is uncondi-tionally converging in Y ; equivalently, if there is no subspace of X isomorphic to c onwhich T is an isomorphism [69, Chapter III, Lemma 3.3.A]. We denote by U ( X, Y ) thespace of unconditionally converging operators from X into Y . A Banach space X has property ( V ) , whenever every unconditionally converging operator T : X → Y is weakly compact; i.e. , when U ( X, Y ) ⊂ W ( X, Y ) for every Banach space Y .Reflexive spaces and C ( K ) -spaces have property ( V ) . Moreover, property ( V ) is inheritedby quotients (but not by closed subspaces). We refer to [69, Section III.3] for additionalinformation on this property.A Banach space X has the Dunford–Pettis property , when for every Banach space Y every weakly compact operator T : X → Y takes weakly convergent sequences of X intoconvergent sequences of Y . It was proved by Grothendieck [66] that C ( K ) -spaces and L ( µ ) for any measure µ have the Dunford–Pettis property. Additionally, L ∞ -spaces and L -spaces have this property. A Banach space X has the Dunford–Pettis property, whenever X ∗ has the Dunford–Pettis property. We refer to [43] for a complete survey on the Dunford–Pettis property.A Schauder decomposition of a Banach space X is a sequence ( X i ) ∞ i =1 of closed non-zerosubspaces of X such that every x ∈ X has a unique representation x = (cid:80) ∞ i =1 x i with x i ∈ X i for every i [113, Definition 1.g.1]. It determines (and is determined by) a sequence ( P i ) ∞ i =1 of non-zero projections on X such that P i P j = 0 for i (cid:54) = j and x = (cid:80) ∞ i =1 P i x foreach x ∈ X ; note that each subspace M i corresponds with the range of the projection P i .This concept is equivalent to the notion of a Schauder basis when the projections haveone-dimensional ranges.Let X be a Banach space. A Markuschevich basis ( M-basis , for short) of X is a family { ( x γ , f γ ) : γ ∈ Γ } in X × X ∗ , which is biorthogonal ( (cid:104) f γ , x γ (cid:105) = δ γ ,γ ), fundamental(span { x γ : γ ∈ Γ } = X ) , and total ( (cid:104) f γ , x (cid:105) = 0 for each γ ∈ Γ implies x = 0 ). Obviously,a Schauder basis with the associated sequence of biorthogonal functionals is an M -basis;however, the concept of an M-basis is much more general: WCG Banach spaces such as c (Γ) for any set Γ , or C ( K ) -spaces, where K is a Valdivia compact space, admit M-bases.For these and other examples, consult, e.g. , [68, Corollary 5.2]. L p -spaces and related concepts Let (cid:54) λ < ∞ and (cid:54) p (cid:54) ∞ . A Banach space X is a L p,λ -space whenever everyfinite-dimensional subspace F of X is contained in another finite-dimensional space G of X whose Banach–Mazur distance to (cid:96) dim Gp is at most λ . A Banach space X is said to be a L p -space if it is a L p,λ -space for some λ . We will be primarily interested in the L ∞ -spacesand the L -spaces. A Banach space X is a Lindenstrauss space , whenever it is a L ∞ ,λ -spacefor every λ > ; equivalently, if X is an isometric predual of some L ( µ ) -space. Spaces ofthe form C ( K ) for some locally compact space K are natural examples of Lindenstraussspaces.We say that a Banach space X is a (cid:103) OL ∞ ,λ -space whenever every finite-dimensionalsubspace of X is contained in another finite-dimensional subspace whose Banach–Mazurdistance to some finite-dimensional C ∗ -algebra is at most λ . The (cid:103) OL ∞ -spaces are the spacewhich are (cid:103) OL λ, ∞ -spaces for some λ (cid:62) . The just-introduced terminology concerning (cid:103) OL ∞ -spaces is not standard as the non-commutative analogues of L ∞ ,λ -spaces—usually termed OL ∞ ,λ -spaces—are defined in termsof the completely bounded Banach–Mazur distance to the set of finite-dimensional C ∗ -algebras. OL ∞ ,λ are (cid:103) OL ∞ ,λ -spaces. A C ∗ -algebra is nuclear if and only if it is OL ∞ ,λ forsome λ > . We refer to [83] for additional information and a more precise definition ofthese spaces. Let Γ be a (possibly uncountable) set and let E be a Banach space that has a normalised,1-unconditional basis ( e γ ) γ ∈ Γ . Given a collection ( X γ ) γ ∈ Γ of Banach spaces, the E -sum of ( X γ ) γ ∈ Γ , denoted by ( (cid:76) γ ∈ Γ X γ ) E , is the set of all tuples ( x γ ) γ ∈ Γ with x γ ∈ X γ for each γ ∈ Γ , and such that (cid:80) γ ∈ Γ (cid:107) x γ (cid:107) e γ ∈ E . Formally, the E -sum depends on the choice of thebasis but when the basis is clear from the context we can afford this abuse of notation.If the basis ( e γ ) γ ∈ Γ is shrinking, the coordinate functionals ( e ∗ γ ) γ ∈ Γ associated to thatbasis form a normalised, 1-unconditional basis of E ∗ . In this case, the map Λ E : ( (cid:77) γ ∈ Γ X ∗ γ ) E ∗ → (cid:16) ( (cid:77) γ ∈ Γ X γ ) E (cid:17) ∗ given by (cid:10) ( x γ ) γ ∈ Γ , Λ E ( f γ ) γ ∈ Γ (cid:11) = (cid:88) γ ∈ Γ (cid:104) x γ , f γ (cid:105) for ( x γ ) γ ∈ Γ ∈ ( (cid:76) γ ∈ Γ X γ ) E and ( f γ ) γ ∈ Γ ∈ ( (cid:76) γ ∈ Γ X ∗ γ ) E ∗ , is an isometric isomorphism (see[104, Section 4] for details). When the basis ( e γ ) γ ∈ Γ is additionally boundedly complete(which implies that E is reflexive), the bidual of ( (cid:76) γ ∈ Γ X γ ) E is naturally isometricallyisomorphic to ( (cid:76) γ ∈ Γ X ∗∗ γ ) E .Let Γ be an infinite set and let { X γ : γ ∈ Γ } be a family of Banach spaces. The (cid:96) ∞ -sumof ( X γ ) γ ∈ Γ , denoted ( (cid:76) γ ∈ Γ X γ ) (cid:96) ∞ (Γ) , is the set of all tuples ( x γ ) i ∈ Γ such that x γ ∈ X γ ( γ ∈ Γ ) and (cid:107) ( x γ ) γ ∈ Γ (cid:107) (cid:96) ∞ (Γ) = sup γ ∈ Γ (cid:107) x γ (cid:107) < ∞ ; it is a Banach space with respect to thisnorm. When X γ = Z for each γ ∈ Γ , we write X = (cid:96) ∞ (Γ , Z ) .Let U be a non-trivial ultrafilter on Γ (assuming that Γ is infinite). Then N U (cid:0) ( X γ ) γ ∈ Γ (cid:1) = (cid:8) ( x γ ) γ ∈ Γ ∈ E : lim γ → U (cid:107) x γ (cid:107) = 0 (cid:9) is a closed subspace of X := ( (cid:76) γ ∈ Γ X γ ) E . The ultraproduct of { X γ : γ ∈ Γ } along U ,denoted [ X γ ] U , is defined as the quotient space X/N U (( X γ ) γ ∈ Γ ) ; when X γ = Z for all γ ∈ Γ , then we call the quotient space an ultrapower of Z (along U ). Let E be a normed space. A subset P ⊂ E is a cone , whenever λP = P for all λ > and P ∩ ( − P ) = { } . An ordered normed space is a normed space E with a distinguished cone P ; we will denote by ( E, P ) an ordered normed space or simply by E if the choice ofthe cone is clear from the context.For x, y ∈ E we write x (cid:54) y as long as y − x ∈ P . If there exists a number λ > such that whenever (cid:54) x (cid:54) y ( x, y ∈ E ) we have (cid:107) x (cid:107) (cid:54) λ (cid:107) y (cid:107) , the cone P is called normal . An order unit of ( E, P ) is an element e ∈ P such that for every x ∈ P there issome n ∈ N with x (cid:54) n · e . A convex subset B of P is called a base for P , whenever foreach non-zero x ∈ P there is a unique real number f ( x ) > such that f ( x ) − x ∈ B . Acone P is well-based in X if it has a bounded base B defined by a some f ∈ X ∗ , that is, B = { x ∈ P : (cid:104) f, x (cid:105) = 1 } . An element x ∈ P is a quasi-interior point of P , when the orderinterval [0 , x ] = { z ∈ E : 0 (cid:54) z (cid:54) x } is linearly dense in X . Whenever E is a Banachlattice or a C ∗ -algebra, by default we consider the (canonical) positive cone E + therein. Definition 2.3.1.
Let ( E, P ) be an ordered normed space. • ( E, P ) has the countable (Riesz) interpolation property , whenever given two se-quences ( x n ) ∞ n =1 , ( y n ) ∞ n =1 with x n (cid:54) y m ( n, m ∈ N ), there exists z ∈ E such that x n (cid:54) z (cid:54) y n ( n ∈ N ). • ( E, P ) has the countable monotone (Riesz) interpolation property , whenever for anynon-increasing sequence ( x n ) ∞ n =1 in E and any non-decreasing sequence ( y n ) ∞ n =1 in E with x n (cid:54) y n ( n ∈ N ), there exists z ∈ E such that x n (cid:54) z (cid:54) y n ( n ∈ N ).These two properties are equivalent in vector lattices but in general they are differentas witnessed by the C ∗ -algebra of × matrices.In C ∗ -algebras the positive cone is automatically normal with λ = 1 ([123, Theorem2.2.5]). Every C ∗ -algebra A decomposes into the real direct sum A = A sa ⊕ R iA sa , where A sa is the self-adjoint part of A , which is naturally a real ordered Banach space. Bydefinition, a C ∗ -algebra A has the countable (monotone) interpolation property, wheneverthe ordered space A sa has this property. The countable monotone interpolation propertyof C ∗ -algebras has been studied in [146, 147].Let A be a C ∗ -algebra decomposed into A sa ⊕ R iA sa . A functional f ∈ A ∗ is self-adjoint ( f ∈ ( A ∗ ) sa ), whenever (cid:104) f, x (cid:105) = (cid:104) f, x ∗ (cid:105) for every x ∈ A and this happens if and only if f restricted to A sa takes real values only. We have the canonical isometric identification ( A sa ⊕ R iA sa ) ∗ ≡ ( A sa ) ∗ ⊕ R i ( A sa ) ∗ . Let Y and Z be Banach spaces. We say that a Banach space X is a twisted sum of Y and Z if there exists a closed subspace M of X so that M is isomorphic to Y and X/M isisomorphic to Z . Note that the notion of a twisted sum is not symmetric.When M is complemented in X , we say that the twisted sum is trivial ; otherwise, thetwisted sum is non-trivial . When the twisted sum is trivial, X is naturally isomorphic tothe direct sum Y ⊕ Z .An exact sequence is a diagram(1) −−−→ Y j −−−→ X q −−−→ Z −−−→ in which X , Y and Z are Banach spaces, and j and q are continuous operators with j injective, q surjective and j ( Y ) = ker q ; thus j ( Y ) is a closed subspace of X and X/j ( Y ) is isomorphic to Z .A twisted sum of Y and Z can be identified with an exact sequence like (1), and thetwisted sum is trivial precisely when the exact sequence splits.Let ℘ be a property of Banach spaces. Then ℘ is a three-space property (or the classof spaces satisfying ℘ has the three-space property ), when ℘ is stable under twisted sums;meaning that if Y and Z in (1) have ℘ , then so has X . We refer to [28] for additionalinformation. Let K be a compact space and let L be a non-empty proper closed subset of K . Thenthe difference K \ L is a locally compact space, and C ( K \ L ) has codimension one in C (( K \ L ) ∞ ) , where ( K \ L ) ∞ is the one-point compactification of K \ L .Thus, if we consider the continuous map ϕ : K → ( K \ L ) ∞ defined by ϕ ( t ) = t for t ∈ K \ L and ϕ ( t ) = ∞ for t ∈ L , we obtain a natural exact sequence(2) −−−→ C ( K \ L ) φ −−−→ C ( K ) R −−−→ C ( L ) −−−→ , where φg = g ◦ ϕ and R is the restriction map f → f | L , which is surjective by the Tietze–Urysohn extension theorem.For a given set Γ , the Čech–Stone compactification β Γ of Γ comprises all ultrafilters on Γ ; this space is topologised by the base { U ∈ β Γ : A ∈ U } ( A ⊆ Γ ). By taking K = β N and L = β N \ N , the exact sequence (2) becomes → c → (cid:96) ∞ → (cid:96) ∞ /c → .More generally, for a given filter F on Γ the set U F consisting of all ultrafilters not extending F is open. Set K U = β Γ \ U F . By the Tietze–Urysohn extension theorem, theclosed subspace(3) c , F = { f ∈ (cid:96) ∞ (Γ) : ∀ ε > ∃ A ∈ F ∀ γ ∈ A | f γ | < ε } ⊆ (cid:96) ∞ (Γ) is isometrically isomorphic to C ( U F ) , the space of continuous functions vanishing at infinityon the locally compact space U F . In particular, we have the following short exact sequence(4) −−−→ c , F φ −−−→ (cid:96) ∞ (Γ) R −−−→ C ( K F ) −−−→ . Let E be a Banach space. A semigroup of operators on E is a map T ( · ) : [0 , ∞ ) → B ( E ) such that • T (0) = I E , the identity on E ; • T ( s + t ) = T ( s ) T ( t ) ( s, t (cid:62) );A semigroup T ( · ) is a C -semigroup whenever it is strongly continuous in the sense thatfor each x ∈ X , lim t → + (cid:107) T ( t ) x − x (cid:107) = 0 .A C -semigroup T ( · ) is uniformly continuous if lim t → (cid:107) T ( t ) − I E (cid:107) = 0 . A semigroup T ( · ) is locally integrable , whenever for each x ∈ E the map t (cid:55)→ T ( t ) x isstrongly measurable and (cid:82) t (cid:107) T ( s ) x (cid:107) d s < ∞ for all t > and x ∈ E .The infinitesimal generator A of a strongly continuous semigroup T ( · ) is defined by A x = lim t → + t ( T ( t ) − I ) x ( x ∈ E ) as long as the limit exists. C -semigroups of operators are a standard tool for solving abstract Cauchy problems.We refer to Lotz [116] for additional information. Grothendieck spaces and their properties often appear to be related to the followingcardinal number p , that is known in the literature as the pseudo-intersection number . Itis defined as the smallest cardinal number λ with the property that for every family A ofsubsets of N such that: • | A | < λ , • for all A , . . . , A n ∈ A ( n ∈ N ), the set A ∩ . . . ∩ A n is infinitethere exists an infinite set B such that B \ A is finite for every A ∈ A . Here B is the pseudo-intersection of A , hence the symbol p .It is known that p is an uncountable cardinal number between ω and the continuumand that Martin’s Axiom is equivalent to p = c ; in particular the value of p cannot bedetermined in ZFC in terms of the aleph numbers. We refer to [54] for further informationconcerning p and its applications in Analysis. In the present chapter we collect the main results concerning Banach spaces with theGrothendieck property as well as some related problems.
Definition 3.0.1.
A Banach space X is Grothendieck (or has the
Grothendieck property )whenever every weakly* convergent sequence in its dual X ∗ is weakly convergent. We begin with a portmanteau characterisation theorem of Grothendieck spaces. Mostof these equivalences were proved in [42, Theorem 1]. We present an easier proof.Let S n , S ∈ B ( X, Z ) ( n ∈ N ). We say that the sequence ( S n ) ∞ n =1 converges to S inthe strong operator topology if ( S n x ) ∞ n =1 converges to Sx for each x ∈ X ; and we saythat ( S n ) ∞ n =1 converges to S in the weak operator topology if ( (cid:104) S n x, z ∗ (cid:105) ) ∞ n =1 converges to ( (cid:104) Sx, z ∗ (cid:105) ) ∞ n =1 for each x ∈ X and z ∗ ∈ Z ∗ . Theorem 3.1.1.
For a Banach space X , the following assertions are equivalent:(1) X is a Grothendieck space; (2) every operator T : X → c is weakly compact;(3) for each separable space Y , every operator T : X → Y is weakly compact.(4) for each WCG space Y , every operator T : X → Y is weakly compact.(5) for each Y with W ∗ SC dual ball, every operator T : X → Y is weakly compact;(6) for each space Z , W ( X, Z ) is sequentially closed in B ( X, Z ) with respect to the weakoperator topology.(7) for each space Z , W ( X, Z ) is sequentially closed in B ( X, Z ) with respect to thestrong operator topology.Proof. The equivalence between clauses (1) and (2) follows easily from the description ofthe operators T : X → c by means of weak ∗ -null sequences in the dual space X ∗ ; theimplications (5) ⇒ (4) ⇒ (3) ⇒ (2) are trivial because WCG Banach spaces have W ∗ SCdual balls [44, Theorem XIII.4].For (1) ⇒ (5) suppose that X is Grothendieck, Y has weak ∗ sequentially compact dualball, and T : X → Y is an operator. Every bounded sequence ( f n ) ∞ n =1 in Y ∗ has a weakly*convergent subsequence ( f n k ) ∞ n =1 . Since ( T ∗ f n k ) ∞ n =1 is weakly* convergent in X ∗ , it is weaklyconvergent. Consequently T ∗ is weakly compact, and by Gantmacher’s theorem, so is T .For (1) ⇒ (6), let ( T n ) ∞ n =1 be a sequence of operators in W ( X, Z ) such ( T n x ) ∞ n =1 is weaklyconvergent in Z for each x ∈ X . By the uniform boundedness principle, ( T n ) ∞ n =1 is norm-bounded and T x = weak- lim T n x defines T ∈ B ( X, Z ) . We have to show that T is weaklycompact.Let Y = span (cid:83) ∞ n =1 { im T n : n ∈ N } and W = (cid:83) ∞ n =1 − n T n ( B X ) . Then W is a relativelyweakly compact subset of Z that generates Y , hence Y is WCG. By (4), since T takesvalues in Y , it is weakly compact.The implication (6) ⇒ (7) is trivial. For (7) ⇒ (2), let us consider an arbitrary operator T : X → c . Let P n denote the projection on c onto the subspace generated by the n firstterms of the unit vector basis ( n ∈ N ), then P n T is compact and converges to T in thestrong topology; hence T is weakly compact. (cid:3) A companion characterisation of Grothendieck spaces may be stated in terms of convexblock subsequences in the dual space (see, e.g. , [30, Proposition 4.2]).
Proposition 3.1.2.
A Banach space X has the Grothendieck property if and only if everyweak*-null sequence in X ∗ admits a convex block subsequence that converges in norm tozero. The above-stated characterisation of Grothendieck spaces still alludes to the propertiesof X ∗ rather than X itself. The first problem stated in [42] remains unsolved: Problem 1.
Does there exist an internal characterisation of Grothendieck spaces?Here ‘internal’ means depending only on the properties of the space X , not on those ofthe dual space X ∗ , or the operators on X .In general, convergence in the strong operator topology does not imply convergence inthe operator norm. However for Grothendieck C ( K ) -spaces the following result is available([35]; see also [11, Theorem 2.1]). Proposition 3.1.3.
Let K be a compact space such that C ( K ) is Grothendieck. Let ( T n ) ∞ n =1 be a sequence of contractions that converges in the strong operator topology. Then ( T n ) ∞ n =1 converges in the norm topology. In [42, Problem 8], Diestel asked if the class of Banach spaces Y such that the equality B ( X, Y ) = W ( X, Y ) holds for every Grothendieck space X coincides with WCG spaces.However, in the Addendum at the end of the paper, he remarked that the space (cid:96) (Γ) with Γ uncountable is a counterexample, and modified the problem as follows. Problem 2.
What class of Banach spaces Y is characterised by the equality B ( X, Y ) = W ( X, Y ) for every Grothendieck space X ?For future reference, let us record the following simple permanence properties of Gro-thendieck spaces. The proofs can be directly derived from the definition of Grothendieckspace. Proposition 3.1.4.
Let X and Y be Banach spaces.(1) If X and Y are Grothendieck spaces, then so is the direct sum X ⊕ Y .(2) If X is a Grothendieck space and there exists a surjective operator T : X → Y ,then Y is Grothendieck. In particular, quotients and complemented subspaces ofGrothendieck spaces are Grothendieck. The subsequent result demonstrates that the Grothendieck property is separably deter-mined.
Proposition 3.1.5.
If every separable subspace W of a Banach space X is contained ina subspace Z of X that is a Grothendieck space, then X is a Grothendieck space.Proof. Observe that an operator T is weakly compact if and only if all restrictions of T toseparable subspaces of X are weakly compact. Let T : X → c be an operator. Then T restricted to a separable subspace W of X must be weakly compact, because the restrictionof T to a Grothendieck superspace of W is weakly compact. The conclusion then followsfrom the equivalence between (1) and (2) in Theorem 3.1.1. (cid:3) The subsequent result is interesting in its own and will be helpful later.
Proposition 3.1.6.
A Banach space X has no quotient isomorphic to c if and only ifevery weakly* convergent sequence in X ∗ has a weakly Cauchy subsequence.Proof. If there exists a weakly* convergent sequence ( f n ) ∞ n =1 in X ∗ which has no weaklyCauchy subsequence then, by Rosenthal (cid:96) theorem and passing to a subsequence, wecan assume that ( f n ) ∞ n =1 is equivalent to the usual basis of (cid:96) . Thus ( g n ) ∞ n =1 defined by g n = f n +1 − f n ( n ∈ N ) is a weak ∗ -null sequence equivalent to the usual basis of (cid:96) , and T x = ( (cid:104) x, g n (cid:105) ) ∞ n =1 defines a surjective operator from X onto c .Conversely, if q : X → c is a quotient map, the adjoint q ∗ takes the standard unit basisof (cid:96) into a weak ∗ -null sequence in X ∗ without a weakly Cauchy subsequence. (cid:3) It is not too difficult to show that if every weakly* convergent sequence in X ∗ is weaklyCauchy subsequence, then X is Grothendieck.The following characterisation of Grothendieck spaces is due to Räbiger [135]. Theorem 3.1.7.
A Banach space X is Grothendieck if and only if X ∗ is weakly sequentiallycomplete and X has no quotient isomorphic to c .Proof. For the direct implication, note that weakly Cauchy sequences are weakly* conver-gent, and surjective operators onto c are not weakly compact.The converse implication easily follows from Proposition 3.1.6. (cid:3) Räbiger [135] noticed that for spaces isomorphic to complemented subspaces of a Banachlattice, being Grothendieck is equivalent to having no quotient isomorphic to c ; let us showthat this characterisation extends to spaces having a local unconditional structure. Proposition 3.1.8.
Suppose that E is a Banach space with a local unconditional structure.Then E is Grothendieck if and only if E has no quotient isomorphic to c .Proof. Suppose that E has a local unconditional structure. This is equivalent to the ex-istence of a Banach lattice F into which E ∗∗ embeds as a complemented subspace [53,Theorem 2.1]. Since E ∗ is complemented in E ∗∗∗ , it is also complemented in the dualBanach lattice F ∗ . Thus, E ∗ is weakly sequentially complete if and only if it does notcontain any subspace isomorphic to c [114, Theorem 1.c.7].If E has no quotient isomorphic to c , then E contains no complemented copy of (cid:96) ,hence E ∗ contains no copies of c [44, Theorem V.10], thus E ∗ is not weakly sequentiallycomplete. (cid:3) Let us revisit some consequences of the following result due to Hagler and Johnson ([67,Theorem 1]).
Theorem 3.1.9.
Let X be a Banach space. Suppose that X ∗ has an infinite-dimensionalclosed subspace without normalised weak ∗ -null sequences. Then X contains a subspaceisomorphic to (cid:96) . A version of the following result appears in [46], mentioning that its proof is similar tothat of Theorem 3.1.9. We can afford a more transparent argument.
Theorem 3.1.10. If X is Grothendieck and T : X → Y is a weakly pre-compact operator,then it is weakly compact.Proof. Given an operator T : X → Y , the DFJP factorisation of T introduced in [38] pro-vides an intermediate Banach space ∆( T ) and two operators j : ∆( T ) → Y and A : X → ∆( T ) such that T = jA . We will require the following two properties of the DFJP factori-sation:(a) T is weakly pre-compact if and only if ∆( T ) contains no copies of (cid:96) ;(b) T ∗ = A ∗ j ∗ is equivalent to the DFJP factorisation of T ∗ ; in particular, ∆( T ∗ ) isisomorphic to ∆( T ) ∗ ; which are [75, Theorem 2.3] and [60, Theorem 1.5].Suppose that T : X → Y is not weakly compact. Then by Gantmacher’s theorem, theadjoint T ∗ is not weakly compact either, so there exists a bounded sequence ( g i ) ∞ i =1 in Y ∗ such that ( T ∗ g i ) ∞ i =1 does not have a weakly convergent subsequence. Since weakly Cauchysequences in dual spaces are weakly* convergent and X is Grothendieck, ( T ∗ g i ) ∞ i =1 failsto have weakly Cauchy subsequences; hence T ∗ is not weakly pre-compact. Passing toa subsequence, by Rosenthal’s (cid:96) -theorem we may suppose that both sequences ( g i ) ∞ i =1 , ( T ∗ g i ) ∞ i =1 are equivalent to the unit vector basis of (cid:96) . In this case, the operator T ∗ is anisomorphism on the closed subspace N generated by ( g i ) ∞ i =1 .Property (b) allows for taking A ∗ j ∗ as the DFJP factorisation of T ∗ . Then j ∗ ( N ) isan infinite-dimensional closed subspace of ∆( T ∗ ) without normalised weak ∗ -null sequences.Indeed, otherwise if ( d i ) ∞ i =1 is such a sequence and X is Grothendieck, ( A ∗ d i ) ∞ i =1 would bea semi-normalised weakly null sequence in T ∗ ( N ) , which is not possible because T ∗ ( N ) isisomorphic to (cid:96) . Since ∆( T ) ∗ ∼ = ∆( T ∗ ) , Theorem 3.1.9 implies that ∆( T ) contains a copyof (cid:96) ; hence T is not weakly pre-compact by property (a). (cid:3) Corollary 3.1.11.
Every non-reflexive Grothendieck space contains a copy of (cid:96) . Corollary 3.1.11 was strengthened in [74], where it was proved that under Martin’s Axiomand the negation of the Continuum Hypothesis (or, actually under p > ω ), every non-reflexive Grothendieck space contains a copy of (cid:96) ( p ) . We may then record the followingresult. Proposition 3.1.12.
A non-reflexive Grothendieck space has density at least p (cid:62) ω . When p = c , then one can deduce that every non-reflexive Grothendieck space has (cid:96) ∞ as a quotient.Corollary 3.1.11 implies that [42, Problem 10] has a negative answer: indeed, the dualof a Grothendieck space cannot be isomorphic to (cid:96) (Γ) . Even more is true, as Haydonproved in [72] that the dual of a non-reflexive Grothendieck space contains a copy of thenon-separable space L ([0 , p ) ≡ L ( { , } p ) . However, when p < c one may ask whethera non-reflexive Grothendieck space of density p exists. This may be indeed so as provedby Brech [20], who used forcing to construct a compact space K for which the space C ( K ) is Grothendieck and has density p < c . The space K arises as the Stone space of a certainBoolean algebra, hence it is totally disconnected.In [42, Problem 5], Diestel asked whether a non-reflexive Grothendieck space containsa copy of (cid:96) ∞ . A negative answer was provided by Haydon in [71] (see Example 4.1.3), buta weaker problem is in place. Problem 3.
Does a non-reflexive Grothendieck space contain a copy of c ?For spaces with property ( V ) a convenient characterisation of the Grothendieck propertyis available (that was noted, e.g. , in [58, Theorem 28]; a further characterisation may befound in [33, Corollary 9]). Proposition 3.1.13.
Let X be a Banach space with property ( V ) . Then X is Grothendieckif and only if it contains no complemented copies of c . Proof. If X is not Grothendieck, every non-weakly compact operator T : X → c is boundedbelow on some subspace M of X isomorphic to c . Moreover, M is complemented in X because T ( M ) is complemented in c . (cid:3) Corollary 3.1.14. If X is isomorphic to a complemented subspace of a dual space withproperty ( V ) , then it is a Grothendieck space.Proof. Dual spaces contain no complemented copies of c . (cid:3) We are now going to describe certain characterisations of Banach spaces with the Gro-thendieck property obtained in [61] in terms of polynomials.Let X be a Banach space and let k ∈ N . We denote by P ( k X ) the space of all k -homogeneous scalar polynomials defined on X . We have P ( X ) = X ∗ . The space P ( k X ) endowed with the norm (cid:107) P (cid:107) = sup x ∈ B X | P ( x ) | is a dual Banach space. Moreover, every P ∈ P ( k X ) has a canonical extension (cid:98) P ∈ P ( k X ∗∗ ) , which is an extension of P by weak ∗ continuity. [61, Theorem] yields a characterisation of Grothendieck spaces in terms ofhomogeneous polynomials. Proposition 3.1.15.
For a Banach space X , the following assertions are equivalent:(a) X has the Grothendieck property;(b) for every k ∈ N , given a sequence ( P n ) ⊂ P ( k X ) with P n ( x ) → for all x ∈ X ,then (cid:98) P n ( x ∗∗ ) → for all x ∗∗ ∈ X ∗∗ ;(c) the same statement as (b) is true for some k . Further characterisations of the Grothendieck property in terms of vector-valued homo-geneous polynomials P ( k X, Y ) may be found in [61] for details. Characterisations in termsof weak-Riemann integrability of weak*-continuous functions are also available ([115]). We begin with an auxiliary lemma from [64, Theorem 2.7] that can be seen as a ‘liftingresult for sequences’ associated to the Grothendieck property.
Lemma 3.2.1.
Let M be a closed subspace of a Banach space X such that X/M has theGrothendieck property, let J : M → X denote the embedding, and let J ∗ : X ∗ → M ∗ beits adjoint. Then every weakly* convergent sequence ( f n ) ∞ n =1 in X ∗ such that ( J ∗ f n ) ∞ n =1 isweakly convergent has a weakly convergent subsequence ( f n k ) ∞ k =1 . The following result was obtained in [64, Corollary 2.8]. We shall employ it for con-structing new examples of Grothendieck spaces.
Proposition 3.2.2.
The class of Grothendieck Banach spaces has the three-space property.Proof.
Suppose that a Banach space X has a closed subspace M such that both M and X/M are Grothendieck spaces. If J : M → X denotes the embedding, then the adjointoperator J ∗ : X ∗ → M ∗ is surjective by the Hahn–Banach theorem.Given a weakly* convergent sequence ( f n ) ∞ n =1 in X ∗ , ( J ∗ f n ) ∞ n =1 is weakly* convergent;hence weakly convergent in M ∗ because M ∗ is Grothendieck. Since X/M is Grothendieck, each subsequence of ( f n ) ∞ n =1 has a weakly convergent subsequence by Lemma 3.2.1, hence ( f n ) ∞ n =1 is weakly convergent, and we conclude that X is Grothendieck. (cid:3) As we will see later, many Grothendieck spaces satisfy the following property that weterm ( V ∞ ) . Definition 3.2.3.
A Banach space X has property ( V ∞ ) when every non-weakly compactoperator T : X → Y is an isomorphism on a subspace of X isomorphic to (cid:96) ∞ .Spaces with property ( V ∞ ) satisfy the following results. Proposition 3.2.4.
Let X and Y be Banach spaces. (i) If X has property ( V ∞ ) , then it is Grothendieck. (ii) If X and Y have property ( V ∞ ) , then so has X ⊕ Y . (iii) If X has property ( V ∞ ) and there exists a surjective operator T : X → Y , then Y hasproperty ( V ∞ ) . In particular, quotients and complemented subspaces of spaces withproperty ( V ∞ ) have also this property.Proof. (i) is a direct consequence of Theorem 3.1.1, and the proofs of (ii) and (iii) areeasy. (cid:3) The space C [0 , has property ( V ) but it is not Grothendieck because it contains com-plemented copies of c . (Actually, a separable Banach space is Grothendieck if and only ifit is reflexive). That cannot happen for dual spaces: Proposition 3.2.5. If X is isomorphic to a complemented subspace of a dual space andit has property ( V ) , then X has property ( V ∞ ) . In particular, X is Grothendieck.Proof. Suppose that a space X is complemented in a dual space and has property ( V ) , andlet T : X → Y be a non-weakly compact operator. Then T is is an isomorphism on a copyof c in X , hence it is an isomorphism on a copy of (cid:96) ∞ in X by a result of Rosenthal [136,Theorem 1.3]. (cid:3) It was proved by Pfitzner ([130]; see Theorem 4.2.1) that C ∗ -algebras have property ( V ) (see [52] for an alternative proof). Consequently, the previous result implies that thespace B ( H ) of bounded operators on a Hilbert space H (and every other von Neumannalgebra) is a Grothendieck space. Thus [42, Problem 2] has positive answer. It may lookcoincidental that known examples of Grothendieck spaces have property ( V ) , hence it isnatural to ask if it is always the case. This problem, which we record below, was originallyraised by Diestel in [43]. Problem 4.
Do Grothendieck spaces have property ( V ) ?We observe that property ( V ) is not a three-space property [27]. Moreover, in light ofProposition 3.2.5, one may propose a weaker problem. Problem 5.
Do dual spaces with the Grothendieck property have property ( V ) ?In view of Theorem 3.1.7, the next result could be helpful to solve Problem 4. Proposition 3.2.6.
A Banach space X has property ( V ) if and only if X ∗ is weaklysequentially complete and for every unconditionally converging operator T : X → Y , theadjoint T ∗ is weakly pre-compact.Proof. Suppose that X has property ( V ) . Then X ∗ is weakly sequentially complete by[69, Propositions 3.3.D and 3.3.F]. Moreover, if T : X → Y is not an isomorphism ona copy of c , then T is weakly compact; hence T ∗ is weakly compact and cannot act asan isomorphism on a copy of (cid:96) .Conversely, if X fails property ( V ) , then there exists a non-weakly compact operator T : X → Y which is not an isomorphism on any copy of c in X . Since T ∗ is not weaklycompact, there exists a bounded sequence ( g n ) ∞ n =1 in Y ∗ such that ( T ∗ g n ) ∞ n =1 has no weaklyconvergent subsequence. Thus X ∗ is not weakly sequentially complete or ( T ∗ g n ) ∞ n =1 hasno weakly Cauchy subsequence. In the latter case, ( g n ) ∞ n =1 has a subsequence generatinga subspace isomorphic to (cid:96) on which T ∗ is an isomorphism, and the result is proved. (cid:3) Räbiger in [135] gave the following partial positive answer to Problem 4.
Proposition 3.2.7.
Every Grothendieck space X isomorphic to a complemented subspaceof a Banach lattice E has property ( V ) .Proof. Since X ∗ is weakly sequentially complete (Theorem 3.1.7), we may suppose that thenorm in E ∗ is order continuous ([121, Theorem 5.1.15]). Let P : E → E be a projectiononto X .By Proposition 3.2.6, it is enough to show that for every unconditionally convergingoperator T : X → Y , the adjoint T ∗ is weakly pre-compact. But T P unconditionallyconverging implies that P ∗ T ∗ is weakly pre-compact ([121, Theorem 3.4.20], hence T ∗ isweakly pre-compact. (cid:3) The subsequent results characterise reflexivity of dual spaces with property ( V ) . Proposition 3.2.8. If X is a Grothendieck space and X ∗ has property ( V ) , then X isreflexive.Proof. If X ∗ is non-reflexive and has property ( V ) , then it contains a copy of c [69,Chapter III, Corollary 3.3.c]. Thus X ∗ is not weakly sequentially complete, hence X is notGrothendieck). (cid:3) The following problem was posed by Diestel [42, Problem 6] is then naturally relatedwith the previously presented ones.
Problem 6.
Suppose that X and X ∗ are Grothendieck. Is X reflexive?However, this would be indeed the case had Problem 5 had affirmative solution.Diestel [42, Problem 7] also asked if X is reflexive when both X and X ∗ are weaklysequentially complete. A negative answer was provided by Bourgain and Delbaen in [19],by showing the existence of an infinite-dimensional L ∞ -space satisfying the Schur property.For each compact Hausdorff space K , the bidual C ( K ) ∗∗ is an injective space, hence itis Grothendieck. More generally, for every C ∗ -algebra A , the bidual A ∗∗ is a von Neumann algebra, hence a Grothendieck space. In this light, Diestel’s [42, Problem 4] appears evenmore natural: Problem 7.
Let X be a Grothendieck space. Is X ∗∗ Grothendieck?
Remark . It is to be noted that in general property ( V ) does not pass to biduals, because X = ( (cid:76) n ∈ N (cid:96) n ) c has property ( V ) , yet X ∗∗ = ( (cid:76) n ∈ N (cid:96) n ) (cid:96) ∞ contains a complemented copyof (cid:96) [80, p. 303], hence it fails property ( V ) .Curiously, even if we knew that (dual) Grothendieck spaces have property ( V ) , it wouldnot be automatic that biduals of such spaces still had property ( V ) . The present section is devoted to results showing that the Grothendieck property pre-vents Banach spaces from admitting certain structures generalising Schauder bases.The following result is due to W. B. Johnson [79].
Theorem 3.3.1.
Every Grothendieck space admitting an M-basis is reflexive.Proof.
Let { ( x i , f i ) : i ∈ I } be an M-basis in a Banach space X , and let Y denote the closedsubspace of X ∗ generated by { f i : i ∈ I } . It is enough to show that Y is reflexive. Indeed,since Y is total over X , Y is weak ∗ -dense in X ∗ . Therefore, if the unit ball B Y is weaklycompact, then it is σ ( X ∗ , X ) -compact, so that it follows from the Krein–Smulian theorem[120, Theorem 2.7.11] that Y is weak ∗ -closed, hence Y = X ∗ .Let ( y n ) ∞ n =1 be a sequence in B Y . Since each y ∈ Y is the norm limit of a sequencein span { f i : i ∈ I } , for each n the set A n = { i ∈ I : (cid:104) y n , x i (cid:105) (cid:54) = 0 } is countable; thus sois (cid:83) ∞ n =1 A n . A standard diagonalisation argument shows that there exists a subsequence ( y n k ) ∞ k =1 such that ( (cid:104) y n k , x i (cid:105) ) ∞ k =1 converges for each i ∈ I . Since ( y n k ) ∞ k =1 is bounded and { x i : i ∈ I } generates a dense subspace of X , ( (cid:104) y n k , x (cid:105) ) ∞ k =1 converges for each x ∈ X . Thus ( y n k ) ∞ k =1 is weak ∗ -convergent to some y ∈ X ∗ . Since X is Grothendieck, ( y n k ) ∞ k =1 is weaklyconvergent to y . Note that y ∈ Y because Y is (weakly) closed. Thus Y is reflexive andthe proof is complete. (cid:3) In relation with Theorem 3.3.1, Diestel [42, Problem 9] asked if a Banach space withan M-basis can contain (cid:96) ∞ . The answer is that it can contain (cid:96) ∞ , but not (cid:96) ∞ (Γ) for anyuncountable set Γ (see [68, Corollary 5.4]).Subsequently, we discuss the non-existence of Schauder decompositions in Banach spaceswith the Grothendieck property. Observe that (cid:96) and (cid:96) ( (cid:96) ∞ ) are Grothendieck spaces (seeProposition 5.4.1) admitting natural Schauder decompositions, so we have to impose anadditional condition to that of being Grothendieck that would prevent the existence of aSchauder decomposition. Proposition 3.3.2.
Let ( P i ) ∞ i =1 be a Schauder decomposition in a Banach space X with theGrothendieck property. Then the sequence of adjoint projections ( P ∗ i ) ∞ i =1 forms a Schauderdecomposition in X ∗ . Proof.
Let S n = P + · · · + P n ( n ∈ N ). Then S n x → x as n → ∞ for each x ∈ X .Therefore, given x ∈ X and f ∈ X ∗ , lim n →∞ (cid:104) S ∗ n f, x (cid:105) = lim n →∞ (cid:104) f, S n x (cid:105) = (cid:104) f, x (cid:105) . Since ( S ∗ n f ) ∞ n =1 is weakly* convergent to f and the space X is Grothendieck, ( S ∗ n f ) ∞ n =1 is weakly convergent to f ; and to finish the proof, it is enough to show that it is normconvergent.The union of ranges D = (cid:83) ∞ n =1 im S ∗ n is a dense subspace of X ∗ such that ( S ∗ n g ) ∞ i =1 is norm -convergent to g for each g ∈ D , and C = sup n ∈ N (cid:107) S n (cid:107) < ∞ by the Banach–Steinhaus theorem. Thus, given f ∈ X ∗ , since we can choose g ∈ D arbitrarily close to f ,the inequality (cid:107) S ∗ k f − S ∗ l f (cid:107) (cid:54) (cid:107) S ∗ k ( f − g ) (cid:107) + (cid:107) S ∗ k g − S ∗ l g (cid:107) + (cid:107) S ∗ l ( g − f ) (cid:107) shows that ( S ∗ n f ) ∞ n =1 is norm-convergent to f . (cid:3) The next two results will be the key in the proof of Proposition 3.3.5.
Lemma 3.3.3.
Let ( P i ) ∞ i =1 be a Schauder decomposition in a Banach space X , and let ( x i ) ∞ i =1 be a sequence with x i ∈ P i ( X ) for each i ∈ N . Then ( x i ) ∞ i =1 is a basic sequence.Proof. This is a standard argument. If x ∈ span { x i : i ∈ N } , then x = (cid:80) ∞ i =1 P i ( x ) ; so it isenough to show that P i ( x ) is a multiple of x i for each i .Otherwise, for some j ∈ N we could find f ∈ im P ∗ j such that (cid:104) f, x j (cid:105) = 0 , however (cid:104) f, P j x (cid:105) (cid:54) = 0 . Since (cid:104) f, x i (cid:105) = 0 for each i ∈ N , (cid:104) f, x (cid:105) = (cid:104) P ∗ j f, x (cid:105) = (cid:104) f, P j x (cid:105) ; a contra-diction. (cid:3) Lemma 3.3.4.
Let ( P i ) ∞ i =1 be a Schauder decomposition in a Banach space X with theGrothendieck property. Then (i) every bounded sequence ( x i ) ∞ i =1 in X with x i ∈ im P i for each i ∈ N is weakly null. (ii) every bounded sequence ( f i ) ∞ i =1 in X ∗ with f i ∈ im P ∗ i for each i ∈ N is weakly null,and span { f i : i ∈ N } is a reflexive subspace.Proof. (i) Recall that S n = P + · · · + P n . If x i ∈ im P i , then x i = ( I − S i − ) x i for i > .Thus, for every f ∈ X ∗ we have |(cid:104) f, x i (cid:105)| = |(cid:104) ( I − S ∗ i − ) f, x i (cid:105)| (cid:54) (cid:107) ( I − S ∗ i − ) f (cid:107) · (cid:107) x i (cid:107) , and, by Proposition 3.3.2, (cid:107) ( I − S ∗ i − ) f (cid:107) → as i → ∞ . Consequently, ( x i ) ∞ i =1 is a weaklynull sequence.(ii) Proceeding as in (i), we can show that ( f i ) ∞ i =1 is weak ∗ null, hence it is weakly nullby the Grothendieck property of X .On the other hand, if span { f i : i ∈ N } were not reflexive, then, by Rosenthal’s (cid:96) -theorem, it would contain a subspace isomorphic to (cid:96) . Since ( f i ) ∞ i =1 is a basic sequence byLemma 3.3.3, we could construct a bounded sequence of successive blocks g j = (cid:80) i ∈ A j c i f i ( j ∈ N, A j ⊂ N ) with no weakly convergent subsequence. Taking Q j = (cid:80) i ∈ A j P i we obtainanother Schauder decomposition ( Q j ) ∞ j =1 of X . Since g j ∈ im Q ∗ j for each j , we would getthat ( g j ) ∞ j =1 is weakly null; a contradiction. (cid:3) The following result was proved by Dean [39]. To see that Dean’s statement and oursare equivalent, observe that it is an easy exercise to show that a Banach space Z has theDunford–Pettis property if and only if given weakly compact operators S : Z → Y and T : X → Z , the product ST is compact. Proposition 3.3.5.
Grothendieck spaces with the Dunford–Pettis property do not admitSchauder decompositions.Proof. If X is Grothendieck and ( P i ) ∞ i =1 is a Schauder decomposition in X , then we canselect a normalised sequence ( f i ) ∞ i =1 in X ∗ with f i ∈ im P ∗ i . Since P ∗ i f i = f i , we can finda bounded sequence ( x i ) ∞ i =1 in X with x i ∈ im P i such that (cid:104) f i , x i (cid:105) = 1 for each i ∈ N .By Lemma 3.3.4, both sequences ( x i ) ∞ i =1 and ( f i ) ∞ i =1 are weakly null; hence X fails theDunford–Pettis property. Indeed, T x = ( (cid:104) f i , x (cid:105) ) ∞ i =1 defines a weakly compact operator T : X → c and ( T x i ) ∞ i =1 does not converge in norm to . (cid:3) We say that a Banach space X has the surjective Dunford–Pettis property if every sur-jective operator from X onto a reflexive Banach space takes weakly convergent sequencesinto convergent sequences. Clearly, spaces with the Dunford–Pettis property have the sur-jective Dunford–Pettis property, but the example described in Proposition 4.6.3 shows thatthe converse implication fails.Leung [107] improved Proposition 3.3.5 as follows. Proposition 3.3.6.
Grothendieck spaces with the surjective Dunford–Pettis property donot admit Schauder decompositions.Proof.
Assume that X is a Grothendieck space that has a Schauder decomposition ( P i ) ∞ i =1 .As in the proof of Proposition 3.3.5, we select bounded sequences ( f ) ∞ i =1 and ( f i ) ∞ i =1 with f i ∈ im P ∗ i , x i ∈ im P i and (cid:104) f i , x i (cid:105) = 1 for each i ∈ N .By Lemma 3.3.4, N = span { f i : i ∈ N } is reflexive, hence ( ⊥ N ) ⊥ = N so X/ ⊥ N isreflexive too. Let Q : X → X/ ⊥ N denote the quotient map. Then the sequence ( x i ) ∞ i =1 isweakly null, but ( Qx i ) ∞ i =1 does not converge in norm to , because (cid:104) f i , Qx i (cid:105) = 1 for each i ∈ N . Consequently, X fails the surjective Dunford–Pettis property. (cid:3) For a Banach space X , we considerAt ( X ) = { f ∈ X ∗ : (cid:107) f (cid:107) = (cid:104) f, x (cid:105) for some x ∈ S X } , the set of all norm-attaining functionals in X ∗ . A well-known result of James establishesthat X is reflexive if and only if At ( X ) = X ∗ .Debs, Godefroy, and Saint-Raymond [40] proved that At ( X ) is not a weak ∗ - G δ subsetof X ∗ when X is separable and non-reflexive. Acosta and Kadets extended this result ([1,Theorem 2.5]) as follows. Proposition 3.4.1. If X is a Banach space and At( X ) is a weak ∗ - G δ subset of X ∗ , then X is Grothendieck. We do not know if the sufficient condition for being a Grothendieck space, presented inProposition 3.4.1, is actually also necessary.
Problem 8.
Let X be a Grothendieck space. Is At( X ) a weak ∗ - G δ subset of X ∗ ? It was Grothendieck himself [66] who proved that for a discrete set Γ , (cid:96) ∞ (Γ) ≡ C ( β Γ) isa Grothendieck space. It follows immediately from the fact that if K is Stonean space , thatis, compact and extremally disconnected, then C ( K ) is injective (hence complemented in (cid:96) ∞ (Γ) , where Γ is the unit ball of C ( K ) ∗ ), so also Grothendieck.We will use the following terminology introduced by Seever [143]. Definition 4.1.1.
A compact (Hausdorff) space K is a G-space whenever C ( K ) has theGrothendieck property.Thus we may say that Stonean spaces are G-spaces. Note that if K is a G-space and L isa continuous image of K , then L need not be a G-space. Indeed, N ∞ is a continuous imageof β N ; and more generally, every compact space K is a continuous image of βK d , where K d is K endowed with the discrete topology. Actually, every Stonean space K continuouslysurjects onto a space L , which is not a G-space, yet every weakly* convergent sequence ofpurely atomic measures on L is weakly convergent [88, Theorem 7.3].Convergent sequences in a compact space K that are non-trivial, in the sense that theyare not eventually constant, give rise to complemented copies of c , hence a G-space cannothave non-trivial convergent sequences. Such a condition is however not sufficient as theproduct of two infinite compact spaces K and L is never a G-space (a more general variantof this result is discussed in Section 5.3; and a further strengthening may be found in[88, Theorem 11.3]; see also [86]). In [87, Theorem 7] it was noticed that C ( K ) is not a Grothendieck space if and only if it is isomorphic to a space C ( L ) with L containinga non-trivial convergent sequence. A strengthening of this observation is available andmay be found. e.g. , in [98, Proposition 6.12]. Proposition 4.1.2.
Let K be a compact space. Suppose that ( µ n ) ∞ n =1 is a weakly* con-vergent sequence in C ( K ) ∗ with no weakly convergent subsequence. Then C ( K ) containsa complemented subspace isomorphic to c .Proof. Let ( µ n ) ∞ n =1 be a weakly* convergent sequence in M ( K ) , the space of Borel measureson K identified with the dual of C ( K ) , that does not have any weakly convergent sequence.As ( µ n ) ∞ n =1 does not have weakly convergent subsequences, it follows from the Eberlein–Smulian theorem together with the Dieudonné–Grothendieck theorem ([44, Theorem 14 inChapter VII]) that there are pairwise disjoint open subsets U n ⊂ K ( n ∈ N ) and δ > such that | µ n ( U n ) | (cid:62) δ . Since ( µ n ) ∞ n =1 is weakly* convergent (to some measure µ ), we may assume passing to a subsequence if necessary, that | ( µ − µ n )( U n ) | > δ/ . Using Urysohn’slemma, for some M > and each n ∈ N we may find a function f n ∈ C ( K ) of norm atmost M whose support is contained in U n and (cid:104) µ − µ n , f n (cid:105) = 1 . Since the functions f n have pairwise disjoint supports ( n ∈ N ) , they span an isometric copy of c . Moreover, theexpression P f = ∞ (cid:88) n =1 (cid:104) µ − µ n , f · U n (cid:105) f n (cid:0) f ∈ C ( K ) (cid:1) defines a projection on C ( K ) onto span { f n : n ∈ N } . (cid:3) In an unpublished note, Plebanek proved that there exists a non-separable compactHausdorff space K that is not a G-space, yet every closed, separable subspace L ⊂ K isa G-space (an exposition of this construction may be found in [14] or [87]).As already mentioned, every Stonean space is a G-space, because C ( K ) is then comple-mented in C ( β Γ) for some set Γ . (A compact space K is Stonean if and only if disjointopen sets in K have disjoint closures; Stonean spaces are precisely Stone spaces of completeBoolean algebras.)Pérez Hernández asked during the Winter School in Abstract Analysis 2017 held inSvratka, Czech Republic, for a characterisation of filters F on N for which the space c , F (see Section 2.5) is complemented in (cid:96) ∞ (hence isomorphic to (cid:96) ∞ by [112]). Leonetti [106]proved that for every filter F such that there exists an uncountable family B ⊂ P ( N ) \ F withthe property that for any two distinct sets N, M ∈ B the union N ∪ M is in F , the subspace c , F is not complemented in (cid:96) ∞ . In [92], the result was strengthened and the question ofwhether the space c , F for a filter satisfying the property distilled by Leonetti is not a Gro-thendieck space was asked. However, for the intersection of finitely many ultrafilters, thecorresponding space has finite codimension in (cid:96) ∞ , and as such, it is a Grothendieck space. Problem 9.
Characterise filters F for which c , F is a Grothendieck space.In 1964, Lindenstrauss [111] studied the C ( K ) -spaces with the property that every op-erator from C ( K ) to a separable Banach space is weakly compact; hence, Grothendieck C ( K ) -spaces, and—using Seever’s terminology—he noticed that F-spaces are G-spaces.The same result was later independently found again by Seever [143]. Remark . Let us list some further examples of G-spaces: • σ -Stonean spaces (Stone spaces of σ -complete Boolean algebras): [4]. • Basically disconnected spaces, also known as Rickart spaces (open σ -compact sub-sets have open closures): [144]. • F-spaces (disjoint open F σ -sets have disjoint closures; equivalently, the Banachlattice C ( K ) has the countable monotone interpolation property): [111, 143]; seealso [146, Theorem 4.6] and [147]. • Weakly Koszmider spaces, which are infinite compact spaces for which every oper-ator T : C ( K ) → C ( K ) is centripetal , that is, for any bounded sequence ( f n ) ∞ n =1 ofdisjointly supported functions in C ( K ) and any sequence ( x n ) in K with f n ( x n ) = 0 for all n ∈ N , we have ( T f n )( x n ) → as n → ∞ . (See [142, Theorem 4.4] for theproof that weakly Koszmider compact spaces are G-spaces.)When K is a weakly Koszmider space such that K \ F is connected for any finiteset F ⊂ K , the Banach space C ( K ) is indecomposable in the sense that each com-plemented subspace of C ( K ) is either finite-dimensional or has finite codimension;in particular such spaces do not contain complemented copies of c .In the literature one can find several constructions of (weakly) Koszmider spaces;see e.g. , [97, 100, 132].We remark in passing that there exist connected F-spaces ( e.g. , the Čech–Stone remain-der β [0 , ∞ ) \ [0 , ∞ ) as observed by Seever [143]), hence not every G-space arises as a Stonespace of a certain Boolean algebra. Such observations led Diestel [42, Problem 3] to restatea problem posed by Lindenstrauss in [111, p. 224] as follows: Problem 10.
Is there an intrinsic characterisation of G-spaces? More precisely, can G-spaces be characterised topologically?A topological space X is a ∆ - space , whenever for every non-increasing sequence ( D n ) ∞ n =1 of subsets of X with empty intersection, there exists a non-increasing sequence ( V n ) ∞ n =1 ofopen subsets of X whose intersection is empty and D n ⊆ V n for every n ∈ N . The notionof a ∆ -space was introduced by Knight [96]. Kąkol and Leiderman observed that infinitecompact ∆ -spaces are not G-spaces ([85, Corollary 3.14]).Next we briefly describe an important example due to Haydon [71].
Example 4.1.3.
In ZFC, there exists a G-space K such that C ( K ) does not containsubspaces isomorphic to (cid:96) ∞ , however it admits a quotient isomorphic to (cid:96) ∞ .The compact space K is the Stone space of a certain algebra A of subsets of N thatcontains the finite sets; the space C ( K ) may be identified with a closed sub- C ∗ -algebra of (cid:96) ∞ generated by the indicator functions of the sets in A . As A contains the finite sets, thealgebra contains the natural copy of c in (cid:96) ∞ .Assuming the Continuum Hypothesis, Talagrand constructed in [148] a G-space L suchthat C ( L ) does not admit any quotients isomorphic to (cid:96) ∞ . The latter condition is equivalentto the fact that C ( L ) does not contain subspaces isomorphic to (cid:96) (Γ) with Γ uncountable.Let (Ω , Σ , µ ) be a measure space. Then the space L ∞ ( µ ) of all µ -essentially boundedscalar-valued functions on Ω is naturally a (complex) Banach lattice (even an AM-space)as well as a commutative C ∗ -algebra, where in either case the lattice/algebra operationsare defined pointwise. It follows from Kakutani’s representation theorem ([114, Theorem1.b.3]) for AM-spaces, or from the Gelfand–Naimark theorem for commutative C ∗ -algebrasthat L ∞ ( µ ) is isometric to a C ( K ) -space. When L ∞ ( µ ) is naturally representable as thedual space of L ( µ ) , it is injective, hence Grothendieck because it is complemented in (cid:96) ∞ (Γ) for some set Γ . There is an exact measure-theoretic condition for µ characterising when L ∞ ( µ ) is a dualspace; measures with this property are called strictly localisable , however we do not requireto invoke the details of this condition here. Example 4.1.4.
Let ν be the counting measure on an uncountable set Γ . Then L ∞ ( µ ) is not a dual space. Indeed, L ∞ ( ν ) is the linear span of (cid:96) c ∞ (Γ) , the subspace of all countablysupported functions on Γ , and the constant function in (cid:96) ∞ (Γ) .The space L ∞ ( ν ) is not injective, but it is Grothendieck.That (cid:96) c ∞ (Γ) is a non-injective Grothendieck space was first remarked by Pełczyński andSudakov ([129, p. 87]; see also [81, Proposition 3.7]). Proposition 4.1.5.
For every (non-negative) measure µ , the space L ∞ ( µ ) is Grothendieck.Proof. As L ∞ ( µ ) is lattice-isometric to C ( K ) for some compact space K , and every boundeddisjoint sequence in L ∞ ( µ ) has a supremum, the same happens in C ( K ) . This means that C ( K ) is a Dedekind σ -complete Banach lattice, which translates into the fact that thecompact K is σ -Stonean. Thus, the Grothendieck property of L ∞ ( µ ) follows from theprevious Remark 2. See also [4]. (cid:3) Let A be a field of sets (a concrete Boolean algebra). The Stone space St A of A coincideswith the maximal ideal space of the Banach algebra B ( A ) of all bounded, scalar-valued, A -measurable functions (endowed with the supremum norm). As B ( A ) is an abstract M -space/commutative C ∗ -algebra, the space C (St A ) is isometrically isomorphic to B ( A ) , sothat we can freely interchange between the two descriptions of the same object.One of the most interesting examples of algebras whose Stone space is not a G-space isthe algebra of Jordan-measurable subsets of the unit interval, that is sets whose boundaryhas Lebesgue measure zero (see, e.g. , [65, Corollary 5.8] for an extension of this fact tomore general Jordan algebras). Schachermayer [141, Proposition 4.6] observed that Stonespaces of algebras expressible as strictly increasing unions of countably many subalgebrasare not G-spaces.In light of the first clause above one may ask about characterisation of those Booleanalgebras (or more concretely, fields of sets) whose Stone spaces are G-spaces. This is indeedan active area of research with a strong set-theoretic flavour.Let us list some examples of algebras whose Stone spaces are G-spaces: • subsequentially complete Boolean algebras (every disjoint sequence in the algebrahas a subsequence that has a least upper bound): [73]. • Boolean algebras with Moltó’s property (f): [122, Corollary 1.4].In [88, Question 7.3], the authors raised the posed the following problem.
Problem 11.
Let A be a Boolean algebra whose Stone space is a G-space. Does thereexist a Boolean subalgebra B of A whose Stone space K fails to be a G-space, yet everyweakly* convergent sequence of purely atomic measures on K converges weakly? Definition 4.1.6.
Let X be a topological space. The 0- Baire functions are the continuous,scalar-valued functions on X ; and for an ordinal number α (cid:62) , the α - Baire functions arethe pointwise limits of sequences of β -Baire functions with β < α . For an ordinal number α , the space B α ( X ) of all bounded Baire- α functions on a topo-logical space X is a commutative C ∗ -algebra when endowed with the supremum normand operations defined pointwise. In particular, B α ( X ) is isometric to some C ( K ) -space.Dashiell Jr. studied the spaces B α ( X ) and proved that they are Grothendieck (see [37]; or[36, Theorem 3.3.9]). Proposition 4.1.7.
For every topological space X and non-zero α < ω , the space B α ( X ) has the Grothendieck property. Let us introduce a related notion for Banach spaces.
Definition 4.1.8.
Let E be a Banach space considered naturally a subspace of E ∗∗ . Wedenote by E w the subspace of E ∗∗ comprising all weak*-limits in E ∗∗ of weakly Cauchysequences in E .It is not difficult to show that if Γ is an infinite set, then c (Γ) w = (cid:96) c ∞ (Γ) . In particular, ( c ) w = (cid:96) ∞ . Moreover, E w satisfies the following properties: Proposition 4.1.9.
Let E be a Banach space. (i) E w is a closed subspace of E . (ii) E = E w if and only if E is weakly sequentially complete. (iii) If E is separable, then E ∗∗ = E w if and only if E contains no copies of (cid:96) .Proof. (i) was proved by McWilliams [119], (ii) is clear, and (iii) was proved by Odell andRosenthal [125]. (cid:3) The following problem arises.
Problem 12.
Characterise Banach spaces E for which E w is Grothendieck.Moreover, is E w Grothendieck when so is E ?The space C ( K ) w has a natural representation in terms of 1-Baire functions ([36, The-orem 3.3.9]). Proposition 4.1.10.
For every compact space K , the space C ( K ) w may be identified withthe space of bounded Baire-1 functions B ( K ) on K , and it is a Grothendieck space. Observe that for every compact space K , we have two Grothendieck spaces B ( K ) and B ( K ) /C ( K ) . In fact, we have potentially many more if we consider the spaces B α ( K ) ofbounded α -Baire functions on K ( (cid:54) α (cid:54) ω ). C ∗ -algebras In Section 4.1, we discussed in detail the question of when commutative (unital) C ∗ -algebras (that is C ( K ) -spaces) are Grothendieck. As for possibly non-commutative C ∗ -algebras, Problem 13 as well as Section 5.3 concerning C ∗ -tensor products, and Proposi-tion 5.5.1 touch upon this topic directly. In this context, Pfitzner’s theorem (Theorem 4.2.1,[130]; see also [52]) is of fundamental importance. Theorem 4.2.1 (Pfitzner’s theorem) . Let A be a C ∗ -algebra and let K ⊆ A ∗ be a boundedset. Then K is not relatively weakly compact if and only if there are δ > and a sequence ( a n ) ∞ n =1 of of pairwise orthogonal, norm-one self-adjoint elements in A such that sup f ∈ K |(cid:104) f, a n (cid:105)| (cid:62) δ. In particular, C ∗ -algebras have property ( V ) . As a consequence, Pfitzner deduced the following result, answering [42, Problem 2].
Corollary 4.2.2.
Each von Neumann algebra (dual C ∗ -algebra) is a Grothendieck space;in particular B ( H ) is Grothendieck for every Hilbert space H . Fernández-Polo and Peralta extended Pfitzner’s theorem to JB*-triples [51].Let us say that a masa is a maximal Abelian sub- C ∗ -algebra of a C ∗ -algebra. The second-named author observed that one can deduce from Theorem 4.2.1 the following result ([91,Proposition 2.5]). Proposition 4.2.3.
Let A be a C ∗ -algebra. Suppose that each masa in A is a Grothendieckspace. Then A is a Grothendieck space.Proof. Let T : A → c be an operator. Assume contrapositively that T is not weaklycompact. Then the adjoint T ∗ is not weakly compact either, so the image T ∗ ( B c ∗ ) ⊂ A ∗ is not relatively weakly compact, where B c ∗ denotes the unit ball of c ∗ . By Theorem 4.2.1there are δ > and a sequence ( a n ) ∞ n =1 of pairwise orthogonal, norm-one self adjointelements in A such that sup f ∈ T ∗ ( B c ∗ ) |(cid:104) f, a n (cid:105)| = sup y ∈ B c ∗ |(cid:104) T ∗ y, a n (cid:105)| = sup y ∈ B c ∗ |(cid:104) y, T a n (cid:105)| (cid:62) δ. Since the elements a n ( n ∈ N ) are self-adjoint and orthogonal, they generate a commu-tative sub- C ∗ -algebra C of A . Let B be a masa containing C . Then T restricted to C (hence also to M ) is not weakly compact. In particular, M is not Grothendieck. (cid:3) A C ∗ -algebra is said to be monotone σ -complete if each upper-bounded, monotone in-creasing sequence of self-adjoint elements has a supremum. The spectrum of a commutative σ -complete C ∗ -algebra is σ -Stonean ( cf . Remark 2) and, vice versa , if K is a σ -Stoneancompact space, then C ( K ) is monotone σ -complete. Saitô and Wright [139] term a C ∗ -algebra Rickart , whenever every masa M of A is monotone σ -complete; in particular, M is Grothendieck because K is a G-space. They noted that each von Neumann algebra,each AW ∗ -algebra, and each monotone σ -complete C ∗ -algebra is Rickart (another proofis presented in [21]). Using Proposition 4.2.3, we have thus subsumed the main result of[139]. Corollary 4.2.4.
Every Rickart C ∗ -algebra is a Grothendieck space. We have already noted that B ( K ) is a commutative C ∗ -algebra. However, the followingproblem is still open ([36, Chapter 6, Question 2]): Problem 13.
Let A be a C ∗ -algebra. Is A w a Grothendieck space?In the non-commutative case, it is not clear whether A w is a sub- C ∗ -algebra of theenveloping von Neumann algebra A ∗∗ .We remark in passing that Chetcuti and Hamhalter [31] characterised C ∗ -algebras sat-isfying the Brooks–Jewett theorem as precisely those C ∗ -algebras that are Grothendieckspaces and whose irreducible representations are finite-dimensional. L ∞ -spaces The C ( K ) -spaces are the best known examples of L ∞ -spaces. They are Lindenstraussspaces; i.e. , they are L ∞ ,λ for every λ > , and they have property ( V ) . That is not truefor all L ∞ -spaces. Example 4.3.1.
Bourgain and Delbaen [19] constructed two families of separable, non-reflexive L ∞ -spaces; hence they are not Grothendieck.The spaces in the first family are hereditarily reflexive, so that they contain no copies of c ; hence they fail property ( V ) .The spaces E in the second one have the Schur property; hence they fail property ( V ) ;moreover they satisfy E = E w .So there are L ∞ -spaces E for which E w is not Grothendieck. Problem 14.
Let E be a Grothendieck L ∞ -space. Is E w a Grothendieck space?For the following result we refer to [10, Proposition 1.9]. Proposition 4.3.2. A L ∞ -space isomorphic to a dual space is injective; hence it has theGrothendieck property. Using this result and the principle of local reflexivity, we can prove provide the followingcharacterisation:
Proposition 4.3.3.
A Banach space X is a L ∞ -space if and only if X ∗∗ is injective. Haydon [70] proved that if the space X ∗∗ is injective then it is isomorphic to (cid:96) ∞ (Γ) forsome set Γ .Since X ∗ is a complemented subspace in X ∗∗∗ , from Proposition 4.3.3 we derive thefollowing consequence: Proposition 4.3.4.
Let Y be a closed subspace of a Banach space X . If Y and X are L ∞ -spaces then so is X/Y .Proof.
Since Y ∗∗ ≡ Y ⊥⊥ and X ∗∗ are injective, X ∗∗ = Y ⊥⊥ ⊕ Z . Thus ( X/Y ) ∗∗ , which isisomorphic to Z , is injective. (cid:3) Some L ∞ -spaces, such as the ones described in Example 4.3.1, fail property ( V ) . There-fore we do not have a good characterisation of those which are Grothendieck. So thefollowing problem was posed in [10]. Problem 15.
Is every L ∞ -space without infinite-dimensional separable complementedsubspace a Grothendieck space?The following result, which is implicit in [111, Appendix], may be relevant for solvingProblem 15. Let us note that every infinite-dimensional, Grothendieck L ∞ -space is non-separable. Proposition 4.3.5. A L ∞ -space X is Grothendieck if and only if every operator from X into a separable space can be extended to any superspace of X .Proof. If X is a Grothendieck L ∞ -space, Y is separable and T : X → Y is an operator,then T is weakly compact. Thus T ∗∗ : X ∗∗ → Y is an extension of T with X ∗∗ injective,and T ∗∗ can be extended to any super-space of X ∗∗ .Conversely, let T : X → c . Since X is contained in (cid:96) ∞ (Γ) for some set Γ and thehypothesis implies that T can be extended to (cid:98) T : (cid:96) ∞ (Γ) → c , T is weakly compact becauseso is (cid:98) T . Hence X is Grothendieck. (cid:3) We state a problem that is a special case of the previous one.
Problem 16.
Let X be a Grothendieck L ∞ -space. Does X have property ( V ) ? Separably injective spaces
In the present section we consider certain subclasses of L ∞ -spaces defined by various extension properties. Definition 4.3.6.
Let Z be a Banach space and let (cid:54) λ < ∞ . We say that Z is(1) λ -injective if for a given a closed subspace Y of a Banach space X , every operator T : Y → Z admits an extension (cid:98) T : X → Z with (cid:107) (cid:98) T (cid:107) (cid:54) λ (cid:107) T (cid:107) ;(2) universally λ -separably injective if it satisfies (1) when Y is separable;(3) λ -separably injective if it satisfies (1) when X is separable. Remark . We can define the classes of injective , universally separably injective , and sepa-rably injective spaces in the same way, but omitting the reference to the bound of the normof the extension. However, it is not difficult to see that every injective space is λ -injectivefor some λ , and the same happens for the other classes; see [10].Obviously injective spaces are universally separably injective, and the latter spaces areseparably injective. Additionally, the following result follows from [10, Proposition 2.8]and Proposition 3.1.13. Proposition 4.3.7.
Separably injective spaces are L ∞ -spaces and have property ( V ) . Con-sequently, they are Grothendieck spaces if and only if they contain no complemented copiesof c . It is well known that a Banach space is -injective if and only if it is isometric to a C ( K ) -space with K extremely disconnected compact, and it is a long-standing open problem ifevery injective space is isomorphic to a -injective space. Proposition 4.3.8.
Universally separably injective spaces have property ( V ∞ ) , hence theyare Grothendieck. Universally separably injective spaces were introduced in [8]; there is an abundance ofexamples of spaces in this class that are not injective spaces. We will point out a few andrefer to [10] for further examples and additional information.
Example 4.3.9.
Let Γ be an uncountable set. The space (cid:96) c ∞ (Γ) is universally separablyinjective because every x ∈ (cid:96) c ∞ (Γ) has countable support. Therefore, if Y is separable and T : Y → (cid:96) c ∞ (Γ) is an operator, then the range of T is contained in a subspace isometric to (cid:96) ∞ , which is -injective.The subsequent proposition ([10, Proposition 2.11]) provides further examples of univer-sally separably injective spaces. Proposition 4.3.10.
Let Y be a closed subspace of X . If X is universally separablyinjective and Y is separably injective, then X/Y is universally separably injective.
Example 4.3.11.
The space (cid:96) ∞ /c is universally separably injective because (cid:96) ∞ is injectiveand c is separably injective.Separably injective spaces are not necessarily Grothendieck: the space c is -separablyinjective. However, every infinite-dimensional separable and separably injective space isisomorphic to c , and it is λ -separably injective for no λ < (see [8]). In fact, we have thefollowing result ([8, Proposition 2.31]). Proposition 4.3.12.
Each -separably injective space has the Grothendieck property. The following problem was stated in [10].
Problem 17.
Suppose that X is a λ -separably injective Banach space for some λ < . Is X Grothendieck?Observe that if a C ( K ) -space is λ -separably injective for some λ < , then it is -separably injective [10, Proposition 2.34]. Here we describe some results of Bourgain in [17] and [18].Let D denote the open unit disc in the complex plane and let T be the unit circle.We denote by A the disc algebra, which is the closed subspace of C ( T ) comprising allfunctions that admit an analytic extension in D . Let m denote the Lebesgue measure on T , L ≡ L ( T ) , and L ∞ ≡ L ∞ ( T ) .By the F. and M. Riesz theorem (see [128]), the annihilator A ⊥ can be identified with aclosed subspace H of L ( T ) , A ⊥ = { µ ∈ C ( T ) ∗ : µ = h · m, h ∈ H } , and ( L /H ) ∗ ≡ H ∞ is a closed subspace of L ∞ ≡ L ∗ that can be identified with the spaceof bounded analytic functions on the unit disc D . Example 4.4.1.
Bourgain proved in [17] that the space H ∞ has property ( V ∞ ) ; hence itis a Grothendieck space, yet it is not a L ∞ -space. For the latter part, observe that H ∞ is not an injective space, because it is not comple-mented in L ∞ ; hence, since H ∞ is a dual space, it cannot be a L ∞ -space by Proposition4.3.2. Example 4.4.2.
The second dual of the disc algebra A ∗∗ has property ( V ∞ ) , yet it is nota L ∞ -space.Indeed, C ( T ) ∗ = L ⊕ V sing , where V sing is the space measures µ on T that are singularwith respect to m . Then we have A ∗ = L /H ⊕ V sing ; hence A ∗∗ = H ∞ ⊕ ∞ V ∗ sing , where V ∗ sing is a dual L ∞ -space, so it is injective.Let E n = C ([0 , n ) denote the space of continuously differentiable functions on the n -dimensional cube.In [18], Bourgain proved that for every n ∈ N the dual space of E n is weakly sequentiallycomplete. Observe that, by Taylor’s theorem, E ∼ = C [0 , ⊕ R ∼ = C [0 , ; hence E ∗∗ is a Grothendieck space. However it is known whether the spaces E n are isomorphic to C ( K ) -spaces for n (cid:62) . These observations lead to the following problem. Problem 18.
Let n (cid:62) . Is E ∗∗ n a Grothendieck space?Should the solution be affirmative, E ∗∗∗ n (hence also E ∗ n ) would be weakly sequentiallycomplete, so that Bourgain’s result would have been subsumed. The fact that a C ( K ) -space with K a Stonean compact is Grothendieck has a lattice-theoretic flavour since these spaces may be abstractly described as Dedekind-completeAM-spaces. Here we collect several examples of Banach spaces that have been provedto be Grothendieck using Banach-lattice techniques, and some results discussing when anordered Banach space—such as a Banach lattice or a C ∗ -algebra (or rather, the real partthereof)—satisfying certain separation property is a Grothendieck space.Throughout this section we fix a complete measure space ( X, Σ , µ ) .In [117, Theorem 1], Lotz proved that a Banach lattice E with weakly sequentiallycontinuous dual and satisfying some technical conditions is a Grothendieck space. Inparticular, the following result ([117, Theorem 3]) was proved. Example 4.5.1.
The space L p, ∞ ( µ ) is Grothendieck for < p < ∞ .Let /p + 1 /q = 1 . The space L p, ∞ ( µ ) , which is also called a weak L p -space , consistsof all real-valued (equivalence classes with respect to the relation of being equal almosteverywhere of) µ -measurable functions f such that { ω : | f ( ω ) | > } is σ -finite and (cid:107) f (cid:107) = sup (cid:26) µ ( B ) /q (cid:90) B | f | d µ : B ∈ Σ , < µ ( B ) < ∞ (cid:27) < ∞ . De Pagter and Sukochev [126] showed that Lotz’ criterion [117, Theorem 1] can beapplied to a large class of Marcinkiewicz spaces M Ψ tha are defined as follows. Let < r (cid:54) ∞ and let Ψ : [0 , r ) → [0 , ∞ ) be a non-zero increasing concave function,continuous on (0 , r ) and satisfying Ψ(0) = 0 . The Marcinkiewicz space M Ψ (0 , r ) is thespace of all real-valued measurable functions f on (0 , r ) such that (cid:107) f (cid:107) M Ψ = sup Let Ψ : [0 , r ) → [0 , ∞ ) be a function as above. Then the Marcinkiewiczspace M Ψ (0 , r ) is Grothendieck if and only if one of the following conditions is satisfied:(A) r = ∞ , lim inf t → Ψ(2 t ) / Ψ( t ) > and lim inf t →∞ Ψ(2 t ) / Ψ( t ) > , or(B) r < ∞ and lim inf t → Ψ(2 t ) / Ψ( t ) > . These results admit some extensions to general measure spaces. We refer to [126] foradditional information.Polyrakis and Xanthos generalised the fact that for a compact F-space K the space C ( K ) is Grothendieck to the framework of general ordered Banach spaces as follows ([134,Theorem 9]). Proposition 4.5.3. Let E be an ordered Banach space whose cone is closed and normal.If E has an order unit and the countable interpolation property, then E is a Grothendieckspace. As already mentioned in Section 2.3, the countable interpolation property and the count-able monotone interpolation property are equivalent for Banach lattices, but in general theyare not. It is thus natural to ask whether countable monotone interpolation property issufficient for the Grothendieck property of a given ordered space. Problem 19. Can we replace ‘the countable interpolation property’ by ‘the countablemonotone interpolation property’ in Proposition 4.5.3?What happens in the case E is a C ∗ -algebra?This problem is particularly relevant for C ∗ -algebras and related structures. Indeed, let A be a C ∗ -algebra. As we have the canonical isometric identification of the dual space of A = A sa ⊕ R iA ∗ sa with ( A sa ) ∗ ⊕ R i ( A sa ) ∗ , A is a Grothendieck space if and only if so is the realBanach space A sa . Consequently, had Problem 19 had affirmative solution, we would havefound a way alternative to invoking Pfitzner’s theorem (Theorem 4.2.1) for establishingthe Grothendieck property of C ∗ -algebras with countable monotone interpolation property.Such algebras include von Neumann algebras ([147, p. 117]) and corona algebras of σ -unital,non-unital algebras ([127, Theorem 3.14.2]).It turns out that the Grothendieck propery may be characterised in terms of well-basedcones ([133, Theorem 15]). Proposition 4.5.4. Let X be a Banach space. Then X is not a Grothendieck space if andonly if there exists a well-based cone P of X ∗ such that the set P = { x ∈ X : (cid:104) f, x (cid:105) (cid:62) f ∈ P ) } has non-empty interior and there exists x ∈ P such that the set (cid:83) ∞ n =1 [ − nx , nx ] is densein X with respect to the seminorm d p ( x ) = sup f ∈ V (cid:104) f, x (cid:105) , where V is the convex hull of ( B X ∗ ∩ P ) ∪ − ( B X ∗ ∩ P ) . We consider the class of Banach spaces X such that the dual space X ∗ is a Grothendieckspace. Clearly reflexive spaces are in this class, as well as L -spaces, because X is a L -space if and only if X ∗ is a L ∞ -space; equivalently, X ∗ is injective (Proposition 4.3.2).Moreover, preduals of von Neumann algebras are in this class too. Problem 20. Find a characterisation of dual Grothendieck spaces.Certainly, weak sequential completeness of X ∗∗ is a necessary condition for X ∗ being aGrothendieck space.Let X ( k ) denote the dual of order k ( k ∈ N ∪ { } ) of a Banach space X ; i.e. , X (0) = X and X ( k +1) = ( X ( k ) ) ∗ . Contreras and Díaz ([34, Corollaries 3.7 and 3.9]) continued thework of Bourgain [17] as follows: Example 4.6.1. The duals of even order A (2 k ) and H ∞ (2 k ) of the disk algebra A and H ∞ are Grothendieck spaces. None of them is a L ∞ -space.From Corollary 5.3.3 we get: Example 4.6.2. For < p < ∞ , the space (cid:96) ∞ (cid:98) ⊗ π (cid:96) p ≡ ( c (cid:98) ⊗ π (cid:96) p ) ∗∗ is Grothendieck.Next we present a construction of Leung [108, Examples 11 and 15] which is a variationof the Schreier space. Proposition 4.6.3. There exists a Banach space F with a shrinking unconditional basis ( e i ) ∞ i =1 , which satisfies the following properties:(1) Both sequences ( e i ) ∞ i =1 and the coefficient functionals ( e ∗ i ) ∞ i =1 are weakly null. Hence F fails the Dunford–Pettis property.(2) F is hereditarily c and F ∗ is hereditarily (cid:96) .(3) F ∗∗ is Grothendieck that is not a L ∞ -space because it fails the Dunford–Pettis prop-erty.(4) F ∗∗ has the surjective Dunford–Pettis property: every surjective operator from F ∗∗ onto a reflexive space maps weakly convergent sequences to convergent sequences. Let us briefly describe the construction of F . Given an integer i (cid:62) , we say that a subset A of N is i - admissible , whenever card A = 2 i and min A (cid:62) i . For (cid:54) p (cid:54) ∞ , we denoteby (cid:107) · (cid:107) p the (cid:96) p -norm in the space c of all finitely supported scalar sequences. Given x = ( a i ) ∞ i =1 ∈ c and an i -admissible set A , we set q A ( x ) = (cid:107) x (cid:107) √ i (with theconvention (cid:107) x (cid:107) √ = (cid:107) x (cid:107) ∞ ) , and define (cid:107) x (cid:107) F = sup { q A ( x ) : i = 0 , , , . . . and A is i -admissible } . The space F is the completion of ( c , (cid:107) · (cid:107) F ) . It is not difficult to check that the standardunit vector basis ( e i ) ∞ i =1 is a monotone unconditional basis of F . We refer to [108] for theproof of the properties of F and its dual spaces.A simple but interesting example is the following one. Example 4.6.4. The space E = ( (cid:76) n ∈ N (cid:96) n ) (cid:96) ∞ (isometric to the bidual of ( (cid:76) n ∈ N (cid:96) n ) c ) hasthe Grothendieck property, yet it is not a L ∞ -space and fails the Dunford–Pettis property,because it contains a complemented copy of (cid:96) .Indeed, E is a complemented subspace of the diret sum (cid:96) ∞ ( (cid:96) ) , which in turn is a quotientof (cid:96) ∞ ( (cid:96) ∞ ) ≡ (cid:96) ∞ so it is a Grothendieck space. In order to construct a projection P onto E with range isomorphic to (cid:96) , we fix a non-principal ultrafilter U on N . Then, for a sequence a = (( a n,k ) nk =1 ) ∞ n =1 ∈ E , we denote a ∞ ,k = lim n → U a n,k , and define P a by ( P a ) n,k = a ∞ ,k . Here we will show how to produce new examples of Grothendieck spaces beginning withthe ones we know. The following result was proved in [63, Proposition 3.1]. Proposition 5.1.1. Let X be a Grothendieck space. If M is a closed subspace of X and X/M is separable, then M is a Grothendieck space.Proof. Let S : M → c be an operator. Since c is separably injective ([10, Theorem 2.3])and the quotient X/M is separable, S admits an extension T : X → c ([10, Proposi-tion 2.5]), which is weakly compact by Theorem 3.1.1. Then S is weakly compact, andTheorem 3.1.1 allows us to conclude that M is Grothendieck. (cid:3) Proposition 4.3.4 provides further information about the examples captured by Propo-sition 5.1.1. Corollary 5.1.2. Let X be a Grothendieck L ∞ -space. If Y is a closed subspace of X with X/Y infinite-dimensional and separable, then Y is Grothendieck, but it is not a L ∞ -space.In particular, Y is not isomorphic to X .Proof. If Y were a L ∞ -space, then X/Y would be too, but it is not since it is reflexive. (cid:3) Corollary 5.1.2 suggests the following problem. Problem 21. Let X be a Grothendieck space without complemented separable, infinite-dimensional subspaces and let Y be a closed subspace of X with X/Y infinite-dimensionalseparable. Is it possible for Y to be isomorphic to X ?Let us invoke another consequence of Proposition 5.1.1 ([63, Theorem 3.3]): Corollary 5.1.3. There exists an uncountable family of pairwise non-isomorphic Grothen-dieck subspaces M of (cid:96) ∞ with (cid:96) ∞ /M infinite-dimensional and separable.Proof. For < p (cid:54) , the space L ≡ L [0 , has a subspace N p isomorphic to L p [113, Corollary 2.f.5]. Then N ⊥ p is a subspace of L ∞ with L ∞ /N ⊥ p isomorphic to L ∗ p .Let U : L ∞ → (cid:96) ∞ be an isomorphim and M p = U ( N ⊥ p ) . Then (cid:96) ∞ /M p ∼ = L ∗ p is infinite-dimensional and separable, hence M p is Grothendieck by Proposition 5.1.1.Let < p, q (cid:54) and suppose that T : M p → M q is an isomorphism. Since (cid:96) ∞ /M p and (cid:96) ∞ /M q are reflexive, by [113, Theorem 2.f.12] there exists an extension ˆ T : (cid:96) ∞ → (cid:96) ∞ of T which is a Fredholm operator. Hence the operator ˆ T induces a Fredholm operator S : (cid:96) ∞ /M p → (cid:96) ∞ /M q implying that L ∗ p and L ∗ q are isomorphic (because either space isisomorphic to their subspaces of finite codimension); hence p = q . (cid:3) Can we improve Proposition 5.1.1 to assuming only that dens X/M < p ? Problem 22. Let X be a Grothendieck Banach space. Suppose that M is a closed subspaceof X such that dens X/M < p . Is M a Grothendieck space?Since separable quotients of Grothendieck spaces are reflexive, the following problemalso arises. Problem 23. Let M be a closed subspace of a Grothendieck space X such that X/M isreflexive. Is M a Grothendieck space? A concrete case: let N be a closed subspace of (cid:96) ∞ such that (cid:96) ∞ /N is isomorphic to (cid:96) (Γ) with Γ uncountable. Is N a Grothendieck space?The space (cid:96) ∞ contains isometric copies of every separable Banach space (in particular,reflexive), so the number of pair-wise non-isomorphic subspaces of (cid:96) ∞ that are Grothendieckis at least continuum. Among reflexive subspaces of (cid:96) ∞ this is also an upper bound as everyreflexive subspace of (cid:96) ∞ is separable. Problem 24. What is the number of pairwise non-isomorphic Grothendieck subspaces of (cid:96) ∞ ; can it be c ?Dow, Gubbi, and Szymański constructed in [50] a family of c pairwise non-homeomor-phic extremally disconnected, separable compact spaces that are topologically rigid. It isknown that every separable extremally disconnected compact space K embeds into β N ([151, Corollary 3.2]). As K is separable, C ( K ) embeds into (cid:96) ∞ . Since the spaces arepairwise non-homeomorphic, by the Banach–Stone theorem, the corresponding spaces ofcontinuous functions are pairwise non-isometric. Since each space K is extremally dis-connected, C ( K ) is injective, hence Grothendieck. Thus, there are c isometry types of Grothendieck subspaces of (cid:96) ∞ . However, each C ( K ) is in this case isomorphic to (cid:96) ∞ , be-cause as observed by Lindenstrauss [112], infinite-dimensional complemented subspaces of (cid:96) ∞ are isomorphic to (cid:96) ∞ .Saitô and Maitland Wright [140, Corollary 20] constructed c sub- C ∗ -algebras of (cid:96) ∞ thatare pairwise non-isomorphic as C ∗ -algebras, and each of these algebras is a monotone σ -complete quotient of the algebra of bounded Borel functions on [0 , by a suitable ideal. Inparticular, they are Grothendieck spaces ( cf . Remark 2 and Section 5.3 for more details).By the Gelfand–Naimark theorem and the Banach–Stone theorem, they are not isometricas Banach spaces. However, we do not know whether they are non-isomorphic as Banachspaces. Recall that the notion of twisted sum of two Banach spaces Y and Z was introduced inSection 2.4. By Proposition 3.2.2, a twisted sum of Grothendieck spaces is Grothendieck.This fact will allow us to obtain further, perhaps exotic, examples of Grothendieck spaces.For this, we shall employ push-out and pull-back diagrams that are standard tools forbuilding twisted sums from given spaces. We refer to [26] and [28] for information on thesetechniques. Example 5.2.1. There exists a Banach space PB which is a non-trivial twisted sum of (cid:96) and (cid:96) ∞ , yet it is not isomorphic to a direct sum of a Hilbert space and a C ( K ) space. Proof. We consider the Banach space Z constructed by Kalton and Peck in [89], which isnot isomorphic to a Hilbert space but it is a twisted sum thereof. Thus we have an exactsequence(5) −−−→ (cid:96) j −−−→ Z q −−−→ (cid:96) −−−→ . Then we take a surjective operator Q : (cid:96) ∞ → (cid:96) , and we can consider the pull-backdiagram(6) −−−→ (cid:96) j −−−→ PB q −−−→ (cid:96) ∞ −−−→ (cid:13)(cid:13)(cid:13) Q (cid:121) (cid:121) Q −−−→ (cid:96) j −−−→ Z q −−−→ (cid:96) −−−→ which is commutative and has both rows exact. In particular, Q is a surjective operatorbecause so is Q .Every operator from C ( K ) into its dual space factors through a Hilbert space ([45,Corollary 2.15 and Theorem 3.5]), hence the same applies to the direct sum C ( K ) ⊕ H of C ( K ) and a Hilbert space H . However, since Z is isomorphic to its dual space ([89,Theorem 6.1]), we can consider the operator Q ∗ Q : PB → PB ∗ that does not factorthrough a Hilbert space. Hence PB is not isomorphic to C ( K ) ⊕ H . In particular, theupper exact sequence is non-trivial. (cid:3) Readers unacquainted with the pull-back diagrams may observe that PB = { ( z, y ) ∈ Z ⊕ ∞ (cid:96) ∞ : qz = Qy } , which is a closed subspace of Z ⊕ ∞ (cid:96) ∞ , and the induced maps Q , q , and j are simplydefined by Q ( z, y ) = z , q ( z, y ) = y , and j x = ( x, , respectively.Next we present a more general construction of twisting a Grothendieck C ( K ) -spaceswith a Hilbert space. Example 5.2.2. Let K be an infinite G-space. Then there exists a Banach space PB which is a non-trivial twisted sum of (cid:96) and C ( K ) and it is not isomorphic to a direct sumof a Hilbert space H and a C ( K ) -space. Moreover, it may be arranged that PB does notcontain isomorphic copies of (cid:96) ∞ . Proof. For the first part, since C ( K ) contains a copy of (cid:96) by Corollary 3.1.11, it admitsa quotient isomorphic to (cid:96) . So we can repeat the argument given in Example 5.2.1.For the second part, if we use as C ( K ) the space given by Haydon in [71] (see Example4.1.3), then PB does not contain copies of (cid:96) ∞ . For this, it is enough to observe thatcontaining no copies of (cid:96) ∞ is a three-space property [28]. (cid:3) The following example of non-trivial twisted sum is based in Proposition 5.1.1. Example 5.2.3. Let M and N be closed subspaces of (cid:96) ∞ with N ⊂ M such that both M/N and (cid:96) ∞ /M are infinite-dimensional and separable. Then there exists a Banach space PO which is a non-trivial twisted sum of M and (cid:96) ∞ /N . Proof. Note that, by Proposition 5.1.1, both M and (cid:96) ∞ /N are Grothendieck spaces.We consider the exact sequence(7) −−−→ N j −−−→ (cid:96) ∞ q −−−→ (cid:96) ∞ /N −−−→ and the embedding J : N → M and form the associated push-out diagram(8) −−−→ N j −−−→ (cid:96) ∞ q −−−→ (cid:96) ∞ /N −−−→ (cid:121) J J (cid:121) (cid:13)(cid:13)(cid:13) −−−→ M j −−−→ PO q −−−→ (cid:96) ∞ /N −−−→ which is a commutative diagram in which both rows are exact.To show that the lower exact sequence is non-trivial; i.e. , that j ( M ) is uncomplementedin PO , it is enough to show that J : N → M does not admit an extension (cid:98) J : (cid:96) ∞ → M .See Lemma 20 of [10, Appendix A]. Indeed, since (cid:96) ∞ /N is reflexive, by [113, Theorem2.f.12], each extension of J to an operator T : (cid:96) ∞ → (cid:96) ∞ is a Fredholm operator (has finite-dimensional kernel and finite-codimensional range). Hence, its range cannot be containedin M . (cid:3) Readers unacquainted with the push-out diagrams may observe that PO = ( M ⊕ (cid:96) ∞ ) /W, where W = { ( J x, − jx ) ∈ M ⊕ (cid:96) ∞ : x ∈ N } ; moreover denoting by Q : M ⊕ (cid:96) ∞ → PO the canonical quotient map, the induced maps j , J , and q are defined by j y = Q ( y, , J z = Q (0 , z ) , and q Q ( y, z ) = qz , respectively. Example 5.2.4. There exists a Banach space PO which is a non-trivial twisted sum ofHaydon’s space C ( K ) in Example 4.1.3 and (cid:96) ∞ /c . Proof. We consider the exact sequence(9) −−−→ c j −−−→ (cid:96) ∞ q −−−→ (cid:96) ∞ /c −−−→ and the embedding J : c → C ( K ) and form the associated push-out diagram(10) −−−→ c j −−−→ (cid:96) ∞ q −−−→ (cid:96) ∞ /c −−−→ (cid:121) J J (cid:121) (cid:13)(cid:13)(cid:13) −−−→ C ( K ) j −−−→ PO q −−−→ (cid:96) ∞ /c −−−→ To show that the lower exact sequence is non-trivial, it is enough to observe that theembedding J : c → C ( K ) does not admit an extension (cid:98) J : (cid:96) ∞ → C ( K ) (see Lemma 20 of[10, Appendix A]), and this follows from the fact that such an extension would fix a copyof (cid:96) ∞ , because J is not weakly compact [113, Proposition 2.f.4], and this is not possiblebecause C ( K ) does not contain copies of (cid:96) ∞ . (cid:3) The fact that property ( V ∞ ) fails the three-space property gives another non-trivialexample of Grothendieck space. We refer to [10, Proposition 6.3] for the details of thisconstruction. Example 5.2.5. There exists a Banach space Z failing property ( V ∞ ) which is a non-trivialtwisted sum of two spaces (cid:96) ∞ (Γ , (cid:96) ) and (cid:96) ∞ which have property ( V ∞ ) .Indeed, there exists an exact sequence(11) −−−→ (cid:96) ∞ (Γ , (cid:96) ) j −−−→ Z q −−−→ (cid:96) ∞ −−−→ such that q is an isomorphism on no copy of (cid:96) ∞ . Thus Z fails property ( V ∞ ) .Note that (cid:96) ∞ (Γ , (cid:96) ) has property ( V ∞ ) because it is a quotient of the space (cid:96) ∞ (Γ , (cid:96) ∞ ) ,which is isometrically isomorphic to (cid:96) ∞ (Γ × N ) ; moreover, (cid:96) ∞ (Γ , (cid:96) ) ⊕ (cid:96) ∞ ∼ = (cid:96) ∞ (Γ , (cid:96) ) andproperty ( V ∞ ) is stable under direct sums. Here we study the possible inheritance of the Grothendieck property by Banach-spacetensor products, which is a rather rare phenomenon, and the Grothendieck property for B ( X ) . Projective tensor product We denote by X (cid:98) ⊗ π Y the projective tensor product of twoBanach spaces X and Y . We start with the following result ([61, Proposition 6]): Proposition 5.3.1. If X (cid:98) ⊗ π Y is Grothendieck, then both X and Y are Grothendieck, andone of them is reflexive.Proof. The first part is clear, since X (cid:98) ⊗ π Y is Grothendieck, and contains complementedcopies of X and Y .Suppose that both X and Y are non-reflexive. By Corollary 3.1.11, both spaces contain (cid:96) . Then both of them have a quotient isomorphic to (cid:96) [45, Corollary 4.16]; hence X (cid:98) ⊗ π Y has a quotient isomorphic to (cid:96) (cid:98) ⊗ π (cid:96) [138, Proposition 2.5], which is separable and non-reflexive; thus X (cid:98) ⊗ π Y is not Grothendieck. (cid:3) Proposition 5.3.1 has a partial converse ([61, Propositions 8 and 9]): Proposition 5.3.2. Suppose that X is Grothendieck and Y is reflexive.(1) If B ( X, Y ∗ ) = K ( X, Y ∗ ) , then X (cid:98) ⊗ π Y is Grothendieck.(2) If X (cid:98) ⊗ π Y is Grothendieck and Y ∗ has the Bounded Compact Approximation prop-erty, then B ( X, Y ∗ ) = K ( X, Y ∗ ) . In particular, we have the following result ([61, Corollary 10]): Corollary 5.3.3. Let p ∈ [1 , ∞ ] . Then (cid:96) ∞ (cid:98) ⊗ π (cid:96) p is a Grothendieck space if and only if < p < ∞ . The dual space of X (cid:98) ⊗ π Y can be naturally identified with B ( X, Y ∗ ) . The followingresult of Holub ([77, Theorem 2]) shows the limitations of Proposition 5.3.1 in producingnon-reflexive Grothendieck spaces. Proposition 5.3.4. Suppose that X and Y are reflexive Banach spaces an one of them hasthe Aproximation property. Then X (cid:98) ⊗ π Y is reflexive if and only if B ( X, Y ∗ ) = K ( X, Y ∗ ) . Injective tensor product We denote by X (cid:98) ⊗ ε Y the injective tensor product of twoBanach spaces X and Y . Our first result concerns the inheritance of the Grothendieckproperty by the injective tensor product ([24, Theorem 4]). Proposition 5.3.5. Suppose that X is a Grothendieck space, Y is reflexive, X ∗∗ or Y ∗∗ has the Approximation property, and B ( X ∗ , Y ) = K ( X ∗ , Y ) . Then X (cid:98) ⊗ ε Y is Grothendieck. In view of Proposition 5.3.1, the following problem was raised in [24, p. 1156]. Problem 25. Suppose that X (cid:98) ⊗ ε Y is Grothendieck. Is X or Y reflexive?We have the following partial answer ([24, Theorem 7]). Proposition 5.3.6. Suppose that Y is a reflexive space and has an unconditional FDD.Then X (cid:98) ⊗ ε Y is Grothendieck if and only if B ( X ∗ , Y ) = K ( X ∗ , Y ) and the space X isGrothendieck. It follows from the previous result that, for < p < ∞ , (cid:96) ∞ (cid:98) ⊗ ε (cid:96) p is not Grothendieck (see[24, p. 1159]). When Y is a reflexive space with the Approximation Property, we have K ( X, Y ) = X ∗ (cid:98) ⊗ ε Y . Let us then record the following consequence ([24, Corollary 8]) ofthat fact. Corollary 5.3.7. Suppose that Y is a reflexive space with an unconditional FDD. Then K ( X, Y ) is Grothendieck if and only if X ∗ is Grothendieck and B ( X, Y ) = K ( X, Y ) . Let < p, q < ∞ . It follows from Corollary 5.3.7 that (cid:96) ∗ p (cid:98) ⊗ ε (cid:96) q is Grothendieck if andonly if B ( (cid:96) p , (cid:96) q ) = K ( (cid:96) p , (cid:96) q ) , but in this case (cid:96) ∗ p (cid:98) ⊗ ε (cid:96) q is reflexive. We may then reiterate thefollowing problem ([24, p. 1158]). Problem 26. Is there a pair of infinite-dimensional Banach spaces X and Y for which thetensor product X (cid:98) ⊗ ε Y is Grothendieck yet it is not reflexive?Khurana [93, Theorem 2] proved that C ( K ) (cid:98) ⊗ ε X (which is isometric to C ( K, X ) , thespace of X -valued continuous functions on K ) is Grothendieck only in two cases: • K is finite and X is a Grothendieck space, • K is a G-space and X is finite-dimensional.Cembranos [29] applied Khurana’s ideas to prove that for every infinite compact Haus-dorff space K and every infinite-dimensional Banach space X , the space C ( K, X ) containsa complemented copy of c , from which Khurana’s result follows immediately.Thus, the answer to Problem 26 is negative when at least one of the spaces X or Y isof the form C ( K ) for an infinite compact Hausdorff space. Indeed, if X = C ( K ) , then X (cid:98) ⊗ ε Y ≡ C ( K, Y ) contains a complemented copy of c . Actually, the key ingredient ofCembranos’ proof is the fact that C ( K ) -spaces have property ( V ) . We can emulate thatproof to establish the following partial result. Proposition 5.3.8. Suppose that X is a non-reflexive space with a subspace isomorphicto c . If X (cid:98) ⊗ ε Y is a Grothendieck space, then Y is finite-dimensional.Proof. Suppose that Y is infinite-dimensional and X contains a sequence ( e n ) ∞ n =1 which is M -equivalent to the standard basis of c for some M > . Let ( f n ) ∞ n =1 be the sequence offixed Hahn–Banach extensions of the coordinate functionals associated to ( e n ) ∞ n =1 on theclosed linear span X of this sequence.Since Y is infinite-dimensional, by the Josefson–Nissenzweig theorem (see [44, ChapterXII]), there is a sequence ( y ∗ n ) ∞ n =1 in the unit sphere of Y ∗ that converges to 0 in the weak*topology, and by [82, Remark III.1] we can find a bounded sequence ( y n ) ∞ n =1 in Y such that (cid:104) y ∗ n , y k (cid:105) = δ n,k ( n, k ∈ N ). Then the map T : X (cid:98) ⊗ ε Y → (cid:96) ∞ given by T ξ = ( (cid:104) ( f n ⊗ y ∗ n ) , ξ (cid:105) ) ∞ n =1 ( ξ ∈ X (cid:98) ⊗ ε Y ) is a bounded linear operator. Moreover, T takes values already in c because |(cid:104) f n , x (cid:105) · (cid:104) y, y ∗ n (cid:105)| (cid:54) M · (cid:107) x (cid:107) · |(cid:104) y, y ∗ n (cid:105)| ( x ∈ X, y ∈ Y ) for some constant M and the sequence ( y ∗ n ) ∞ n =1 is weak*-null. The operator T is not uncon-ditionally converging because T ( e n ⊗ y n )( k ) = δ n,k ( n, k ∈ N ), and this is a contradiction, because X (cid:98) ⊗ ε Y is Grothendieck, so every operator from that space into c is weakly com-pact, hence by the Orlicz–Pettis theorem, unconditionally converging. (cid:3) Let us observe that a positive answer to Problem 3 implies positive answer to Problem 26.Indeed, X is isomorphic to a complemented subspace of X (cid:98) ⊗ ε Y , so as such it is a Grothen-dieck space. Being non-reflexive, it would contain a copy of c , hence by Proposition 5.3.8, Y must be finite-dimensional.We close this section by noting that the aforestated results concerning preservation ofthe Grothendieck property by injective/projective tensor products resonate in the theoryof Banach lattices and their tensor products (positive injective, Fremlin, Wittstock tensorproducts, etc.). Defining these notions formally is beyond the scope of the present paper—instead we refer directly to the relevant papers: [22, 78, 110, 152]. C ∗ -tensor products The second-named author [91], inspired by Cembranos’ result [29],proved that an infinite-dimensional C ∗ -algebra A , which is a Grothendieck space cannotbe decomposed as a C ∗ -tensor product B ⊗ γ B of two infinite-dimensional C ∗ -algebras B , B ([91, Theorem 1.1]). Proposition 5.3.9. Let A be a C ∗ -algebra, and let B ⊗ γ B be a C ∗ -tensor product oftwo infinite-dimensional C ∗ -algebras. If A ∼ = B ⊗ γ B , then A is not Grothendieck.Sketch of the proof. There is a minimal C ∗ -tensor product ⊗ min , which for commutative C ∗ -algebras coincides with the injective tensor product of Banach spaces.The *-homomorphisms between C ∗ -algebras have always closed range, so there is al-ways a surjective homomorphism B ⊗ γ B → B ⊗ min B . In particular, if B ⊗ γ B isGrothendieck, then so is B ⊗ min B . The minimal tensor product ‘respects’ subalgebras,and the injective tensor product of Banach spaces ‘respects’ closed subspaces. However,every infinite-dimensional C ∗ -algebra contains a subspace isomorphic to c (every infinite-dimensional C ∗ -algebra contains a self-adjoint element a with infinite spectrum ([84, Ex.4.6.12], so by the spectral theorem, C ∗ ( σ ( a )) ⊂ A is an infinite-dimensional commutativesub- C ∗ -algebra), hence Proposition 5.3.8 applies. (cid:3) In particular, B ( (cid:96) ) and the Calkin algebra B ( (cid:96) ) / K ( (cid:96) ) do not admit such decompo-sitions. It is still not known whether there exists a reasonable Banach-space cross-norm γ and two Banach spaces B , B such that B ( (cid:96) ) is isomorphic as a Banach space to thetensor product B ⊗ γ B . Spaces of bounded operators The space B ( E ) of all bounded operators on E contains E ∗ (cid:98) ⊗ ε E as a closed subspace, that can be identified with the approximable operators, andalso contains complemented copies of E and E ∗ . So the next problem is a special case ofProblem 6. Problem 27. Suppose that B ( E ) is Grothendieck. Is E reflexive?The converse implication fails: Example 5.3.10. For < p < ∞ , the spaces E = ( (cid:76) n ∈ N (cid:96) n ) (cid:96) p , E = T , the Tsirelsonspace, or E = B p , the p th Bearnstein space are reflexive, yet B ( E ) is not a Grothendieckspace. We refer to [12, 90] for details. Problem 28. Suppose that E is super-reflexive. Is B ( E ) Grothendieck?What happens for E = (cid:96) p , < p < ∞ , p (cid:54) = 2 ?As Banach spaces, B ( (cid:96) p ) and B ( L p ) are isomorphic; in fact, for every separable, infinite-dimensional L p -space E , B ( (cid:96) p ) ∼ = B ( E ) ∼ = (cid:0) ∞ (cid:77) n =1 B ( (cid:96) np ) (cid:1) (cid:96) ∞ ; see [6, Section 2]. For p = 2 , this result is attributed to Lindenstrauss and Haagerup (see[32]), and the proof relies on the Pełczyński decomposition method. This implies that theGrothendieck property of B ( (cid:96) p ) and that of of B ( L p ) are equivalent.We observe that it is an open problem whether B ( E ) may be reflexive for an infinite-dimensional (reflexive) space E . It is known that B ( E ) is non-reflexive when E has theBounded Approximation Property, because in this case K ( E ) ∗∗ ≡ B ( E ) with the inclusion K ( E ) → B ( E ) being the canonical embedding. For this reason, should Problem 28 havepositive answer, there would be no super-reflexive analogue of the Argyros–Haydon space[5], that is, there would be no super-reflexive space E with a Schauder basis on which everyoperator is of the form λI E + S , where λ is a scalar, I E is the identity operator on E , and S ∈ K ( E ) . Bombal [15] noticed that for < p < ∞ , if X is a Grothendieck space, then so is (cid:96) p ( X ) .Let us state a more general result that involves direct sums with respect to more generalunconditional bases. Proposition 5.4.1. Let E be a reflexive Banach space with a 1-unconditional basis ( e γ ) γ ∈ Γ ,and let X γ ( γ ∈ Γ) be a family of Banach spaces. Then the following conditions areequivalent:(1) each space X γ ( γ ∈ Γ) is Grothendieck;(2) ( (cid:76) γ ∈ Γ X γ ) E is Grothendieck.Proof. Let us denote X = ( (cid:76) γ ∈ Γ X γ ) E . The implication (2) ⇒ (1) is clear as each space X γ is isomorphic to a complemented subspace of X . For the converse implication, let usfix a weak*-null sequence ( f n ) ∞ n =1 in X ∗ . Then lim n →∞ (cid:10) ( x γ ) γ ∈ Γ , Λ E ( f nγ ) γ ∈ Γ (cid:11) = (cid:88) γ ∈ Γ (cid:104) x γ , f nγ (cid:105) = 0 (cid:0) ( x γ ) γ ∈ Γ ∈ X ) , where Λ E is the map defined in Section 2.2.Thus, lim n →∞ (cid:104) x γ , f nγ (cid:105) = 0 for each γ ∈ Γ and all x γ ∈ X Γ . As X γ is Grothendieck, lim n →∞ (cid:104) x ∗∗ γ , f nγ (cid:105) = 0 for each γ ∈ Γ and all x ∗∗ γ ∈ X ∗∗ Γ . As E is reflexive, the basis ( e γ ) γ ∈ Γ is both shrinking and boundedly complete, hence ( (cid:76) γ ∈ Γ X γ ) E is naturally isometricallyisomorphic to ( (cid:76) γ ∈ Γ X ∗∗ γ ) E .The susbspace c (Γ , X ∗∗ ) of all finitely supported X ∗∗ -valued tuples is norm-dense in ( (cid:76) γ ∈ Γ X ∗∗ γ ) E . So we may restrict ourselves to the elements G ∈ c (Γ , X ∗∗ ) for testingthe Grothendieck property. Indeed, it follows from the above arguments that for such G we have lim n →∞ (cid:104) G, ( f nγ ) γ ∈ Γ (cid:105) = 0 . Then, by the uniform boundedness principle applied to ( f nγ ) γ ∈ Γ viewed as an element of ( (cid:76) γ ∈ Γ X ∗∗∗ γ ) E ∗ , we conclude the result. (cid:3) A Banach space X contains uniformly complemented copies of (cid:96) n ( n ∈ N ) if and only if X has non-trivial type ([45, Theorem 13.3]). We have already seen in Remark 1 a specialcase of the following result. Proposition 5.4.2. If ( (cid:76) n ∈ N X n ) (cid:96) ∞ is Grothendieck, then there is no λ > such thateach space X n contains a λ -complemented copy of (cid:96) n .Proof. If each X n contains a λ -complemented copy of (cid:96) n , then ( (cid:76) n ∈ N X n ) (cid:96) ∞ contains a com-plemented copy of (cid:96) [80, p. 303]. (cid:3) In [107, Theorem 3], Leung proved that if E is a countably order-complete Banachlattice satisfying certain technical conditions, then (cid:96) ∞ (Γ , E ) is a Grothendieck space. Asa consequence, he derived the following fact ([107, Theorem 6]): Example 5.4.3. If φ is an Orlicz function such that the Orlicz space L φ ( µ ) has weaklysequentially complete dual space, then (cid:96) ∞ (Γ , L φ ( µ )) is Grothendieck.The special case of (cid:96) ∞ (Γ , L p ( µ )) was proved in [135].A cardinal number λ is real-valued , whenever there exists an atomless probability measureon the power-set of λ . The existence of such a cardinal number cannot be proved in ZFCas it implies that ZFC is consistent. Assuming that a real-valued cardinal number exists,we denote by m r the smallest real-valued measurable cardinal.Leung and Räbiger [109, Theorem p. 55] obtained the following result. Theorem 5.4.4. Let Γ be a set with | Γ | < m r and let ( E γ ) γ ∈ Γ be a family of Banachspaces. Suppose that E = ( (cid:76) γ ∈ Γ E γ ) (cid:96) ∞ (Γ) has property ( V ) . Then E is a Grothendieckspace if and only if each space E γ is Grothendieck.In particular, E is Grothendieck, whenever(1) each space E γ ( γ ∈ Γ) is a Grothendieck Lindenstrauss space; or(2) each space E γ ( γ ∈ Γ) is a Grothendieck C ∗ -algebra.Moreover, if no real-valued cardinal exists, the result holds for any index set Γ . Clause (1) is [109, Corollary 2], whereas clause (2) follows from the fact that E isnaturally a C ∗ -algebra when so is each E γ ( γ ∈ Γ ) and Pfitzner’s theorem asserting that C ∗ -algebras have property ( V ) (Theorem 4.2.1).Let us reiterate the question asked by Kucher ([102, Conjecture 1]). Problem 29. Suppose that E is super-reflexive. Is (cid:96) ∞ ( E ) a Grothendieck space? The case of spaces of vector-valued function on atomless measure spaces appears to bemore restrictive. Proposition 5.4.5. Suppose that µ is an atomless measure.(1) If < p < ∞ and the Bochner space L p ( µ, X ) is Grothendieck, then the space X isreflexive.(2) If L ∞ ( µ, X ) is Grothendieck, then X has non-trivial type and it is reflexive.Proof. We give the proof when µ is the Lebesgue measure on the unit interval, which isessentially valid in the general case.Let < p (cid:54) ∞ . If X is a non-reflexive Grothendieck space, then X ∗ contains a sequence ( f n ) ∞ n =1 equivalent to the unit vector basis of (cid:96) (Corollary 3.1.11). Let ( r n ) ∞ n =1 be a sequenceof Rademacher functions and /p + 1 /q = 1 ( q = 1 for p = ∞ ). We consider the simplefunctions ϕ n ( t ) = r n ( t ) f n in L q ( µ, X ∗ ) , regarded naturally a subspace of L p ( µ, X ) ∗ . Itis not difficult to check that ( ϕ n ) ∞ n =1 is a weak ∗ null sequence in L p ( µ, X ) ∗ equivalentto the unit vector basis of (cid:96) (see the proof of [41, Theorem 1]), hence L p ( µ, X ) is notGrothendieck.In the case p = ∞ , observe that L ∞ ( µ, X ) contains a complemented copy of (cid:96) ∞ ( X ) .Having non-trivial type is equivalent to non-containment of uniformly complemented copiesof (cid:96) n ( n ∈ N ). Thus X must have non-trivial type when L ∞ ( µ, X ) is Grothendieck ( cf .Remark 1). (cid:3) The former clause of Proposition 5.4.5 is due to Díaz [41], whereas the latter one maybe found in [45, Theorem 13.3]. A Banach space E is super-reflexive , whenever every Banach space that is finitely rep-resentable in E is reflexive. It is well known that a Banach space E is super-reflexiveif and only if every ultrapower of E with respect to a countably incomplete ultrafilter isreflexive [75, Proposition 6.4]. This fact provides non-trivial examples of ultrapowers thatare Grothendieck spaces. Problem 30. Can an ultrapower of a reflexive space be Grothendieck without being re-flexive?By Proposition 5.5.1, there exist ultraproducts of finite dimensional spaces which areGrothendieck non-reflexive spaces: just take { (cid:96) n ∞ : n ∈ N } .If { X γ : γ ∈ Γ } is a family of Banach spaces such that ( (cid:76) γ ∈ Γ X γ ) (cid:96) ∞ is Grothendieck and U is an ultrafilter on Γ , then the ultraproduct [ X γ ] U is Grothendieck being a quotient of ( (cid:76) γ ∈ Γ X γ ) (cid:96) ∞ (Γ) .Let us revisit further examples of Banach spaces whose ultraproducts are Grothendieckspaces. Proposition 5.5.1. Let U be a countably incomplete ultrafilter over a set Γ and let X γ ( γ ∈ Γ) be Banach spaces.(1) If X γ are C ∗ -algebras ( γ ∈ Γ) , then [ X γ ] U is a Grothendieck space.(2) If the ultraproduct [ X γ ] U is a (cid:103) OL ∞ -space, then it is a Grothendieck space.(3) If the ultraproduct [ X γ ] U has property ( V ) , then it is a Grothendieck space.Proof. Clause (1) has been already noticed in [91, Proposition 1.2(ii)]. It follows from theconjunction of Theorem 4.2.1 (ultraproducts of C ∗ -algebras are naturally C ∗ -algebras) andthe fact that ultraproducts of Banach spaces over countably incomplete ultrafilters do notcontain complemented copies of c ([9, Proposition 3.3]). The conclusion then follows fromProposition 3.1.13.As for (2), by Proposition 3.1.5, it is enough to show that every separable subspace ofthe ultraproduct is contained in a Grothendieck subspace. The proof closely emulates thatof [9, Proposition 3.2].Suppose that the ultraproduct [ X γ ] U is a (cid:103) OL ∞ ,λ + ε -space for some λ (cid:62) and all ε > ,and let W be a separable subspace of [ X γ ] U . Let D be a countable linearly dense, linearlyindependent subset of W . For each d ∈ D , let ( d γ ) γ ∈ Γ ∈ ( (cid:76) γ ∈ Γ X γ ) (cid:96) ∞ (Γ) be a representativeof d . We write D as a strictly increasing union of finite sets D n and set W n = span D n .Thus, there is a finite-dimensional subspace F n of [ X γ ] U that contains W n and a C ∗ -algebra A n such that d BM ( F n , A n ) (cid:54) λ + n .We fix a basis B n of F n that contains the set D n . For n ∈ N and b ∈ B n , we choosea representative ( b γ ) γ ∈ Γ ∈ ( (cid:76) γ ∈ Γ X γ ) (cid:96) ∞ (Γ) , however we insist that ( b γ ) γ ∈ Γ = ( d γ ) γ ∈ Γ aslong as b ∈ D . We then define ( F n ) γ as span { b γ : b ∈ B n } ⊂ X γ .As the ultrafilter U is countably incomplete, we may find a sequence of sets I n all in U whose intersection is empty. For each n set ˜ J n = (cid:110) γ ∈ Γ : d BM (( F n ) γ , A ) (cid:54) λ + n for some C ∗ -algebra A (cid:111) ∩ I n and J n = ˜ J ∩ . . . ∩ ˜ J n ( n ∈ N ). We have J n ∈ U for all n . Now, S ⊆ (cid:2) F sup { k ∈ N : γ ∈ J k } (cid:3) U and the latter space has the Banach–Mazur distance at most λ to the ultraproduct of C ∗ -algebras, which is a Grothendieck space itself.For (3), note that the ultraproduct [ X γ ] U contains no complemented copy of c by [9,Proposition 3.3]. (cid:3) Contreras and Díaz [34, Section 3] proved that all the ultrapowers of the disk algebra A and H ∞ have property ( V ) . Since ultrapowers of Banach spaces over countably incompleteultrafilters do not contain complemented copies of c ([9, Proposition 3.3]), the followingresult is a consequence of Proposition 3.1.13. Proposition 5.5.2. All ultrapowers of A and H ∞ over countably incomplete ultrafiltershave the Grothendieck property. Ultrapowers can be applied to obtain non-separable reflexive quotients of (cid:96) ∞ . Indeed,since (cid:96) q is a quotient of (cid:96) ∞ for (cid:54) q < ∞ (see the proof of Corollary 5.1.3), (cid:96) ∞ ( (cid:96) q ) isa quotient of (cid:96) ∞ ≡ (cid:96) ∞ ( (cid:96) ∞ ) . Therefore, if U is a non-principal ultrafilter on N , then theultrapower [ (cid:96) p ] U is a non-separable L p ( µ ) -space [75] and a quotient of (cid:96) ∞ . Thus, Problem23 admits the following special case. Problem 31. Let U be a non-principal ultrafilter on N and let Q : (cid:96) ∞ → [ (cid:96) p ] U be a sur-jective operator ( (cid:54) p < ∞ ). Does the kernel of Q have the Grothendieck property? Here we briefly describe some additional results that could be worth to know for peopleinterested in the Grothendieck property, but we have chosen not to treat in detail. We say that an operator T : X → Y is Grothendieck if the adjoint operator T ∗ takesweak ∗ -convergent sequences in Y ∗ to weak-convergent sequences in X ∗ . Obviously, X isGrothendieck if and only if the identity on X is a Grothendieck operator.Pietsch [131, 3.2.6] proved that an operator T ∈ B ( X, Y ) is Grothendieck if and only iffor every S ∈ B ( Y, Z ) with separable range, the product ST is weakly compact. Note alsothat the class of Grothendieck operators is an operator ideal in the sense of [131], whichis surjective ( S surjective operator and ST Grothendieck implies T Grothendieck) andclosed (the norm-limit of a convergent sequence of Grothendieck operators is Grothendieck).This is a consequence of the fact that the class of weakly compact operators shares theseproperties.A subset C of a Banach space Y is said to be a Grothendieck subset if T ( C ) is relativelyweakly compact for every operator T : X → c . It is not difficult to prove the followingresult (see, e.g. , [57, Proposition 1]). Proposition 6.1.1. An operator T : X → Y is Grothendieck if and only if it takes B X into a Grothendieck subset of Y . We do not know whether the ideal of Grothendieck operators has the factorisation prop-erty. Problem 32. Does every Grothendieck operator factorise through a Banach space withthe Grothendieck property?Domański, Lindström, and Schlüchterman proved that if T is a Grothendieck operatorand S is a compact operator, then the tensor-product operator T (cid:98) ⊗ ε S defined on theinjective tensor product of the domains of the respective operators is still Grothendieck.Since the injective tensor product of two Grothendieck spaces need not be Grothendieck(for example, (cid:96) (cid:98) ⊗ ε (cid:96) contains a complemented copy of c ), tensoring the identity operators on such spaces provides counterexamples to tensorial stability of the ideal of Grothendieckoperators.In [62], Grothendieck holomorphic functions between complex spaces E and F wheredefined as those holomorphic functions f : E → F for which each x ∈ E has a neighbour-hood V x such that f ( V x ) is a Grothendieck subset, and they where characterised in termsof factorisation as follows: Proposition 6.1.2. ([62, Theorem 6]) A holomorphic function f : E → F is Grothendieckif and only if there exists a Banach space G , a holomorphic function g : E → G , anda Grothendieck operator T : G → F such that f = T ◦ g . Krulišová (née Bendová) introduced in [13] a quantitative version of the Grothendieckproperty using the following two measures δ w and δ w ∗ of ‘non-weak-Cauchyness’ and ‘non-weak*-Cauchyness’, respectively, defined for bounded sequences ( f n ) ∞ n =1 in the dual ofa Banach space X : • δ w (( f n ) ∞ n =1 ) = sup x ∗∗ ∈ B X ∗∗ inf n ∈ N sup k,l (cid:62) n |(cid:104) x ∗∗ , f k − f l (cid:105)| , • δ w ∗ (( f n ) ∞ n =1 ) = sup x ∈ B X inf n ∈ N sup k,l (cid:62) n |(cid:104) f k − f l , x (cid:105)| . Definition 6.2.1. Let λ (cid:62) . A Banach space X is a λ - Grothendieck space , whenever forevery for every bounded sequence ( f n ) ∞ n =1 in X ∗ we have δ w (( f n ) ∞ n =1 ) (cid:54) λ · δ w ∗ (( f n ) ∞ n =1 ) . Krulišová proved ([13, Theorem 4.1]) that (cid:96) ∞ (Γ) is a 1-Grothendieck space for any set Γ (hence every 1-injective space is 1-Grothendieck too), however not every Grothendieckspace is λ -Grothendieck for some λ (cid:62) ([13, Theorem 2]); this was obtained by formingan (cid:96) -sum of λ n -Grothendieck spaces with λ n → ∞ as n → ∞ . Lechner extended in [105]the results about the 1-Grothendieck property of 1-injective spaces by proving that forevery subsequentially complete Boolean algebra A (see Section 4.1) the space C (St A ) is1-Grothendieck.Moreover, Krulišová introduced in [101] a quantitative version of property ( V ) as follows:given λ (cid:62) , a Banach space has quantitative property ( V ) (with constant λ ), whenever forevery Banach space Y and every operator T : X → Y , one has γ ( T ) (cid:54) λ · uc( T ) , where • γ ( T ) is the measure relative weak non-compactness of T : γ ( T ) = sup (cid:8) | lim n →∞ lim m →∞ (cid:104) f m , x n (cid:105) − lim m →∞ lim n →∞ (cid:104) f m , x n (cid:105)| : x n ∈ T ( B X ) ( n ∈ N ) and ( f m ) ∞ m =1 is a sequence in B X ∗ so that both limits exist (cid:9) • uc( T ) measures how far is the operator T from being unconditionally converging: uc( T ) = sup (cid:8) ca(( n (cid:88) i =1 T x i ) ∞ n =1 : ( x n ) ∞ n =1 ⊂ X, sup f ∈ B X ∗ ∞ (cid:88) n =1 |(cid:104) f, x n (cid:105)| (cid:54) (cid:9) , where ca(( y n ) ∞ n =1 ) = inf n ∈ N sup k,l (cid:62) n (cid:107) y k − y l (cid:107) (cid:0) ( y n ) ∞ n =1 ⊂ Y (cid:1) . Using these notions, Krulišová refined Pfitzner’s theorem (Theorem 4.2.1) by showing that C ∗ -algebras have the quantitative property ( V ) ([101, Theorem 4.1]). Moreover, she provedthe following counterpart of Proposition 3.2.5 ([101, Theorem 5.1]). Proposition 6.2.2. Every dual space with the quantative property ( V ) is a λ -Grothendieckspace for some λ (cid:62) . An alternative approach to quantify the Grothendieck property was taken in [30], wherethe following quantity G ( X ) was defined for any Banach space X : G ( X ) = sup ( f n ) ∞ n =1 ⊆ B X ∗ weak ∗ null inf ( g n ) ∞ n =1 ∈ cbs(( f n ) ∞ n =1 ) lim sup n →∞ (cid:107) g n (cid:107) . Here cbs(( f n ) ∞ n =1 ) stands for the family of all convex block subsequences of ( f n ) ∞ n =1 .)One can use Proposition 3.1.2 to prove that X is a Grothendieck space if and only if G ( X ) = 0 ([30, Theorem 4.5]). It is easy to see that G ( c ) = 1 . A Banach lattice (or more generally, an ordered Banach space) E has the positive Gro-thendieck property , if every positive weakly* convergent sequence in E ∗ is weakly conver-gent.As in the case of the Grothendieck property, a Banach lattice E has the positive Groth-endieck property if and only if every positive operator T : E → c is weakly compact.Moreover, the positive Grothendieck property is preserved by positive surjective operators.The space c is a paradigm example of a Banach lattice failing the positive Grothendieckproperty. The positive Grothendieck property is much weaker than the usual one as forexample c , the space of convergent sequences (in which c has codimension one) has thepositive Grothendieck property because every positive functional on c attains its norm at N ∈ c .Wnuk ([150, Proposition 2.12]) proved the following characterisation of Banach latticeswith the positive Grothendieck property. Proposition 6.3.1. For a Banach lattice E , the following assertions are equivalent:(1) E has the positive Grothendieck property;(2) for every non-reflexive Banach lattice F with order-continuous norm there is nopositive surjective operator T : E → F ;(3) There is no positive surjective operator T : E → c . Kühn proved that for Archimedean ordered Banach spaces (for example, Banach latticeswith order unit) with countable Riesz interpolation property the Grothendieck and posi-tive Grothendieck properties are equivalent ([103, 1. Proposition]; see also [121, Theorem5.3.13]). Koszmider and Shelah ([99, Lemma 2.2]) obtained a handy condition characterisingStone spaces K for which C ( K ) has the positive Grothendieck property. Proposition 6.3.2. Let A be a Boolean algebra. Then the space C (St A ) has the posi-tive Grothendieck property if and only if given an antichain { A n : n ∈ N } in A , ε > ,and a bounded sequence ( µ n ) ∞ n =1 of bounded, finitely additive signed measures on A with | µ n ( A n ) | > ε , there exists A ∈ A such that the scalar sequence ( µ n ( A )) ∞ n =1 fails to converge. C -semigroups of operators on Grothendieck spaces In the present section, we discuss results that may be considered extensions of Proposi-tion 3.3.5. Definition 6.4.1. A Banach space X has the Lotz property , whenever every C -semigroupof operators on X is uniformly continuous.The following result is due to Lotz [116]. Proposition 6.4.2. Let X be a Grothendieck space with the Dunford–Pettis property.Then X has Lotz property. Therefore, Grothendieck C ( K ) -spaces have the Lotz property. In [124], van Neervenproved a partial converse to the above theorem for Banach lattices. Proposition 6.4.3. ([124, Theorem 2]) Let E be a Banach lattice with a quasi-interiorpoint. Then the following assertions are equivalent:(1) E has the Lotz property,(2) E is a Grothendieck space with the Dunford–Pettis property,(3) E is isomorphic to a C ( K ) -space with the Grothendieck property. In [108, Examples 13 and 15] (see also Proposition 4.6.3), Leung constructs a Banachspace E with an unconditional basis—hence E ∗∗ is a Banach lattice—such that E ∗∗ hasthe Lotz property [108, Corollary 11] and it is Grothendieck, but fails the Dunford–Pettisproperty.Atalla proved that for a contractive operator T : E → E on a Grothendieck space E , thesequence of Cesàro means (cid:0) n (cid:80) nk =1 T k (cid:1) ∞ n =1 converges strongly if and only if the norm closureand the weak* closure of the range of I E ∗ − T ∗ coincide ([7, Theorem 2.2]) and noted thatthe hypothesis of the Grothendieck property of E cannot be dropped ([7, Examples 2.3])as witnessed by certain contractive Markov operators on C [0 , . Shaw ([145, Theorem2]) extended and improved this result in the setting of locally integrable semigroups ofoperators on Grothendieck spaces. Proposition 6.4.4. Let E be a Grothendieck space and let T ( · ) be a locally integrablesemigroup of operators on E . Set S ( t ) x = (cid:82) t T ( s ) x d s ( x ∈ E, t > . Then T ( · ) isstrongly ergodic, which means that • lim t →∞ t S ( t ) x for all x ∈ E ,if and only if the following conditions are satisfied: • lim sup t →∞ t (cid:107) S ( t ) (cid:107) < ∞ . • lim t →∞ t T ( t ) S ( u ) x = 0 for all x ∈ E and u > . • the norm closure and the weak* closure of span (cid:8) (cid:91) t> im ( T ( t ) ∗ − I E ∗ ) (cid:9) coincide.If T ( · ) is a C -semigroup with infinitesimal generator A , then the final condition may bereplaced by coincidence of the norm and the weak*-closures of the range of A ∗ . The final condition in Proposition 6.4.4 is certainly redundant when E is reflexive andindeed the result recovers Masani’s result in this setting [118]. The definition of a Grothendieck space naturally extends to topological vector spaces,however it is perhaps more natural in the context of (Hausdorff) locally convex spaces asthey have non-trivial (continuous) bidual spaces.Given a topological vector space E , we denote by E ∗ the (continuous) dual space of E , and by E ∗∗ the bidual of E ; that is, the dual of ( E ∗ , β ( E ∗ , E )) , where β ( E ∗ , E ) is thestrong topology. Definition 6.5.1. A (locally convex) topological vector space E is Grothendieck wheneverthe σ ( E ∗ , E ) –sequential and σ ( E ∗ , E ∗∗ ) -sequential convergences coincide in equicontinuoussubsets of E ∗ .Research concerning locally convex Grothendieck spaces has been conducted since 1980sin parallel to the Banach-space framework; we only highlight the most basic properties ofGrothendieck spaces in this context and some results related to spaces of vector-valuedcontinous functions on topological spaces that are counterparts of the results for C ( K, E ) -spaces presented in Section 4.1.As observed by Freniche ([55, Proposition 2.3]), locally convex Grothendieck spaces havethe following stability properties: • A locally convex space E is Grothendieck if and only if so is every dense linearsubspace F ⊂ E . • Let T : E → F be a continuous linear operator such that for every bounded subset B of F there is a bounded subset C of E so that B ⊆ T ( C ) . If E is Grothendieck,then so is F . • If E is an inductive limit (in the category of locally convex spaces) of a sequence ( E n ) ∞ n =1 of Grothendieck spaces, and if every bounded subset of E is contained insome E n , then E is Grothendieck.In [55], the definition of a G-space was extended to arbitrary topological spaces: a topo-logical space X is a G-space , whenever for every compact subset K ⊂ X the Banach space C ( K ) is Grothendieck. For a locally convex space E , C ( X, E ) (the space of E -valued continuous functions on X endowed with the compact-open topology) is Grothendieck if and only if for every compactsubset K ⊂ X the space C ( K, E ) is Grothendieck ([55, Theorem 2.4]). Moreover, if X is a G-space and E is a strict inductive limit of Fréchet–Montel spaces, then C ( X, E ) isGrothendieck. Additionally, Khurana proved that for a compact G-space K and a Montelspace E the space C ( K, X ) is Grothendieck [94].For completely regular spaces X containing infinite compact subsets, further character-isations of Grothendieck C ( X, E ) -spaces have been obtained in [95]. Further variations ofthis result were obtained in [48], where it was proved that for a completely regular space E containing an infinite compact subset and a non-Montel Fréchet space, the space C ( X, E ) contains a complemented copy of c .Valdivia proved that Corollary 3.1.11 extends to Fréchet spaces ([149, Theorem 1]). Proposition 6.5.2. Let E be a Fréchet space. If E is a non-reflexive Grothendieck space,then it contains an isomorphic copy of (cid:96) . In the literature concerning Fréchet spaces, the Grothendieck property is often consideredin tandem with the Dunford–Pettis property; spaces satisfying both properties are termed GDP spaces . Their systematic treatment may be found, e.g. , in [2, 3, 16]; for non-Fréchetspaces see, e.g. , [56]. We conclude the paper by reiterating the accumulated open problems in the form of a con-cise list. The numbering below corresponds to the numbering of Problems used earlier inthe paper.(1) Does there exist an internal characterisation of Grothendieck spaces?(2) What class of Banach spaces Y is characterised by the equality B ( X, Y ) = W ( X, Y ) for every Grothendieck space X ?(3) Does a non-reflexive Grothendieck space contain a copy of c ?(4) Do Grothendieck spaces have property ( V ) ?(5) Do dual spaces with the Grothendieck property have property ( V ) ?(6) Suppose that X and X ∗ are Grothendieck. Is X reflexive?(7) Let X be a Grothendieck space. Is X ∗∗ Grothendieck?(8) Let X be a Grothendieck space. Is At( X ) a weak ∗ - G δ subset of X ∗ ?(9) Characterise filters F for which c , F is a Grothendieck space.(10) Is there an intrinsic characterisation of G-spaces? More precisely, can G-spaces becharacterised topologically?(11) Let A be a Boolean algebra whose Stone space is a G-space. Does there exista Boolean subalgebra B ⊂ A whose Stone space K fails to be a G-space yet everyweakly* convergent sequence of purely atomic measures on K converges weakly? (12) Characterise Banach spaces E for which E w is Grothendieck. Is E w Grothendieckwhen so is E ?(13) Let A be a C ∗ -algebra. Is A w a Grothendieck space?(14) Let E be a L ∞ -space which is Grothendieck. Is E w a Grothendieck space?(15) Is every L ∞ -space without infinite-dimensional separable complemented subspacea Grothendieck space?(16) Let X be a Grothendieck L ∞ -space. Does X have property ( V ) ?(17) Suppose that a Banach space X is λ -separably injective for some λ < . Is X Grothendieck?(18) Let n (cid:62) . Set E n = C ([0 , n ) . Is E ∗∗ n a Grothendieck space?(19) Can we replace ‘the countable interpolation property’ by ‘the countable monotoneinterpolation property’ in Proposition 4.5.3? What happens in the case E is a C ∗ -algebra?(20) Find a characterisation of dual Grothendieck spaces.(21) Let X be a Grothendieck space without complemented separable, infinite-dimensio-nal subspaces and let Y be a closed subspace of X with X/Y infinite-dimensionalseparable. Is it possible for Y to be isomorphic to X ?(22) Let X be a Grothendieck Banach space. Suppose that M is a closed subspace of X such that dens X/M < p . Is M a Grothendieck space?(23) Let M be a closed subspace of a Grothendieck space X such that X/M is reflexive.Is M a Grothendieck space? A concrete case: let N be a closed subspace of (cid:96) ∞ such that (cid:96) ∞ /N is isomorphic to (cid:96) (Γ) with Γ uncountable. Is N a Grothendieckspace?(24) What is the number of pairwise non-isomorphic Grothendieck subspaces of (cid:96) ∞ ; canit be c ?(25) Suppose that X (cid:98) ⊗ ε Y is Grothendieck. Is X or Y reflexive?(26) Is there a pair of infinite-dimensional Banach spaces X and Y for which the tensorproduct X (cid:98) ⊗ ε Y is Grothendieck yet it is not reflexive?(27) Suppose that B ( E ) is Grothendieck. Is E reflexive?(28) Suppose that E is super-reflexive. Is B ( E ) Grothendieck? What happens when E = (cid:96) p , < p < ∞ , p (cid:54) = 2 ?(29) Suppose that E is super-reflexive. Is (cid:96) ∞ ( E ) a Grothendieck space?(30) Can an ultrapower of a reflexive space be Grothendieck without being reflexive?(31) Let U be a non-principal ultrafilter on N and let Q : (cid:96) ∞ → [ (cid:96) p ] U be a surjectiveoperator ( (cid:54) q < ∞ ). Does the kernel of Q have the Grothendieck property?(32) Does every Grothendieck operator factorise through a Banach space with the Gro-thendieck property? References [1] A.D. Acosta; V. Kadets. A characterization of reflexive spaces . Math. Ann. 349, No. 3 (2011), 577–588.[2] A.A. Albanese; J. Bonet; W.J. Ricker. Grothendieck spaces with the Dunford–Pettis property . Posi-tivity 14 (2010), 145–164.[3] A.A. Albanese; E.M. Mangino. Some permanence results of the Dunford–Pettis and Grothendieckproperties in lcHs , Funct. Approximatio, Comment. Math. 44, No. 2 (2011), 243–258.[4] T. Andô. Convergent sequences of finitely additive measures . Pacific J. Math. 11 (1961), 395–404.[5] S.A. Argyros; R. Haydon. A hereditarily indecomposable L ∞ -space that solves the scalar-plus-compactproblem . Acta Math. 206 (2011), 1–54.[6] A. Arias; J.F. Farmer. On the structure of tensor products of (cid:96) p -spaces . Pacific J. Math., 175 (1996),13–37.[7] R.E. Atalla. On the ergodic theory of contractions . Revista Colombiana de Matemáticas, 10 (1976),75–81.[8] A. Avilés; F. Cabello Sánchez; J.M.F. Castillo; M. González; Y. Moreno. On separably injectiveBanach spaces. Adv. Math. 234 (2013), 192–216.[9] A. Avilés; F. Cabello Sánchez; J.M.F. Castillo; M. González; Y. Moreno. On ultrapowers of Banachspaces of type L ∞ . Fundamenta Math. 222 (2013), 195–212.[10] A. Avilés; F. Cabello Sánchez; J.M.F. Castillo; M. González; Y. Moreno. Separably injective BanachSpaces. Lecture Notes in Math. 2132. Springer-Verlag, 2016.[11] D. Bárcenas; L.G. Mármol. On C(K) Grothendieck spaces . Rend. Circ. Mat. Palermo 54 (2005),209–216.[12] K. Beanland; T. Kania; N.J. Laustsen. The algebras of bounded operators on the Tsirelson andBaernstein spaces are not Grothendieck spaces. Houston J. Math. 45 (2019), 553–566.[13] H. Bendová. Quantitative Grothendieck property . J. Math. Anal. 412 (2014), 1097–1104.[14] W. Bielas. On convergence of sequences of Radon measures , Praca semestralna nr 2 (semestr zimowy2010/11), ssdnm.mimuw.edu.pl/pliki/prace-studentow/st/pliki/wojciech-bielas-2.pdf .[15] F. Bombal. Operators on vector sequence spaces. London Math. Soc. Lecture Notes 140 (1989),94–106.[16] J. Bonet; W.J. Ricker. Schauder decompositions and the Grothendieck and Dunford–Pettis propertiesin Köthe echelon spaces of infinite order . Positivity, 11 (2007), 77–93.[17] J. Bourgain. H ∞ is a Grothendieck space. Studia Math. 75 (1983), 193–216.[18] J. Bourgain. On weak completeness of the dual of spaces of analytic and smooth functions . Bull. Soc.Math. Belg. Sér. B, 35 (1983), 111–118.[19] J. Bourgain; F. Delbaen. A class of special L ∞ -spaces. Acta Math. 145 (1980), 155–176.[20] C. Brech. On the density of Banach spaces C ( K ) with the Grothendieck property. Proc. Amer. Math.Soc. 134 (2006), 3653–3663.[21] J.K. Brooks; K. Saitô; J.D.M. Wright. Operators on σ -complete C ∗ -algebras . Quart. J. Math. (Ox-ford) 56, Issue 3 (2005), 301–310.[22] Q. Bu. On Kalton’s theorem for regular compact operators and Grothendieck property for positiveprojective tensor products. Proc. Amer. Math. Soc. 148 (2020), 2459–2467.[23] Q. Bu; G. Emmanuele. The projective and injective tensor products of L p [0 , and X being Grothen-dieck spaces. Rocky Mount. J. Math. 35 (2005), 713–726.[24] Q. Bu; D. Ji; X. Xue. The Grothendieck property for injective tensor products of Banach spaces. Czech. Math. J. 60 (2010), 1153–1159.[25] Q. Bu; Y. Li. New examples of non-reflexive Grothendieck spaces. Houston J. Math. 43 (2017),569–575. [26] F. Cabello Sánchez; J.M.F. Castillo. Homological methods in Banach space theory , Cambridge Studiesin Advanced Mathematics 193, Cambridge Univ. Press 2021.[27] J.M.F. Castillo; M. González. Properties ( V ) and ( u ) are not three-space properties. Glasgow Math.J. 36 (1994), 297–299.[28] J.M.F. Castillo; M. González. Three-space problems in Banach space theory. Lecture Notes in Math.1667, Springer-Verlag 1997.[29] P. Cembranos, C ( K, E ) contains a complemented copy of c , Proc. Amer. Math. Soc. 91 (1984),556–558.[30] D. Chen; T. Kania; Y. Ruan. Quantifying properties ( K ) and ( µ s ) . Preprint (2021), arXiv:2102.00857 .[31] E. Chetcuti; J. Hamhalter. A noncommutative Brooks-Jewett theorem. J. Math. Anal. Appl. 355,No. 2 (2009), 839–845.[32] E. Christensen; A.M. Sinclair. Completely bounded isomorphisms of injective von Neumann algebras .Proc. Edinburgh Math. Soc. (2) 32 (1989), 317–327.[33] R. Cilia; G. Emmanuele. Pelczynski’s property ( V ) and weak* basic sequences . Quaest. Math. 38,No. 3 (2015), 307–316.[34] M.D. Contreras; D. Díaz. Some Banach space properties of the duals of the disk algebra and H ∞ . Michigan Math. J. 46 (1999), 123–141.[35] Th. Coulhon. Suites d’opérateurs sur un espace C ( K ) de Grothendieck . C.R. Acad. Sci. Paris, 298(1984) 13–15.[36] H.G. Dales; F.K. Dashiell, Jr.; A.T.-M. Lau; D. Strauss. Banach Spaces of Continuous Functions asDual Spaces . CMS Books Math., Springer, Cham, 2016.[37] F.K. Dashiell, Jr. Nonweakly compact operators from order-Cauchy complete C ( S ) lattices, withapplications to Baire classes. Trans. Amer. Math. Soc. 266 (1981), 397–416.[38] W.J. Davis; T. Figiel; W.J. Johnson; A. Pełczyński. Factoring weakly compact operators. J. Funct.Anal. 35 (1974), 397–411.[39] D.W. Dean. Schauder decompositions in ( m ) . Proc. Amer. Math. Soc. 18 (1967), 619–623.[40] G. Debs; G. Godefroy; J. Saint Raymond. Topological properties of the set of norm-attaining linearfunctionals . Can. J. Math. 47 (1995), 318–329.[41] D. Díaz. Grothendieck’s property in L p ( µ, X ) . Glasgow Math. J. 37 (1995), 379–382.[42] J. Diestel. Grothendieck spaces and vector measures. In J. Diestel. “Vector and operator valuedmeasures and applications” (Proc. Sympos., Alta, Utah, 1972), pp. 97–108. Academic Press, 1973.[43] J. Diestel. A survey of results related to the Dunford–Pettis property. Proceedings of the Conferenceon Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C.,1979), pp. 15–60, Contemp. Math. 2, Amer. Math. Soc. Providence, 1980.[44] J. Diestel. Sequences and series in Banach Spaces. Springer-Verlag, 1984.[45] J. Diestel; H. Jarchow; A. Tonge. Absolutely summing operators. Cambridge Univ. Press, 1995.[46] J. Diestel; C.J. Seifert. The Banach–Saks ideal, I. Operators acting on C (Ω) . Commentationes Math.(Tomus especialis in honorem W. Orlicz) I (1978), 109–118.[47] J. Diestel; J.J. Uhl, Jr. Vector measures. Math. Surveys, 15. Amer. Math. Soc., 1977.[48] P. Domański; L. Drewnowski. Fréchet spaces of continuous vector-valued functions: Complementabil-ity in dual Fréchet spaces and injectivity . Studia Math. 102, No. 3 (1992), 257–267.[49] P. Domański; M. Lindström; G. Schlüchtermann. Grothendieck operators on tensor products . Proc.Am. Math. Soc. 125, No. 8 (1997), 2285–2291.[50] A. Dow; A.V. Gubbi; A. Szymański, Rigid Stone spaces with ZFC . Proc. Amer. Math. Soc. 102, No.3 (1988), 745–748.[51] F.J. Fernández-Polo; A.M. Peralta. Weak compactness in the dual space of a JB*-triple is commu-tatively determined , Math. Scand. 105 (2009), 307–319.[52] F.J. Fernández-Polo; A.M. Peralta. A short proof of a theorem of Pfitzner , Quart. J. Math. Oxford61 (2010), 329–336. [53] T. Figiel; W.B. Johnson; L. Tzafriri. On Banach lattices and spaces having local unconditional struc-ture, with applications to Lorentz function spaces, J. Approx. Theory 13 (1975), 395–412.[54] D. Fremlin. Consequences of Martin’s Axiom . Cambridge Tracts in Math. 84, Cambridge UniversityPress (1984).[55] F.J. Freniche. Grothendieck locally convex spaces of continuous vector valued functions . Pac. J. Math.120 (1985), 345–355.[56] S. Gabriyelyan; J. Kąkol. Dunford–Pettis type properties and the Grothendieck property for functionspaces . Rev. Mat. Complut. 33, No. 3 (2020), 871–884.[57] I. Ghenciu. The weak Gelfand–Phillips property in spaces of compact operators . Comm. Math. Univ.Carolinae 58, No. 1 (2017), 35–47.[58] I. Ghenciu; P. Lewis. Completely continuous operators . Colloq. Math. 126, No. 2 (2012), 231–256.[59] G. Godefroy; P. Saab. Quelques espaces de Banach ayant les propriétés (V) ou (V*) de A. Pełczyński .C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), 503–506.[60] M. González. Dual results of factorization for operators. Ann. Acad. Sci. Fennicae 18 (1993), 3–11.[61] M. González; J.M. Gutiérrez. Polynomial Grothendieck properties. Glasgow Math. J. 37 (1995), 211–219.[62] M. González; J.M. Gutiérrez. Surjective factorization of holomorphic mappings. Comment. Math.Univ. Carolinae 43 (2000), 469–476.[63] M. González; F. León-Saavedra; M.P. Romero de la Rosa. On (cid:96) ∞ -Grothendieck subspaces. J.Math.Anal. Appl. 497 (2021) 124857, 5 pp.[64] M. González; V.M. Onieva. Lifting results for sequences in Banach spaces. Math. Proc. CambridgePhilos. Soc. 105 (1989), 117–121.[65] W.H. Graves; R.F. Wheeler. On the Grothendieck and Nikodym properties for algebras of Baire,Borel and universally measurable sets . Rocky Mountain J. Math. 13 (1983), no. 2, 333–354.[66] A. Grothendieck. Sur les applications linéaires faiblement compactes d’espaces du type C ( K ) . Canad.J. Math. 5 (1953), 129–173.[67] J. Hagler; W.B. Johnson. On Banach spaces whose dual balls are not weak ∗ sequentially compact. Israel J. Math. 28 (1977), 325–330.[68] P. Hájek; V. Montesinos; J. Vanderwerff; V. Zizler. Biorthogonal systems in Banach spaces. CMSBooks in Math. 26; Springer-Verlag, 2008.[69] P. Harmand; D. Werner; W. Werner. M-ideals in Banach spaces and Banach algebras. Lecture Notesin Math. 1547; Springer-Verlag, 1993.[70] R. Haydon. On dual L -spaces and injective bidual Banach spaces. Israel J. Math. 31 (1978), 142–152.[71] R. Haydon. A non-reflexive Grothendieck space that does not contain (cid:96) ∞ . Israel J. Math. 40 (1981),65–73.[72] R. Haydon. An unconditional result about Grothendieck spaces . Proc. Amer. Math. Soc. 100. (1987),no. 3, 511–516.[73] R. Haydon. Boolean rings that are Baire spaces . Serdica Math. J. 27 (2001), 91–106.[74] R. Haydon; M. Levy; E. Odell. On sequences without weak* convergent convex block subsequences .Proc. Amer. Math. Soc., 100 (1987) 94–98.[75] S. Heinrich. Ultraproducts in Banach space theory. J. Reine Angew. Math. 313 (1980), 72–104.[76] S. Heinrich. Closed operator ideals and interpolation. J. Funct. Anal. 35 (1980), 397–411.[77] J.R. Holub. Reflexivity of L ( E, F ) . Proc. Amer. Math. Soc. 39 (1973), 175–177.[78] D. Ji; M. Craddock; Q. Bu. Reflexivity and the Grothendieck property for positive tensor products ofBanach lattices-I . Positivity 14 (2010), 59–68.[79] W.B. Johnson. No infinite-dimensional P space admits a Markuschevivh basis . Proc. Amer. Math.Soc., 26 (1970), 467–468.[80] W.B. Johnson. A complementary universal conjugate Banach space and its relation to the approxi-mation problem . Israel J. Math. 13 (3-4) (1972), 301–310. [81] W.B. Johnson; T. Kania; G. Schechtman. Closed ideals of operators on and complemented subspacesof Banach spaces of functions with countable support . Proc. Amer. Math. Soc., 144 (2016), 4471–4485.[82] W.B. Johnson; H.P. Rosenthal. On w ∗ -basic sequences and their application to the study of Banachspaces. Studia Math. 43 (1972), 77–92.[83] M. Junge; N. Ozawa; Z.-J. Ruan. On OL ∞ structures of nuclear C*-algebras . Math. Ann., 325(2003), 449–483.[84] R.V. Kadison; J.R. Ringrose. Fundamentals of the Theory of Operator Algebras, Vol. I, ElementaryTheory , Pure and Applied Math., Vol. 100 Academic Press, New York, 1983.[85] J. Kąkol; A. Leiderman. A characterization of X for which spaces C p ( X ) are distinguished and itsapplications . arXiv:2011.14299 (2021), to appear in Proc. Amer. Math. Soc. [86] J. Kąkol; W. Marciszewski; D. Sobota; L. Zdomskyy. On complemented copies of the space c inspaces C p ( X × Y ) . Preprint arxiv.org/abs/2007.14723 (2020), 29 pp.[87] J. Kąkol; A. Moltó. Witnessing the lack of the Grothendieck property in C ( K ) -spaces via convergentsequences Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 4, Paper No. 179(2020), 7 pp.[88] J. Kąkol; D. Sobota; L. Zdomskyy. The Josefson–Nissenzweig theorem, Grothendieck property, andfinitely-supported measures on compact spaces . Preprint arxiv.org/abs/2009.07552 (2020), 57 pp.[89] N.J. Kalton; N.T. Peck. Twisted sums of sequence spaces and the three space problem. Trans. Amer.Math. Soc. 255 (1979), 1–30.[90] T. Kania. A reflexive Banach space whose algebra of operators is not a Grothendieck space. J. Math.Anal. Appl. 401 (2013), 242–243.[91] T. Kania. On C*-algebras which cannot be decomposed into tensor products with both factors infinite-dimensional , Quart. J. Math. (Oxford) 66 (2015), 1063–1068.[92] T. Kania. A letter concerning Leonetti’s paper ‘Continuous Projections onto Ideal Convergent Se-quences’ , Results Math., 12 (2019), 4 pp.[93] S.S. Khurana. Grothendieck spaces , Illinois J. Math., 22 (1978), 79–80.[94] S.S. Khurana. Grothendieck spaces. II. J. Math. Anal. Appl. 159, No. 1 (1991), 202–207.[95] S.S. Khurana; J. Vielma. Grothendieck spaces. III. Simon Stevin 67 (1993), 81–85.[96] R.W. Knight, ∆ -Sets . Trans. Amer. Math. Soc. 339 (1993), 45–60.[97] P. Koszmider. Banach spaces of continuous functions with few operators , Math. Ann. 330 (2004),151–183.[98] P. Koszmider. Set-theoretic methods in Banach space theory . University of Wrocław lecture notes,2010, ssdnm.mimuw.edu.pl/pliki/wyklady/skrypt_PKoszmider.pdf .[99] P. Koszmider; S. Shelah. Independent families in Boolean algebras with some separation properties .Algebra Univers. 69 (2013), 305–312.[100] P. Koszmider; S. Shelah; M. Świętek. There is no bound on sizes of indecomposable Banach spaces Adv. Math. 323 (2018), 745–783.[101] H. Krulišová. C ∗ -algebras have a quantitative version of Pełczyński’s property ( V ) . Czech. Math. J.67, No. 4 (2017), 937–951.[102] O.V. Kucher. The Grothendieck property in the space (cid:96) ∞ ( E ) and the weak Banach–Saks property in c ( E ) . J. Math. Sci. 96 (1999), 2828–2833.[103] B. Kühn. Schwache Konvergenz in Banachverbänden . Arch. Math. 35 (1980), 554–558.[104] N.J. Laustsen. Matrix multiplication and composition of operators on the direct sum of an infinitesequence of Banach spaces , Math. Proc. Cambridge Philos. Soc. 131 (2001) 165–183.[105] J. Lechner. C ( K ) -spaces . J. Math. Anal. Appl. 446 (2017), 1362–1371.[106] P. Leonetti. Continuous projections onto ideal convergent sequences . Results Math., 73 (2018), 5 pp.[107] D.H. Leung. Weak ∗ convergence on higher duals of Orlicz spaces. Proc. Amer. Math. Soc. 103 (1988),797–800. [108] D.H. Leung. Uniform convergence of operators and Grothendieck spaces with the Dunford–Pettisproperty . Math. Z. 197 (1988), 21–32.[109] D.H. Leung; F. Räbiger. Complemented copies of c in (cid:96) ∞ -sums of Banach spaces. Illinois J. Math.34 (1990), 52–58.[110] Y. Li; Q. Bu. New examples of non-reflexive Grothendieck spaces , Houston J. Math., 43 (2017),569–575.[111] J. Lindenstrauss. On the extension of operators with range in a C ( K ) -space , Proc. Amer. Math. Soc.15 (1964), 218–225.[112] J. Lindenstrauss. On complemented subspaces of m . Israel J. Math., 5 (1967), 153–156.[113] J. Lindenstrauss; L. Tzafriri. Classical Banach Spaces I. Springer-Verlag, 1977.[114] J. Lindenstrauss; L. Tzafriri. Classical Banach Spaces II. Springer-Verlag, 1979.[115] N.A. Lone. On the weak-Riemann integrability of weak*-continuous functions . Mediterr. J. Math. 14(2017), 7 pp.[116] H.P. Lotz. Uniform convergence of operators on L ∞ and similar spaces , Math. Z. 190 (1985), no. 2,207–220.[117] H.P. Lotz. Weak convergence in the dual of weak L p . Israel J. Math. 176 (2010), 209–220.[118] P. Masani. Ergodic theorems for locally integrable semigroups of continuous linear operators on aBanach space . Adv. Math., 21 (1976), 202–228.[119] R.D. McWilliams. A note on weak sequential convergence . Pacific J. Math 12 (1962), 333–335.[120] R.E. Megginson. An Introduction to Banach Space Theory. Springer-Verlag, 1998.[121] P. Meyer-Nieberg. Banach lattices. Springer-Verlag, 1991.[122] A. Moltó. On the Vitali–Hahn–Saks theorem . Proc. Roy. Soc. Edinburgh Sect. A 90 (1981), 163–173.[123] G.J. Murphy. C ∗ -algebras and operator theory , Academic Press, Inc., Boston, MA, 1990.[124] J.M.A.M. van Neerven. A converse of Lotz’s theorem on uniformly continuous semigroups. Proc.Am. Math. Soc. 116 (1992), 525–527.[125] E. Odell; H.P. Rosenthal. A double dual characterization of separable Banach spaces containing (cid:96) .Israel J. Math. 20 (3-4) (1975), 375–384.[126] B. de Pagter; F.A. Sukochev. The Grothendieck property in Marcinkiewicz spaces. Indag. Math. 31(2020), 791–808.[127] G.K. Pedersen. C ∗ -algebras and their automorphism groups . Edited by S. Eilers and D. Olesen. 2ndedition. Amsterdam: Elsevier/Academic Press, 2018.[128] A. Pełczyński. Banach spaces of analytic functions and absolutely summing operators. CBMS Reg.Conf. 30, Amer. Math. Soc. 1977.[129] A. Pełczyński; V.N. Sudakov. Remark on non-complemented subspaces of the space m ( S ) . Colloq.Math. 19 (1962), 85–88.[130] H. Pfitzner. Weak compactness in the dual of a C ∗ -algebra is determined commutatively. Math. Ann.298 (1994), 349–371.[131] A. Pietsch. Operator ideals. North-Holland, Amsterdam, 1980.[132] G. Plebanek. A construction of a Banach space C ( K ) with few operators. Topology Appl. 143 (2004),217–239.[133] I.A. Polyrakis; F. Xanthos. Cone characterization of Grothendieck spaces and Banach spaces con-taining c . Positivity, 15, No. 4 (2011), 677–693.[134] I.A. Polyrakis; F. Xanthos. Grothendieck ordered Banach spaces with an interpolation property . Proc.Amer. Math. Soc. 141 (2013), 1651–1661.[135] F. Räbiger. Beiträge zur Strukturtheorie der Grothendieck-Räume. Sitzungsberichte der HeidelbergerAkademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse 85-4, 78 pp. Springer-Verlag, 1985.[136] H.P. Rosenthal. On relatively disjoint families of measures, with some applications to Banach spacetheory. Studia Math. 37 (1970), 13–36. [137] H.P. Rosenthal. On injective Banach spaces and the spaces L ∞ ( µ ) for finite measures µ . Acta Math.124 (1970), 205–248.[138] R.A. Ryan. Introduction to tensor products of Banach Spaces. Springer-Verlag, 2002.[139] K. Saitô; J.D. Maitland Wright. C ∗ -algebras which are Grothendieck spaces . Rend. Circ. Mat.Palermo 52 (2003), 141–144.[140] K. Saitô; J.D. Maitland Wright. On classifying monotone complete algebras of operators . RicercheMat. 56 (2007), 321–355.[141] W. Schachermayer. On some classical measure-theoretic theorems for non-sigma-complete Booleanalgebras. Dissertationes Math. (Rozprawy Mat.) 214 (1982), 33 pp.[142] I. Schlackow. Centripetal operators and Koszmider spaces . Topology Appl. 155 (2008), 1227–1236.[143] G.L. Seever. Measures on F -spaces. Trans. Amer. Math. Soc. 133 (1968), 267–280.[144] Z. Semadeni. On weak convergence of measures and σ -complete Boolean algebras . Colloq. Math., 12(1964), 229–233.[145] S.-Y. Shaw. Ergodic theorems for semigroups of operators on a Grothendieck space. Proc. JapanAcad. 59 (A) (1983), 132–135.[146] R.R. Smith; D.P. Williams. The decomposition property for C ∗ -algebra . J. Oper. Theory 16 (1986),51–74.[147] R.R. Smith; D.P. Williams. Separable injectivity for C ∗ -algebras . Indiana Univ. Math. J. 37, No. 1(1988), 111–133.[148] M. Talagrand. Un nouveau C ( K ) qui possède la propriété de Grothendieck. Israel J. Math. 37 (1980),181–191.[149] M. Valdivia. Fréchet spaces with no subspaces isomorphic to (cid:96) . Math. Japon. 38 (1993), 397–411.[150] W. Wnuk. On the dual positive Schur property in Banach lattices . Positivity 17 (2013), 759–773.[151] R.G. Woods. Characterizations of some C ∗ -embedded subspaces of β N . Pac. J. Math. 65 (1976),573–579.[152] S. Zhang; Z. Gu; Y. Li. On Positive Injective Tensor Products Being Grothendieck Spaces . Indian J.Pure Appl. Math. 51 (2020), 1239–1246.(M. González) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Can-tabria, Avda. de los Castros s/n, 39071-Santander, Spain Email address : [email protected] (T. Kania) Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha1, Czech Republic and Institute of Mathematics and Computer Science, Jagiellonian Uni-versity, Łojasiewicza 6, 30-348 Kraków, Poland Email address ::