The SHAI property for the operators on L^p
William B. Johnson, N. Christopher Phillips, Gideon Schechtman
aa r X i v : . [ m a t h . F A ] F e b The SHAI property for the operators on L p ∗ W. B. Johnson, † N. C. Phillips, ‡ and G. Schechtman § Abstract
A Banach space X has the SHAI (surjective homomorphisms are injective) propertyprovided that for every Banach space Y , every continuous surjective algebra homomor-phism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAIproperty. The condition is satisfied for L p (0 ,
1) for 1 < p < ∞ , spaces with symmetricbases that have finite cotype, and the Schatten p -spaces for 1 < p < ∞ . Following Horvath [9], we say that a Banach space X has the SHAI (surjective homomor-phisms are injective) property provided that for every Banach space Y , every surjectivecontinuous algebra homomorphism from the space L ( X ) of bounded linear operators on X onto L ( Y ) is injective, and hence by Eidelheit’s [6] classical theorem, X is isomorphic asa Banach space to Y . The continuity assumption is redundant by an automatic continuitytheorem of B. E. Johnson [5, Theorem 5.1.5]. The spaces ℓ p for 1 ≤ p ≤ ∞ are known tohave the SHAI property [9, Proposition 1.2], as do some other classical spaces [9], [10], butthere are many spaces that do not have the SHAI property [9]. Our research on the SHAI ∗ AMS subject classification: 47L20, 46E30. Key words: Ideals of operators, L p spaces, SHAI property † Supported in part by NSF DMS-1900612 ‡ Supported in part by NSF DMS-1501144 and by Simons Foundation Collaboration Grant for Mathe-maticians § Supported in part by the Israel Science Foundation L p = L p (0 ,
1) hasthe SHAI property. A consequence of our main results, Corollary 1.6, is that for 1 < p < ∞ ,the space L p has the SHAI property. We do not know whether L has the SHAI property.The space L ∞ does have the SHAI property because L ∞ is isomorphic as a Banach space to ℓ ∞ [2, Theorem 4.3.10].Before stating our theorems, we need to review the notion of an unconditional Schauderdecomposition of a Banach space X . A family ( E α ) α ∈ A of closed subspaces of X is calledan unconditional Schauder decomposition for X provided every vector x in X has a uniquerepresentation x = P α ∈ A x α , where the convergence is unconditional and, for each α ∈ A ,the vector x α is in E α . Notice that by uniqueness of the representation, E α ∩ E β = { } when α = β , and there are idempotents P α on X such that P α X = E α and P α P β = 0 for α = β . It is known that the P α are in L ( X ). Moreover, for any subset B of A , the net { P α ∈ F P α : F ⊂ B finite } is bounded in L ( X ) and converges strongly to an idempotent P B that has range span α ∈ B E α . The suppression constant of the decomposition is then defined tobe sup { (cid:13)(cid:13)P α ∈ F P α (cid:13)(cid:13) : F ⊂ A finite } . Note that k P B k is bounded by this suppression constantfor all subsets B of A . In practice, this theorem is rarely used, since typically one constructsthe idempotents P α and checks the uniform boundedness of the aforementioned nets andverifies the statement about the ranges of the strong limits of the nets. Finally, observe thata collection ( e α ) α ∈ A forms an unconditional Schauder basis for X if and only if ( E α ) α ∈ A isan unconditional Schauder decomposition of X , where E α = K e α ( K is the scalar field). Inthe sequel, we will most often use an unconditional Schauder decomposition E α where each E α is finite dimensional. Such a decomposition is called an unconditional FDD. FDD standsfor finite dimensional decomposition. Schauder decomposititions and FDDs are discussed inthe monograph [13, Section 1.g]. Schauder bases, type/cotype theory, and other conceptsfrom Banach space theory that are used in this paper are treated in the textbook [2].A concept that is particularly relevant for us is that of bounded completeness. An uncon-ditional Schauder decomposition ( E α ) α ∈ A for X is said to be boundedly complete providedthat whenever x α ∈ E α and { (cid:13)(cid:13)P α ∈ F x α (cid:13)(cid:13) X : F ⊂ A finite } is bounded, then the formal sum P α ∈ A x α converges in X , which is the same as saying that the net { P α ∈ F x α : F ⊂ A finite } converges. A convenient condition that obviously guarantees bounded completeness is thatthe decomposition has a disjoint lower p estimate for some p < ∞ . The decomposition( E α ) α ∈ A is said to have a disjoint lower; respectively, upper; p estimate provided that thereis C < ∞ so that whenever x , . . . , x n are finitely many vectors in X such that for every α ∈ A there is at most one i with 1 ≤ i ≤ n for which P α x i = 0, we have for x = P ni =1 x i the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ C n X i =1 k x i k p ! /p ; respectively, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C n X i =1 k x i k p ! /p . It is easy to see that the decomposition ( E α ) α ∈ A has a disjoint lower p estimate with constant2 if and only if whenever F , . . . , F n are disjoint finite subsets of A and x is in X , then k x k ≥ C n X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X α ∈ F j P α x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p /p , where, as usual, P α is the idempotent associated with the decomposition. Important forus is the following observation, which is very easy to prove. Suppose that ( E α ) α ∈ A is anunconditional Schauder decomposition for a subspace X of a Banach space Y . Assume thatthe idempotents ˜ P α associated with the decomposition extend to commuting idempotents P α from Y onto E α and that the net { P α ∈ F P α : F ⊂ A finite } is bounded in L ( Y ). If ( E α ) α ∈ A is a boundedly complete unconditional Schauder decomposition of X , then for each subset B of A , the net { P α ∈ F P α : F ⊂ B finite } converges strongly in L ( Y ) to an idempotent P B whose range is the closed linear span of the spaces E α for α ∈ B (which, by abuse ofnotation, we abbreviate to span { E α : α ∈ B } ) and P B extends the basis projection from X onto span { E α : α ∈ B } . In particular, X is complemented in Y . Conversely, if X isknown to be complemented in Y , then such extensions P B of the basis projections ˜ P B from X onto span { E α : α ∈ B } obviously exist even when the decomposition is not boundedlycomplete. In general, to guarantee that X is complemented in Y , something is needed otherthan having commuting extensions P α with { P α ∈ F P α : F ⊂ A finite } uniformly bounded:consider X = c , Y = ℓ ∞ , and the unit vector basis of c .From the definitions of type and cotype, it is clear that if X has type p and cotype q ,then every unconditional Schauder decomposition for X has a disjoint upper p estimate anda disjoint lower q estimate, where the constants depend only on the suppression constant ofthe decomposition and the type p and cotype q constants of X . In particular, if 1 < p ≤ L p has adisjoint upper p estimate and a disjoint lower 2 estimate, while if 2 ≤ p < ∞ , then everyuncondtional Schauder decomposition for a subspace of a quotient of L p has a disjoint upper2 estimate and a disjoint lower p estimate [2, Theorem 6.2.14].The observation in the following lemma will be used for transferring information from Y to X when there is a surjective homomorphism from L ( Y ) onto L ( X ). Lemma 1.1
Suppose that ( E α ) α ∈ A is an unconditional decomposition for X that has adisjoint lower p estimate with ≤ p < ∞ , and let Y ⊇ X . Then there is a constant C < ∞ such that if A , . . . , A n are disjoint subsets of A and P A j is the basis projection onto E A j = span { E α : α ∈ A j } and T , . . . , T n are operators in L ( Y ) , then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 T i P i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C n X i =1 k T i k q ! /q , where /p + 1 /q = 1 . roof: Suppose x ∈ X . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 T i P i x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n X i =1 k T i kk P i x k ≤ n X i =1 k T i k q ! /q n X i =1 k P i x k p ! /p ≤ C n X i =1 k T i k q ! /q k x k , where the constant C is the disjoint lower p constant of ( E α ) α ∈ A .A family of sets is said to be almost disjoint provided the intersection of any two of them isfinite. Definition 1.2
Suppose that ( E n ) ∞ n =1 is an unconditional FDD for a Banach space X . Wesay that ( E n ) has property ( { N α : α < c } of infinite sets of natural numbers such that for each α < c , X is isomorphic to the closedlinear span of the subspaces E n for n ∈ N α . Subsymmetric bases are obvious examples of FDDs that have property ( L p has property ( < p < ∞ . Proposition 1.3
Let ( E n ) ∞ n =1 be an FDD for a Banach space X . Assume that ( E n ) hasproperty ( { N α : α < c } of infinite subsets ofthe natural numbers. For F ⊂ N , let P F be the basis projection from X onto the closedlinear span E F of the subspaces E n for n ∈ F . Suppose that Φ is a non zero, non injectivecontinuous homomorphism from L ( X ) onto a Banach algebra A . Then for each α < c , Φ( P N α ) is a non zero idempotent in A . Moreover, there is a constant C < ∞ such that if F is any finite subset of [ α < c ] , then (cid:13)(cid:13)P α ∈ F Φ( P N α ) (cid:13)(cid:13) A ≤ C . If A is a subalgebra of L ( Y ) forsome Banach space Y , then (Φ( P N α )) α Since, for each α , the range of P N α is isomorphic to X , and Φ is not zero, Φ( P N α ) isa non zero idempotent in A . Suppose that F is a finite subset of { α : α < c } . Take a finiteset S of natural numbers so that N α ∩ N β ⊂ S for all distinct α, β in F . For α ∈ F , let Q α = P N α \ S be the basis projection from X onto span { E n : n ∈ N α \ S } . The kernel of Φ is4 non trivial ideal in L ( X ) and hence contains the finite rank operators. Since P N α − Q α isa finite rank operator, Φ( P N α ) = Φ( Q α ) for each α ∈ F . But the projections Q α , for α ∈ F ,are projections onto the closed spans of disjoint subsets of the FDD ( E n ) ∞ n =1 , so (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ F Φ( Q α ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X α ∈ F Q α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k Φ k ≤ C k Φ k , where C is the suppression constant of ( E n ). The last statement is now obvious.With the preliminaries out of the way, we state the main theorem in this article. Theorem 1.4 Let ( E n ) ∞ n =1 be an unconditional FDD for a Banach space X . Assume that ( E n ) ∞ n =1 has property ( ( E n ) ∞ n =1 has a disjoint lower p estimate forsome p < ∞ . Then X has the SHAI property. Proof: Suppose, for contradiction, that Φ is a non injective continuous homomorphismfrom L ( X ) onto L ( Y ) for some non zero Banach space Y . We continue with the set upin Proposition 1.3, where property ( E n ) is witnessed by an almost disjoint family { N α : α < c } of infinite subsets of the natural numbers, and for F ⊂ N , the basis projectionfrom X onto the closed linear span E F of { E n : n ∈ F } is denoted by P F .We claim that to get a contradiction it is enough to prove that the subspace Y iscomplemented in Y . Indeed, if Y is complemented in Y , then L ( Y ) is isomorphic as aBanach algebra to a subalgebra of L ( Y ). However, defining Y α = Φ( P N α ) Y for α < c , weknow that ( Y α ) α 1, and hence the density character of L ( Y ), whence also of L ( Y ), is at least2 c . However, since X is separable, the density character of L ( X ) is at most c (actually, equalto c since X has an uncondtional FDD), so L ( Y ) cannot be a continuous image of L ( X ).This completes the proof of the claim.To show that Y must be complemented in Y , we use the fact proved in Proposition1.3 that there is a constant C such that for every finite subset F of { α : α < c } we have (cid:13)(cid:13)P α ∈ F Φ( P N α ) (cid:13)(cid:13) L ( Y ) ≤ C . It was remarked in the introduction to this section that thiscondition guarantees that Y is complemented in Y when ( Y α ) α For < p < ∞ , the space L p has the SHAI property. Proof: In view of Theorem 1.4, it is enough to prove that the Haar basis for L p hasproperty ( { N α : α < c } be a continuum of almost disjoint infinite subsets of thenatural numbers N . Define for α < cX α = span (cid:8) h n,i : n ∈ N α and 1 ≤ i ≤ n (cid:9) , where { h n,i : n = 0 , , . . . and 1 ≤ i ≤ n } is the usual (unconditional) Haar basis for L p (0 , {| h n,i | : 1 ≤ i ≤ n } is the set of indicator functions of thedyadic subintervals of (0 , 1) that have length 2 − n . By the Gamlen–Gaudet theorem [7], X α is isomorphic to L p with the isomorphism constant depending only on p . Remark 1.7 Although our proof that L p has the SHAI property is simple enough, it isstrange. The “natural” way of proving that a space X has the SHAI property is to verifythat for any non trivial closed ideal I in L ( X ), the quotient algebra L ( X ) / I contains nominimal idempotents. (An idempotent P is called minimal provided P = 0 and the onlyidempotents Q for which P Q = QP = Q are P and 0. Rank one idempotents in L ( X )are minimal.) This suggests the following problem, which is related to the known problemwhether every infinite dimensional complemented subspace of L p is isomorphic to its square. Problem 1.8 Is there a non trivial closed ideal I in L ( L p ) for which L ( L p ) / I has a minimalidempotent? If there is a positive answer to Problem 1.8, the witnessing ideal I cannot be containedin the ideal of strictly singular operators. This is because every infinite dimensional comple-mented subspace of L p contains a complemented subspace that is isomorphic either to ℓ p orto ℓ [11], and the fact that idempotents in L ( X ) / I lift to idempotents in L ( X ) when I isan ideal that is contained in L ( X ) [4]. Problem 1.9 Does L have the SHAI property? Here we present some more examples of spaces with property ( L p has the SHAI property,7ut we show that at least some of the known examples of such spaces do. Along the way westate and prove some permanence properties of ( L p have the SHAI property when 1 < p < ∞ .This was known for ℓ and ℓ p and proved above for L p . The case of ℓ p ⊕ ℓ follows easilyfrom Theorem 1.4. That the remaining classical complemented subspace of L p , ℓ p ( ℓ ), the ℓ p sum of ℓ , has ( Definition 2.1 Suppose that ( E n ) ∞ n =1 is an unconditional FDD for a Banach space X and K is a positive constant. We say that ( E n ) has property ( K provided thereis an almost disjoint continuum { N α : α < c } of infinite sets of natural numbers such thatfor each α < c , X is K -isomorphic to the closed linear span of { E n : n ∈ N α } . Note that if ( E n ) ∞ n =1 has property ( K . Nevertheless, we need this quantitative notion for the full generality of Proposition 2.2.Recall that if ( e i ) is an unconditional basis for some Banach space Y and X i , for i =1 , , . . . , is a Banach space, ( L ∞ i =1 X i ) Y is the space of sequences ¯ x = ( x , x , . . . ) whosenorm, k ¯ x k = (cid:13)(cid:13) P ∞ i =1 k x i k · e i (cid:13)(cid:13) Y , is finite. We denote the subspace of ( L ∞ i =1 X i ) Y of allsequences of the form (0 , . . . , , x i , , . . . ) by X i ⊗ e i . Proposition 2.2 For i = 1 , , . . . let ( E in ) ∞ n =1 be an unconditional FDD for a Banach space X i , all satisfying property ( K . Then for each subsymmetric basis ( e i ) of some Banach space Y , the unconditional FDD ( E in ⊗ e i ) ∞ i,n =1 of ( L ∞ i =1 X i ) Y satisfies ( ( E in ) ∞ n =1 have disjoint lower p estimates with uniformconstant and ( e i ) also has such an estimate, then ( L ∞ i =1 X i ) Y has the SHAI property. Proof: For each i , let { N iα : α < c } be an almost disjoint continuum of infinite sets ofnatural numbers such that for every α < c , X α is K -isomorphic to the closed linear spanof the subspaces E in for n ∈ N α . Also, let { N α : α < c } be an almost disjoint continuum ofinfinite sets of natural numbers. Then (cid:8) ( i, n ) : i ∈ N α and n ∈ N iα (cid:9) is a continuum of almost disjoint subsets of N × N . It is easy to see that this continuumsatisfies what is required of the unconditional FDD ( E in ⊗ e i ) ∞ i,n =1 to satisfy ( E in ) ∞ n =1 have disjoint lower p estimates with uniform constant and ( e i ) alsohas such an estimate, then the FDD ( E in ⊗ e i ) ∞ i,n =1 clearly has a disjoint lower p estimate aswell, so the SHAI property follows from Theorem 1.4.8 emark 2.3 Note that the proof above works with only notational differences if we dealwith only finitely many X i (and here we do not need to assume the uniformity of the ( X and Y has an unconditional FDD with ( X ⊕ Y .As we said above, this takes care of the space ℓ p ( ℓ ). The first non classical complementedsubspace of L p is the space X p of Rosenthal [15]. We recall its definition. Let p > w = ( w i ) ∞ i =1 be a bounded sequence of positive real numbers. Let ( e i ) ∞ i =1 and ( f i ) ∞ i =1 be theunit vector bases of ℓ p and ℓ . Let X p, ¯ w be the closed span of ( e i ⊕ w i f i ) ∞ i =1 in ℓ p ⊕ ℓ . Ifthe w i are bounded away from zero, then X p, ¯ w is isomorphic to ℓ . If P ∞ i =1 w pp − i < ∞ , then X p, ¯ w is isomorphic to ℓ p . If one can split the sequence ¯ w into two subsequences, one boundedaway from zero and the other such that the sum of the pp − powers of its elements converges,then X p, ¯ w is isomorphic to ℓ p ⊕ ℓ . Rosenthal proved that in all other situations one getsa new space, isomorphically unique (i.e., any, two spaces corresponding to two choices of¯ w with this condition are isomorphic). Moreover, X p,w is isomorphic to a complementedsubspace of L p . The constants involved (isomorphisms and complementations) are boundedby a constant depending only on p . This common (class of) space(s) is denoted by X p . For1 < p < X p is defined to be X ∗ p/ ( p − . Proposition 2.4 Let p ∈ (1 , ∞ ) \ { } . Then X p has ( Proof: Let p > 2. Write N as a disjoint union of finite subsets σ j for j = 1 , , . . . , with | σ j | → ∞ . For i ∈ σ j put w i = | σ j | − p p , so w i → j , P i ∈ σ j w pp − i = 1. Set E j = span ( e i ⊕ w i f i ) i ∈ σ j . It follows that for any infinite subsequence of the unconditionalFDD ( E j ), the closed span of this subsequence is isomorphic to X p . The FDD is unconditionaland, as it lives in L p , has a lower p estimate. So the result in this case follows from Theorem1.4. The case 1 < p < X p and the classical complemented subspaces of L p , Rosenthal [15] lists afew more isomorphically distinct spaces that are isomorphic to complemented subspaces of L p when p ∈ (1 , ∞ ) \ { } . Using the discussion above one can easily show that they allhave ( B p . It is the ℓ p sumof spaces X i each having a 1-symmetric basis, and thus having ( X i is isomorphic to ℓ , but the isomorphism constant tends to infinity as i → ∞ . ByProposition 2.2, B p has ( L p for p ∈ (1 , ∞ ) \ { } was constructed in [16]. We recall the simple construction. Given two9ubspaces X and Y of L p (Ω) with 1 ≤ p ≤ ∞ , X ⊗ p Y denotes the subspace of L p (Ω ) thatis the closed span of all functions of the form h ( s, t ) = f ( s ) g ( t ) with f ∈ X and g ∈ Y . Itis easy to see (and was done in [16]) that the isomorphism class of X ⊗ p Y depends only onthe isomorphism classes of X and Y and that, if X and Y are complemented in L p (Ω), then X ⊗ p Y is complemented in L p (Ω ). More generally, if X , X , Y , Y are subspaces of L p (Ω)and T i ∈ L ( X i , Y i ), then T ⊗ p T ∈ L ( X ⊗ p X , Y ⊗ p Y ). Note also that if ( E in ) ∞ n =1 is anunconditional FDD for X i for i = 1 , 2, then ( E n ⊗ p E m ) ∞ n,m =1 is an unconditional FDD for X ⊗ p X . This follows from iterating Khinchine’s inequality.With a little abuse of notation we denote by X p some isomorph of X p that is comple-mented in L p [0 , Y = X p , and for n = 2 , , . . . , let Y n = Y n − ⊗ p X p . From the aboveit is clear that the spaces Y n are complemented (alas, with norm of projection dependingon n ) in some L p space isometric to L p [0 , Y n are isomorphically different. That all the spaces Y n have ( ⊗ p satisfies Conditions (1) and (2) inProposition 2.5 for the class of all m tuples of subspaces of L p ( µ ) spaces. Proposition 2.5 Assume that X , . . . , X m are Banach spaces, each of which has an uncon-ditional FDD satisfying ( Y ⊗ · · · ⊗ Y m denote an m fold tensor product endowed withnorm defined on some class of m tuples of Banach spaces with the following two properties:1. If T i ∈ L ( Y i , Z i ) for i = 1 , . . . , m , then T ⊗ · · · ⊗ T m : Y ⊗ · · · ⊗ Y m → Z ⊗ · · · ⊗ Z m is bounded.2. If Y i has an unconditional FDD ( F in ) ∞ n =1 for each i , then ( F n ⊗ · · · ⊗ F mn m ) ∞ n ,...,n m =1 isan unconditional FDD for the completion of Y ⊗ · · · ⊗ m Y m .Then, if we assume in addition that ( X , . . . , X m ) is in this class, the completion of X ⊗· · · ⊗ X m has an unconditional FDD with ( Proof: For each i = 1 , . . . , m , let ( E in ) ∞ n =1 be an unconditional FDD for a Banach space X i such that there is an almost disjoint continuum { N iα : α < c } of infinite sets of N such thatfor each α < c , X i is isomorphic to the closed linear span of the spaces E in for n ∈ N iα .Consider the continuum { N α × · · · × N mα : α < c } 10f subsets of N m . This is an almost disjoint family whose cardinality is the continuum.Property (2) of the tensor norms we consider guarantees that ( E n ⊗ · · · ⊗ E mn m ) ∞ n ,...,n m =1 isan unconditional FDD for the completion of X ⊗ · · · ⊗ X m . Property (1) implies that foreach α < c , the closed linear span of( E n ⊗ · · · ⊗ E mn m ) ( n ,...,n m ) ∈ N α ×···× N mα is isomorphic to the completion of X ⊗ · · · ⊗ X m . Remark 2.6 Note that in general Property (1) does not imply Property (2). The Schattenclasses C p for p = 2 are examples of tensor norms that satisfy (1) but not (2).We note that it is clear from Proposition 2.5 that if X , . . . , X m are subspaces of L p for1 ≤ p < X ⊗ p · · · ⊗ p X m has ( L p for 1 ≤ p < p > 2, up to isomorphism it includes only ℓ p and ℓ .) Thus the class of tensor products above includes, for example, ℓ p ( ℓ p ( . . . ( ℓ p m ) . . . ))whenever p ≤ p < p < · · · < p m ≤ Problem 2.7 Suppose p ∈ (1 , ∞ ) \ { } and let X be a complemented subspace of L p . Does X have the SHAI property? What if, in addition, X has an unconditional basis? What if,in addition, X is one of the ℵ spaces constructed in [1]? We complete this section with a discussion of another class of classical Banach spaces thathave property ( C p ofcompact operators T on ℓ for which the eigenvalues of ( T ∗ T ) / are p -summable. We treatthe case 1 < p < ∞ but remark afterwards how one can prove that C (trace class operatorson ℓ ) has the SHAI property. Neither C nor its predual C ∞ (compact operators on ℓ ) hasan unconditional FDD [12] and hence these spaces do not have property ( p = 2 because C , being isometrically isomorphic to ℓ , has already beendiscussed.First, consider the subspace T p of C p consisting of the lower triangular matrices in C p .Here we include p = 1 and p = ∞ but exclude p = 2. Neither T p nor C p has an unconditionalbasis [12], but T p has an obvious unconditional FDD ( E n ); namely, E n = span ≤ j ≤ n e n ⊗ e j ;that is, a matrix is in E n if and only if the only non zero terms are in the first n entriesof the n -th row. Since multiplying all entries in a row by the same scalar of magnitudeone is an isometry on C p , ( E n ) is even 1-unconditional. If M is an infinite subset of N ,let T p ( M ) be the closed span in T p of ( E n ) n ∈ M . Since ( E n ) is 1-uncondtional, T p ( M ) is11orm one complemented in T p and, similarly, T p is isometric to a norm one complementedsubspace of T p ( M ). The space T p is isomorphic to ℓ p ( T p ) [3, p. 85], so the decompositionmethod [2, Theorem 2.2.3] shows that T p is isomorphic to T p ( M ). Thus every almost disjointfamily of infinite subsets of N witnesses that T p has property ( < p < ∞ , T p is complemented in C p via the projection that zeroes out the entries that lie above thediagonal [14], [8], from which it follows easily [3] that T p is isomorphic to C p . We recordthese observations in Proposition 2.8. Proposition 2.8 For ≤ p ≤ ∞ , the space T p has property ( . Moreover, for < p < ∞ ,the space C p has property ( . As we mentioned above, it can be proved that C and C ∞ have the SHAI property eventhough neither has an unconditional FDD. However, the C p norms for 1 ≤ p ≤ ∞ are whatKwapie´n and Pe lczy´nski [12] call unconditional matrix norms; i.e., the norm (cid:13)(cid:13)(cid:13)P i,j a i,j e i,j (cid:13)(cid:13)(cid:13) ofa linear combination P i,j a i,j e i,j of the natural basis elements ( e i,j ) ∞ i,j =1 , is equivalent (in ourcase even equal) to the norm of P i,j ε i δ j a i,j e i,j for all sequences of signs ( ε i ) ∞ i =1 and ( δ j ) ∞ j =1 .One can define a variation of property ( C p , for 1 ≤ p ≤ ∞ , satisfy this property, and prove a versionof Theorem 1.4. This shows that C has the SHAI property (and gives an alternative proofalso for C p for 1 < p < ∞ ). This variation of Theorem 1.4 does not apply to C ∞ , which doesnot have finite cotype, and we do not know whether C ∞ has the SHAI property. 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