Uniqueness of Unconditional Bases in Banach Spaces

We prove a general result on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis. 1. I n t r o d u c t i o n A Banach space with an uncondit ional basis is said to have a unique uncondit ional basis if any two normalized uncondit ional bases are equivalent after a permutation. It is well-known tha t e2 has a unique uncondit ional basis (cf. [17]) and a classic result of Lindenstrauss and Petczynski [18] asserts tha t the spaces el and co also have unique uncondit ional bases; later Lindenstrauss and Zippin [21] showed tha t this is the complete list of spaces with symmetr ic bases for which the unconditional basis is unique. Subsequently Edelstein and Wojtaszczyk [10] showed tha t direct sums of el, e2 and co also have unique uncondit ional bases. In 1985, Bourgain, Casazza, Lindenstrauss and Tzafriri [3] studied the classification problem for such spaces. Their main results showed that e I (e2) , C 0 ( e l ) , e I (C0) , C0(e2) and 2-convexified Tsirelson T (2) have unique uncondit ional bases but tha t e2 (el) * Both authors were supported by NSF Grant DMS-9201357. Received August 24, 1995 141 142 P.G. CASAZZA AND N. J. KALTON Isr. J. Math. and g2(Co) do not. Based on their results a complete classification looks hopeless. We also remark that a recent example of Gowers [12] may be easily shown to have unique unconditional basis. Thus there are many "pathological" spaces with unique unconditional basis. In this paper we will give (Theorem 3.5) a simple and, we feel, useful characterization of complemented unconditional basic sequences in Banach sequence spaces which are not sufficiently Euclidean (i.e. do not have uniformly complemented g~'s). This theorem is the discrete analogue of Theorem 8.1 of [14]; in fact the basic arguments are very similar to those given in [16] and [14], but we have opted to present a self-contained proof here. We then use this result and the recent work of Wojtaszczyk [26] to give some more examples of fairly natural spaces with unique unconditional basis. In Section 5, we introduce the class of leftand right-dominant bases and use this notion to show that the Nakano space g(Pn) has a unique unconditional basis if pn $1 and (Pn -P2n)log n is bounded (there is a dual result if p~ 1" ec). We also show that Tsirelson space T has a unique unconditional basis (a question raised in [3] p. 62). In Section 6, we use similar techniques to show that certain complemented subspaces of Orlicz sequence spaces have unique unconditional bases. Based on these examples we are able to resolve Problem 11.2 (p. 104) of [3] by showing that there is a space with unique unconditional basis with a complemented subspace (spanned by a subsequence of the basis) failing to have unique unconditional basis. Also in Section 4, we use Theorem 3.5 to give a contribution to the problem of uniqueness of unconditional bases in finite-dimensional spaces. Specifically, we prove that in any class of finite-dimensional lattices so that ~ is not complementably and disjointly representable, the unconditional basis is almost unique; for a more precise statement see Theorem 4.1. We remark that the techniques developed here using Theorem 3.5 can be used successfully to obtain other results on uniqueness. In particular we plan to study unconditional bases in c0-products in a later publication. Since the arguments in such spaces are considerably more complicated, it seemed, however, appropriate to restrict attention here to some simple applications. 2. D e f i n i t i o n s a n d n o t a t i o n We will take the viewpoint that an unconditional basis in a Banach space X confers the structure of an atomic Banach lattice on X. We will thus adopt Vol. 103, 1998 UNIQUE NE S S OF U N C O N D I T I O N A L BASES 143 the language and s t ructure of Banach lattices. It is well-known tha t a separable Banach latt ice can be regarded as a Khthe function space. We will in general use the same nota t ion as in [16]. Let f~ be a Polish space (i.e. a separable complete metric space) and let # be a a-finite Borel measure on ft. We denote by L0(#) the space of all Borel measurable functions on f~, where we identify functions differing only on a set of measure zero; the natura l topology of L0 is convergence in measure on sets of finite measure. An admissible norm is then a lower-semi-continuous map f --+ IIf]l from Lo(#) to [0, c~] such that : (a) lice/I[ -]c~]l]f[[ whenever ~ 9 R, f 9 Lo. (b) [If + glt <-Ilfll + []gl[, for f, g 9 Lo. ( c ) I[/ll Ibll, whenever ]f] <_ ]g[ a.e. (almost everywhere). (d) Ilfil < o ~ for a dense set of f 9 n0. (e) Ilfl[ = 0 if and only if f = 0 a.e. A Khthe function space on (•, p) is defined to be a dense order-ideal X in L0(#) with an associated admissible norm ]l ]Ix such tha t if Xmax = {f : [l/[]x < ~ } then either: (1) X = Xmax (X is m a x i m a l ) o r (2) X is the closure of the simple functions in Xm~z (X is m i n i m a l ) . Any order-continuous Khthe function space is minimal. Also any Khthe function space which does not contain a copy of Co is bo th maximal and minimal. If X is an order-continuous Khthe function space then X* can be identified with the Khthe function space of all f such tha t II/[Ix* = sup f l f g l d # < oc. IlgllxK_l d X* is always maximal. A Khthe function space X is said to be p-convex (where 1 < p < oo) if there is a constant C such tha t for any f l , . . . , fn 9 X we have n n I1(~-~ Ifi[P)l/pllx 0 and any f l , . . . , fn E X we have n n I1(}-~ Ifilq)l/qllx >c(~-~ Ilkllq) 1/q i 1 i = 1 X is said to have a lower q-es t imate if for some c > 0 and any dis joint f l , . 9 9 , fn E X, n n In ~ kllx _> ~(}-~. Ilkll~)l/". i=1 i=1 A Banach space X is said to be of (Rademacher ) t ype p (1 < p < 2) if the re is a cons tan t C so t ha t for any X l , . . . , xn E X, n n Ave II Y~x~l l < c(}-~ IlxillP) ~/p ei=d:l i=1 i=1 and X is of co type q (2 _< q < co) if for some c > 0 and any X l , . . . , xn E X we have n n Ave II ~E~x~ll > c(}--~ Ilzillq) l/q. ei=:t:l i=1 i=1 We recall t ha t a Banach la t t ice has nontr iv ia l co type (i.e. has co type q < ec for some q) if and only if it has nontr iv ia l concavi ty (i.e. is q-concave for some q < ee). If X is a Banach la t t ice which has nontr iv ia l concavi ty then there is a cons tan t C = C ( X ) so t ha t for any x l , . . . , x~ E X we have I ( A v e ~ n n c ~=• II ~k~kll2) '/~ -< I1(~ Ixkl2)l/211x 0: E F n ( A I [ x ( n ) [ ) <_ 1}. n=l In the case Fn = F for all n we have the Orlicz space f.F. If g(F,,) is separable or has finite cotype then the canonical basis vectors form an uncondit ional basis of g(F,); otherwise they form an uncondit ional basis of their closed linear span h(F,). We refer to [19] for the basic propert ies of modular sequence spaces. One special case is to take Fn(t) = t p~ where 1 _< Pn < oo. This is often called a Nakano space and we denote it g(p~), f(Pn) is separable if and only if sup Pn < CV. I t m a y also be shown tha t if pn > 1 for all n and sup Pn < oo then g(p~)* = f(q~) where p~l + q~l = 1. If suppn = c~ then we write h(ps) for the closed linear span of the basis vectors, and we have h(p~)* = g(qn). Let (un) and (v~) be two uncondit ional basic sequences. We say tha t (Un) and (v~) are p e r m u t a t i v e l y e q u i v a l e n t if there is a pe rmuta t ion rr of N so tha t (us) and (v~(~)) are equivalent. We say tha t (us) is e q u i v a l e n t t o i t s s q u a r e if (us) is pe rmuta t ive ly equivalent to the basis {(Ul, 0), (0, Ul) , (U2 ,0 ) , . . . } of [Un] @ [Un]. A Banach space X with an uncondit ional basis has a u n i q u e u n c o n d i t i o n a l b a s i s if any two normalized uncondit ional bases are pe rmuta t ive ly equivalent. We remark tha t there is an impor tan t Cantor Bernstein type principle which helps de te rmine whether two uncondit ional bases are permuta t ive ly equivalent: if (u,~) is pe rmuta t ive ly equivalent to some subset of (v~) and if (Vs) is pe rmuta t ively equivalent to some subset of (Us) then (Un) and (Vs) are pe rmuta t ive ly equivalent. We are grateful to P. Woj taszczyk for drawing our a t ten t ion to this principle, which appears explicitly in [27] and is used in [26]. We are indebted to C. Bessaga for the information tha t the Can to r -Berns te in principle was used implicit ly earlier by Mityagin in [22]. A Banach space X is called s u f f i c i e n t l y E u c l i d e a n if there is a constant M so tha t for any n there are opera tors S: X --+ ~ and T: ~ ' -+ X so tha t S T = Ie~ and IISIIIITII <_ M. We will say tha t X is a n t i E u c l i d e a n if it is not sufficiently Euclidean. A Banach latt ice X is called s u f f i c i e n t l y l a t t i c e E u c l i d e a n if there is a cons tant M so t ha t for any n there are opera tors S: X --+ ~ ' and T: ~ ' --+ X so tha t 146 P.G. CASAZZA AND N. J. KALTON Isr. J. Math. S T =Ieg and lISIlliTII < M, and such tha t S is a lattice homomorphism. This is equivalent to asking that ~2 is finitely representable as a complemented sublatt ice of X. We will say tha t X is latt ice anti -Eucl idean if it is not sufficiently lattice Euclidean. We use the same terminology for an unconditional basic sequence, which we regard as inducing a lattice s tructure on its closed linear span. Final ly if X is a family of Banach lattices we say that 2( is su f f i c i en t ly latt ice Eucl idean if there is a constant M so tha t for any n there exists X E A" and operators S: X --+ ~ and T: f~ --~ X so that S T = / e ~ and [[SII[[T][ _< M, and such tha t S is a lattice homomorphism. If 2r is not sufficiently lattice Euclidean we will say tha t it is l a t t i c e a n t i E u c l i d e a n . 3. C o m p l e m e n t e d uncondi t ional basic sequences The main results of this section are Theorems 3.4 and 3.5, which show tha t complemented lattice anti-Euclidean uncondit ional basic sequences in an ordercontinuous Banach lattice or Banach sequence space take a part icularly simple form. LEMMA 3.1: Let X be a Banach sequence space and suppose ( u l , . . . ,Un) are disjoint elements of X+, and (u l , . , n) are disjoint in X+. Suppose that M > 1 is a constant such that n n n 72 [[ ~ _ a j u j I l x ~ M(~_~[aj[2) 1/2 and [[ ~~aju;[ Ix , ~ l~/l(~~[aj[2) 1/2 j----1 j = l j = l j = l whenever al , . 9 , an E R. Suppose further that