Banachspaceswithoutapproximationpropertiesoftypep.pdf

arxiv:1003.0085v2 math.fa 25 apr 2010 banach spaces without approximation properties of type p oleg reinovand qaisar latif abstract. the main purpose of this note is to show that the question posed in the paper of sinha d.p. and karn a.k. (“compact operators which factor through subspaces of lpmath. nachr. 281, 2008, 412-423; see the very end of that paper) has a negative answer, and that the answer could be obtained, essentially, in 1985 after the papers 2, 3 by reinov o.i. have been appeared in 1982 and in 1985 respectively. the main purpose of this note is to show that the question posed in 1 (see the very end of that paper) has a negative answer, and that the answer could be obtained, essentially, in 1985 after the papers 2, 3 have been appeared in 1982 and in 1985 respectively. for the sake of completeness (and since the article 3 is now quite diffi cult to get) we will reconstruct some of the facts from 3, maybe, in a somewhat changed form (a translation of 3 with some remarks can be found in arxiv:1002.3902v1 math.fa). all the notation and terminology, we use, are from 2, 4, or 5 and more or less standard. for the general theory of absolutely p-summing, p-nuclear and other operator ideals, we refer to 6. we will keep, in particular, the following standard notations. if a is a bounded subset of a banach space x, then (a) is the closed absolutely convex hull of a; xais the banach space with “the unit ball“ (a); a: xa x is the canonical embedding. for b x, it is denoted by b and b kk the closures of the set b in the topology and in the norm k k respectively. when it is necessary, we denote by kkxthe norm in x. other notations: p, qnp, np, ipare the ideals of absolutely p-summing, quasi-p-nuclear, p-nuclear, strictly p-integral operators, respectively; xbpy is the complete tensor product associated with np(x,y ); xb b py = x y p (i.e. the closure of the set of fi nite dimensional operators in p(x,y ). finally, if p 1,+ then pis the adjoint exponent. the research was supported by the higher education commission of pakistan. ams subject classifi cation 2000: 46b28. spaces of operators; tensor products; approximation properties . key words: p-nuclear operators, approximation properties, compact approximation, nuclear tensor norms, tensor products. 1 banach spaces without approximation properties of type p2 now, we recall the main defi nition from 3 of the topology pof the p-compact convergence, which is just the same as the p-topology in 11, where it was proved, essentially, that that t kd p(x,y ) if and only if t qnp(x,y ). for banach spaces x,y, the topology pof p-compact convergence in the space p (y,x) is the topology, a local base (in zero) of which is defi ned by sets of type k,= u p(y,x) : p(uk) 0, k = (k) a compact subset of y. proposition 1 3. let r be a linear subspace in p(y,x), containing y x. then (r,p)is isomorphic to a factor space of the space xbpy. more precisely, if (r,p), then there exists an element z = p 1 xn yn xbpy such that (u) = trace u z, u r.() on the other hand, for every z xbp y the relation () defi nes a linear continuous functional on (r,p). proof. let be a linear continuous functional on (r,p ). then one can fi nd a neighborhood of zero k= k, such that is bounded on it: u k, |(u)| 6 1. we may assume that = 1. consider the operator uk: yk k y u x. since the mapping kis compact, uk qnp(yk,x). put k(uk) = (u) for u r. on the linear subspace rk= ?v qn p(yk,x) : v = uk ? of the space qnp(yk,x), the linear functional kis bounded: if v = uk rkand p(v ) 6 1, then |k(v )| = |(u)| 6 1. therefore, kcan be extended to a linear continuous functional e on the whole qnp(yk,x); moreover, because of the injectivity of the ideal qnp, considering x as a subspace of some space c(k0), we may assume that e qnp(yk,c(k0). let us mention that e (juk) = k(uk) = (u)(1) (here j is an isometric embedding of x into c(k0). furthermore, since qnp(yk,c(k0)= ip(c(k0),(yk) ), we can fi nd an oper- ator : c(k0) (yk), for which e (a) = trace a, a (yk) c(k0). let an (yk) c(k0), p(an juk) 0. then e (juk) = lim trace an.(2) consider the operator k : c(k0) (yk) k y. since ip, and kis compact, we have k np(c(k0),y ) = c(k0) b py. let p 1 n yn c(k0)bpy be a representation of the operator k. put z = p j(n) yn. the element z generates an operator kj from x to y. we will show now that 1 defi nition 4.3 from 1: “given a compact subset k x we defi ne a seminorm | |k on p(x,y ) given by |t|k= infd p(tiz) : iz : z x as above. the family of seminorms |k: k x is compact determines a locally convex topology pon p(x,y ).“; for details, and for the defi nition of maps iz, see 1, lemma 4.2. banach spaces without approximation properties of type p3 trace u z = e (juk) (note that u z is an element of the space xbx, so the trace is well defi ned). we have: trace u z = trace ?x j(n) uyn ? = x hj(n),uyni = x hn,juyni = trace ju k = trace(juk) , (3) where (juk) : c(k0) (yk) k y u x j c(k0). since p(an juk) 0, then p(a n (juk) 0. moreover, if a := an= pn 1 wmfm (yk) c(k0), then trace a n = x m hwm,fmi = x m hwm,fmi = trace a. hence, trace(juk) = lim trace a n = lim trace an. now, it follows from (3) and (2) that e (juk) = trace u z. finally, we get from (1): (u) = trace u z. thus, the functional is defi ned by an element of xbpy. inversely, if z xbp y, put (u) = trace u z for u r (the trace is defi ned since u z xbx). we have to show that the linear functional is bounded on a neighborhood k,of zero in p. for this, we need the following fact, which proof is rather standard (see 3, lemma 1.2): if z xbqy, then z xbqyk, where k = (k) is a compact in y.now, let k be a compact subset of y, for which z xbpyk. if u k,1, then p(uk) 2 below) and secondly the negative answer to the question in the end of the paper 1 (whether for p 1,2) every banach space has the approximation property of type p of 12). or, generally, we have: theorem. let 1 p 6= 2 . then there is a (refl exive) banach space that fails to have the approximation property of type p of 1. references 1 sinha d.p., karn a.k.: compact operators which factor through subspaces of lp, math. nachr. 281(2008), 412-423. 2 o.i. reinov: approximation properties of order p and the existence of non-p-nuclear operators with p-nuclear second adjoints, math. nachr. 109(1982), 125-134. 3 o.i. reinov: approximation of operators in banach spaces, application of functional analysis in the approximation theory (kgu, kalinin) (1985), 128-142. 4 o.i. reinov: disappearance of tensor elements in the scale of p-nuclear operators, theory of operators and theory of functions (lgu) 1(1983), 145-165. 5 o. reinov: approximation properties of order p and the existence of non-p-nuclear operators with p-nuclear second adjoints, doklady an