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Schauder Equivalence Relations BLAST2015, University of North Texas Schauder equivalence relations BLAST2015, University of North Texas Xin Ma Department of Mathematics, Texas A&M University Xin Ma (Texas A&M University) June. 12, 2015 1 / 21 Plan of my talk 1 Introduction 2 Reducibility and non-reducibility 3 Structure theorems Xin Ma (Texas A&M University) June. 12, 2015 2 / 21 Now, Let X ,Y be Polish spaces and E,F be equivalence relations on X and Y , respectively. Definition a Borel reduction from E to F , denoted by E ≤B F , is a Borel function θ : X ! Y such that for all x; y 2 X , xEy () θ(x)F θ(y). We write E ∼B F if both E ≤B F and F ≤B E hold. We also write E <B F if E ≤B F but not F ≤B E . Xin Ma (Texas A&M University) June. 12, 2015 3 / 21 Banach space and Schauder equivalence relations Definition Given a Banach space X , a Schauder basis fxng of X means that any 1 P x 2 X can be expanded by the form x = anxn for a unique fang in RN. n=1 A sequence fxng is a basis of its closed linear span is called a basic sequence. Any separable Banach space is a Polish space, we consider Schauder equivalence relations defined as follows. This definition is due to Ding. Definition (Ding, 2013) For fxng is a basic sequence of Banach space X , we denote coef (X ; (xn)) P to be the set of all a = (an) 2 RN such that anxn converges. Define the equivalence relation E(X ; (xn)) by, for any x; y 2 RN, (x; y) 2 E(X ; (xn)) () x − y 2 coef (X ; (xn)): Xin Ma (Texas A&M University) June. 12, 2015 4 / 21 Previous work So far, A lot of important results and essential tools have been investigated. z Dougherty and Hjorth (1999,2000) proved that for any p; q 2 [1; 1], RN=lp ≤B RN=lq () p ≤ q and RN=lp and RN=c0 are Borel incomparable. z M´atrai(2008) introduced a kind of lp−like equivalence relations. He + defined a relation on [0; 1]N that for any f : [0; 1] ! R , P xEf y () f (jx(n) − y(n)j) < 1: He embedded any a liner chain i<! of the order P(!)=Fin into the set of the equivalence relations between RN=lp and RN=lq, where 1 ≤ p < q < 1. Yin pointed out that M´atrai'sequivalence relations can be considered as a kind of Schauder equivalence relations generated by Orlicz sequence spaces. Xin Ma (Texas A&M University) June. 12, 2015 5 / 21 z Ding (2011 and 2012) introduced another kind of lp−like equivalence relations E((Xn); p) for a sequence of pseudo-metric spaces (Xn; dn); n < ! with p ≥ 1. He set that for any x; y 2 Πn<!Xn, P (x; y) 2 E((Xn); p) () dn(x(n); y(n)) < 1; He found that the n<! reducibility between such equivalence relations is closely related to H¨older(α) embedding. His theorem provides a lot of results of equivalence relations related to classical Banach spaces of the form E((Lr ); p) and E((c0); q) z Yin (2013) moved further than M´atraiso that he embedded whole P(!)=Fin into the set of the equivalence relations between RN=lp and RN=lq to show the reducibility order of Borel equivalence relations between RN=lp and RN=lq are rather complex. Xin Ma (Texas A&M University) June. 12, 2015 6 / 21 Some notations concerning basic sequences • Let fxng be a basic sequence in a Banach space X . A sequence of p Pj+1 non-zero vectors fuj g in X of the form uj = anxn, with fang pj +1 scalars and p1 < p2 < ··· an increasing sequence of integers, is called a block basic sequence or briefly a block basis of fxng. 1 P • we say fxng unconditional if x = anxn converges unconditionally. n=0 1 P Here, a sum xn converges unconditionally means that the sum n=0 1 P xπ(n) converges for every permutation π of the integers. n=0 Xin Ma (Texas A&M University) June. 12, 2015 7 / 21 • For two basic sequence fxng and fyng in X and Y , respectively, if coef (X ; (xn)) ⊂ coef (Y ; (yn)), we say that fxng dominates fyng, denoted by fxng fyng. We say fxng lower semi-homogeneous if any normalized block bases of fxng dominates fxng. • An unconditional basic sequence fxng is called subsymmetric if for any subsequence fxkn g of fxng, coef(X ; (xn)) = coef(X ; (xkn )). Xin Ma (Texas A&M University) June. 12, 2015 8 / 21 Reducibility and non-reducibility Now, two natural questions are raised: • What does the hierarchy of the reducibility about Schauder equivalence relations look like? • What is the relations between Schauder equivalence relations and their underlying basic sequences? The two following theorems have solved some cases of well-known Banach spaces especially for the ones with a subsymmetric schauder basis. Xin Ma (Texas A&M University) June. 12, 2015 9 / 21 Theorem (M.,2014) Let Y be a Banach space, with fyng being a Schauder basis. If X is a subspace of Y , then there is a subspace Z of X , with a basis fzng, such that E(Z; (zn)) ≤B E(Y ; (yn)). Furthermore, If X is a subspace of Y , with a normalized subsymmetric basic sequence fxng, then, E(X ; (xn)) ≤B E(Y ; (yn)). Theorem (M.,2014) Both X and Y are Banach spaces. fxng and fyng are normalized basic sequences of X and Y , respectively. If two following conditions hold: (1) for all subsequences fxbn g of fxng, fxbn g does not dominate the fyng, and (2) fyng is lower semi-homogeneous, then E(X ; (xn)) B E(Y ; (yn)). Xin Ma (Texas A&M University) June. 12, 2015 10 / 21 Using the theorem above, we obtain following results. Corollary (Dougherty and Hjorth (1999,2000)) for any p; q 2 [1; 1], RN=lp ≤B RN=lq () p ≤ q and RN=lp and RN=c0 are Borel incomparable. Corollary Let X be a Banach space, which admits a normalized subsymmetric basis fxng. Then E(X ; (xn)) is a minimum element, in the order of the ≤B , of the set fE(X ; (yn)): fyngis a Schauder basis of X g. Xin Ma (Texas A&M University) June. 12, 2015 11 / 21 The following cases concerning Orlicz sequence spaces hM and Lorentz sequence spaces d(w; p): Corollary M and N are Orlicz functions. If two following conditions hold: (1) hM * hN , and (2) N satisfies ∆0 conditions, then RN=hM B RN=hN . Corollary For any p ≥ 1, RN=lp <B RN=d(w; p) Xin Ma (Texas A&M University) June. 12, 2015 12 / 21 Structure Theorems Due to the work of Rosendal, we know that RN=l1 is the largest Kσ equivalence relation. In the case of Schauder equivalence relations, Ding has given a characterization of Schauder equivalence relations which are below RN=l1. Theorem (Ding, 2015) For any Schauder equivalence relation E(X ; (xn)), The followings are equivalent: • E(X ; (xn)) is generated by boundedly complete basis (xn) • coef (X ; (xn)) is Fσ • E(X ; (xn)) ≤B RN=l1 Xin Ma (Texas A&M University) June. 12, 2015 13 / 21 Boundedness Fact For any Schauder equivalence relation E • (Pelczynski, essentially) There is an universal space U2 with a Schauder basis (vn) such that E ≤B E(U2; (vn)). • (Hjorth) If E ≤B RN=l1 , then E sB RN=l1 . • (M,) If E ≤B RN=c0, then E sB RN=c0. 1 • (Ding) On c0, there is a universal Schauder basis cs such that all Schauder equivalence relations E on c0 witnesses that 1 E ≤B E(c0; cs ). Xin Ma (Texas A&M University) June. 12, 2015 14 / 21 Basis of Schauder equivalence relations We say subclass B of a class C of equivalence relations is a basis of C if for any E 2 C, there is a F 2 B such that F ≤B E. Thus another natural question raised here: What can we say about a basis of the Schauder equivalence relations? In particular, Whether RN=l1 and RN=c0 form a basis of all Schauder equivalence relations? Xin Ma (Texas A&M University) June. 12, 2015 15 / 21 The way to address on this problem is to use a famous space in Banach space theory: Tsirelson space. The followings are some key properties of the space. Fact (Cassaza, Johnson, and Tzafriri) For Tsirelson's space T , we mean the space which is the dual of the original space constructed by Tsirelson. It is an example of spaces which contains no isomorphic copies of c0 and lp,p ≥ 1. Theorem (Cassaza, Johnson, and Tzafriri) p Pj+1 Let fyj g in T of the form yj = antn, with fang scalars, is a normalized pj +1 block basic sequence of ftng, Then for every choice of natural numbers pj < kj ≤ pj+1, and every sequence of scalars fbng, we have: 1 X X X k b t k ≤ k b y k ≤ 18k b t k 3 j kj j j j kj j j j . Xin Ma (Texas A&M University) June. 12, 2015 16 / 21 Theorem (Casazza, Johnson and Tzafriri) The natural basis ftng in Tsirelson space T is lower semi-homogeneous. Then we reduce the situation about equivalence relations generated by unconditional basic sequences to that generated by the action of ideals, which are regarded as Polish groups in 2!.
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