2017/05/31 by Florent Baudier, F. Baudier, Gilles Lancien +3 · 24 citations
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Banach space #Conjecture #Hilbert space #Lipschitz continuity #Point processes and geometric inequalities #Rigidity (electromagnetism) #Separable space #math.FA #math.GR #math.KT #math.MG #msc:05C63 #msc:20F65 #msc:46B20 #msc:46B85 #msc:46T99
paper · pdf · doi:10.1090/jams/899
published in Journal of the American Mathematical Society 31(3), 699-717 (American Mathematical Society) · changes from v1: new title, expanded abstract, introduction partially rewritten, AMS Early View is available for AMS members only in the Journal of the AMS
openalex publication_date 2018/02/08 · arxiv created 2018/02/09 · arxiv updated 2018/02/13 · openalex created_date 2018/03/06 · openalex updated_date 2026/08/06
The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson’s original space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">T^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Every Banach space that is coarsely embeddable into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">T^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must be reflexive, and all of its spreading models must be isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">c0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">T^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coarsely contains neither <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">c0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript p"> <mml:semantics> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">ℓ p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-bracket 1 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p∈ [1,∞ )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>T</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">T^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c 0"> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">c0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Also, a purely metric characterization of finite dimensionality is obtained.