Bounded sequences in noncommutative bimodules of 𝜏-measurable operators

We establish a noncommutative version of a result due to Lindenstrauss and Tzafriri [ <italic>Classical Banach spaces. I</italic> , Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 92, Springer-Verlag, Berlin–New York, 1977]. Precisely, every bounded sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{x_i\}_{i=1}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in a noncommutative quasi-Banach <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bimodule <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper E subset-of upper L Subscript p Baseline left-parenthesis script upper M comma tau right-parenthesis plus script upper M"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">E</mml:mi> </mml:mrow> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi> τ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {E}\subset L_{p}(\mathcal {M},\tau )+\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> stands for a semifinite von Neumann algebra), <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , having order continuous quasi-norm either satisfies that there exists a constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">c&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that, for every choice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace a Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{a_i\}_{i=1}^\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of scalars, <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline double-vertical-bar sigma-summation Underscript i equals 1 Endscripts r Subscript i Baseline left-parenthesis t right-parenthesis a Subscript i Baseline x Subscript i Baseline double-vertical-bar Subscript script upper E Baseline d t greater-than-or-equal-to c left-parenthesis sigma-summation Underscript i equals 1 Overscript n Endscripts StartAbsoluteValue a Subscript i Baseline EndAbsoluteValue squared right-parenthesis Superscript 1 slash 2 Baseline comma n equals 1 comma 2 comma midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo> ∫ </mml:mo> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow>