Twisting non-commutative L<sup>psup> spaces

详细信息    查看全文

• 作者：Fé ; lix Cabello S&aacute ; nchez<sup>asup><sup> ; sup><sup>fcabello@unex.es" class="auth_mail" title="E-mail the corresponding authorsup><sup> ; sup> ; Jesú ; s M.F. Castillo<sup>asup><sup> ; sup><sup>castillo@unex.es" class="auth_mail" title="E-mail the corresponding authorsup> ; Stanisław Goldstein<sup>bsup><sup> ; sup><sup>goldstei@math.uni.lodz.pl" class="auth_mail" title="E-mail the corresponding authorsup> ; Jesú ; s Su&aacute ; rez de la Fuente<sup>csup><sup> ; sup><sup>jesus@unex.es" class="auth_mail" title="E-mail the corresponding authorsup>
• 关键词：46L52 ; 46M15 ; 46B70
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：14 May 2016
• 年：2016
• 卷：294
• 期：Complete
• 页码：454-488
• 全文大小：609 K

文摘

The paper makes the first steps into the study of extensions (“twisted sums”) of noncommutative <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si1.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=437d0cd87cc21f3f3e496072d55e0cca" title="Click to view the MathML source">L<sup>psup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>Lpsup>span>span>span>-spaces regarded as Banach modules over the underlying von Neumann algebra <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si2.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=4285a449ccceed0f783fe23fde95a2b2" title="Click to view the MathML source">Mspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Mspan>span>span>. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss, saying that whenever one finds a given Banach space Y   as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si3.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=daca727b3d878949f7880afe8076daf2" title="Click to view the MathML source">0⟶Y⟶X⟶Y⟶0span><span class="mathContainer hidden"><span class="mathCode">scroll">0stretchy="false">⟶Ystretchy="false">⟶Xstretchy="false">⟶Ystretchy="false">⟶0span>span>span>.

sp0020">For semifinite algebras, considering <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si4.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=3722ddfe4567c9a6e25dc5b53e7909c1" title="Click to view the MathML source">L<sup>psup>=L<sup>psup>(M,τ)span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>Lpsup>=sup>Lpsup>stretchy="false">(script">M,τstretchy="false">)span>span>span> as an interpolation space between <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si2.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=4285a449ccceed0f783fe23fde95a2b2" title="Click to view the MathML source">Mspan><span class="mathContainer hidden"><span class="mathCode">scroll">script">Mspan>span>span> and its predual <span id="mmlsi5" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si5.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=7d85ae9623adb58d300234549b106981" title="Click to view the MathML source">M<sub>⁎sub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>script">M⁎sub>span>span>span> one arrives at a certain self-extension of <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081600089X&_mathId=si1.gif&_user=111111111&_pii=S000187081600089X&_rdoc=1&_issn=00018708&md5=437d0cd87cc21f3f3e496072d55e0cca" title="Click to view the MathML source">L<sup>psup>span><span class="mathContainer hidden"><span class="mathCode">scroll">sup>Lpsup>span>span>span> that is a kind of noncommutative Kalton&ndash;Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented.

sp0030">For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one may find (nontrivial) self-extensions of bimodules in all cases is left open.