Twisting non-commutative L^p spaces

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• 作者：Fé ; lix Cabello Sá ; nchez^a ; ^{fcabello@unex.es" class="auth_mail" title="E-mail the corresponding author} ; ; Jesú ; s M.F. Castillo^a ; ^{castillo@unex.es" class="auth_mail" title="E-mail the corresponding author} ; Stanisław Goldstein^b ; ^{goldstei@math.uni.lodz.pl" class="auth_mail" title="E-mail the corresponding author} ; Jesú ; s Suá ; rez de la Fuente^c ; ^{jesus@unex.es" class="auth_mail" title="E-mail the corresponding author}
• 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 L^p$L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra M$\mathcal{M}$. 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 0⟶Y⟶X⟶Y⟶0$0⟶Y⟶X⟶Y⟶0$.

For semifinite algebras, considering 6e25dc5b53e7909c1" title="Click to view the MathML source">L^p=L^p(M,τ)$L^p=L^p(\mathcal{M},\tau )$ as an interpolation space between M$\mathcal{M}$ and its predual M_⁎${\mathcal{M}}_{⁎}$ one arrives at a certain self-extension of L^p$L^p$ that is a kind of noncommutative Kalton–Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented.

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.