Twisting non-commutative Lp spaces

The paper makes the first steps into the study of extensions (“twisted sums”) of noncommutative LpLp-spaces regarded as Banach modules over the underlying von Neumann algebra MM. 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⟶00⟶Y⟶X⟶Y⟶0.For semifinite algebras, considering Lp=Lp(M,τ)Lp=Lp(M,τ) as an interpolation space between MM and its predual M⁎M⁎ one arrives at a certain self-extension of LpLp 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.