For semifinite algebras, considering 6e25dc5b53e7909c1" title="Click to view the MathML source">Lp=Lp(M,τ) as an interpolation space between M and its predual M⁎ one arrives at a certain self-extension of Lp 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.