The noncommutative Kalton-Peck spaces (Q680721)

Consider two Banach spaces \(Z, X\) and a short exact sequence \(0\rightarrow X\overset{i}{\rightarrow}Z\overset{\pi}{\rightarrow}X\rightarrow0\) which is supposed to be nontrivial in the sense that the image of \(i\) is not complemented in \(Z\). (It is also said that \(Z\) is an extension of \(X\) by \(X\).) For \(X=\ell_2\), the existence of \(Z\) and such a nontrivial exact sequence is a formal way to say that being a Hilbert space is not a three-space property (or ``that there exist twisted Hilbert spaces''), as has been proved in [Math. Scand. 36, 199--210 (1975; Zbl 0314.46015)] by \textit{P. Enflo} et al. This result was followed by systematic studies leading to several generalizations like the Kalton-Peck spaces \(Z_p\) which give nontrivial extensions in the case \(X=\ell_p\), \(0< p <\infty\) [\textit{N. J. Kalton} and \textit{N. T. Peck}, Trans. Am. Math. Soc. 255, 1--30 (1979; Zbl 0424.46004)]. Subsequently, the existence of twisted sums was proved in the case that \(X\) is the classical Lebesgue space \(L_p\), the Schatten class \(S_p\) or the non-commutative \(L_p(\mathcal{M},\tau)\)-space associated to a semifinite von Neumann algebra \(\mathcal{M}\) with trace \(\tau\). In these cases, \(i\) and \(\pi\) above are even module homomorphisms if one considers the just mentioned spaces as (bi-)modules over \(\ell_\infty\), \(L_\infty\), \(B(H)\), \(\mathcal{M}\), respectively. The main purpose of the paper under review is to complete the picture by finally treating the general case in which \(X\) is an arbitrary noncommutative \(L_p\)-space, i.e., the Haagerup \(L_p(\mathcal{M})\)-space associated to an arbitrary von Neumann algebra \(\mathcal{M}\) considered as a bimodule over \(\mathcal{M}\), and this for \(0< p<\infty\). To wit, there are a bimodule noncommutative Kalton-Peck space \(Z_p(\mathcal{M})\) and an exact sequence \(0\rightarrow L_p(\mathcal{M})\overset{i}{\rightarrow}Z_p(\mathcal{M})\overset{\pi}{\rightarrow}L_p(\mathcal{M})\rightarrow0\) such that \(i\) and \(\pi\) are module homomorphisms. As for the construction, similarly as in the case of \(\ell_p\) of Kalton and Peck [loc. cit.], \(Z_p(\mathcal{M})\) is defined by means of a map \(\Phi:X\rightarrow Y\subset W\), where \(X\), \(Y\) are (quasi-)Banach modules over a Banach algebra \(\mathcal{M}\) and \(W\) is a module over \(\mathcal{M}\) containing \(Y\). This map is a bicentralizer from \(X\) to \(Y\) which means that the difference \(\Phi(afb)-a\Phi(f)b\) is in \(Y\) with \(\|\Phi(afb)-a\Phi(f)b\|_Y\leq C\|a\|_{\mathcal{M}}\|f\|_X\|b\|_{\mathcal{M}}\) uniformly in \(a,b\in \mathcal{M}\), \(f\in X\) for a constant \(C\). If \(X=Y=L_p\) (as throughout the paper), then \(\Phi\) is automatically quasilinear, which means that it is homogeneous and that there is a uniform constant \(Q\) such that, for \(f,g\in X\), the difference \(\Phi(f+g)-\Phi(f)-\Phi(g)\) is in \(Y\) with \(\|\Phi(f+g)-\Phi(f)-\Phi(g)\|_Y\leq Q(\|f\|_X+\|g\|_X)\). For the case \(p>1\), complex interpolation is used in a crucial way, following the (known) technique of scaling the intermediate spaces not by \(\theta\in[0,1]\) but by the numbers in an open set of \(\mathbb{C}\) conformally equivalent to the open unit disc. Then the case \(0< p\leq1\) is settled by ``transferring'' centralizers on \(L_p\) for \(p>1\) to centralizers on \(L_q\) for \(0 < q < p\). The paper is written in an appealing way.