The noncommutative Kalton–Peck spaces

Abstract

For every von Neumann algebra $\mathcal{M}$ and $0<p<\infty$ we construct a nontrivial exact sequence of $\mathcal{M}$-bimodules and homomorphisms $0\to L_p\to Z_p(\mathcal{M})\to L_p\to 0$, where $L_p$ is the Haagerup $L_p$ space built over $\mathcal{M}$. The middle space $Z_p(\mathcal{M})$ can be seen as a noncommutative version of the Kalton–Peck space $Z_p$.