math.FA — Functional Analysis

The operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into a noncommutative $\mathrm{L}^1$-space

Contributed by Cédric ARHANCET

We prove that Pisier's operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into the predual of a $\mathrm{QWEP}$ von Neumann algebra, answering a longstanding question introduced by Pisier and later by Haagerup and Musat. The embedding is obtained as an ultraproduct of free probabilistic models $h_k = \Lambda^{\frac{1}{2}}c_k\Lambda^{\frac{1}{2}}$ realized in the free group factor $\mathrm{VN}(\mathbb{F}_\infty)$, with the trace and circular family scaled according to the mass of the truncated measure $t^{-\frac{5}{3}} \mathrm{d} t$.