Some classes of smooth bimodules over II${}_1$ factors and their associated 1-cohomology spaces

We study several classes of Banach bimodules over a II${}_1$ factor $M$, endowed with topologies that make them “smooth” with respect to $L^p$-norms implemented by the trace on $M$. Letting $M\subset \text{B}=\text{B}(L^{2}M)$, and $2\le p<\mathrm{\infty }$, we consider: $(1)$ the space $\text{B}(p)$, obtained as the completion of $\text{B}$ in the norm $$\text{vertiii}{T}_p:=\sup \{\mathrm{\mid }\phi (T)\mathrm{\mid }\mid \phi \in {\text{B}}^{\ast },\sup \{\mathrm{\mid }\phi (xYz)\mathrm{\mid }\mid Y\in (\text{B}{)}_1,x,z\in M\cap (L^{p}M{)}_1\}\le 1\};$$ $(2)$ the subspace $\text{K}(p)\subset \text{B}(p)$, obtained as the closure in $\text{B}(p)$ of the space of compact operators $\text{K}(L^{2}M)$; $(3)$ the space ${\text{K}}_p\subset \text{B}$ of operators that are $\text{vertiii}{\cdot }_p$-limits of bounded sequences of operators in $\text{K}(L^{2}M)$. We prove that ${\text{K}}_p$ are all equal to the {\it $\tau$-rank-completion} of $\text{K}(L^{2}M)$ in $\text{B}$, defined by \begin{align} $\text{q}$\K_M:= {K\in \B(L^2M) \mid & \exists K_n \in \K(L^2M), p_n\in \mathcal P(M), \nonumber \ & \lim_n |p_n(K-K_n)p_n|= 0, \lim_nτ(1-p_n)=0}. \nonumber \end{align} We show that any separable II${}_1$ factor $M$ admits non-inner derivations into $\text{q}{\text{K}}_M$, but that any derivation $\delta :M\to \text{q}{\text{K}}_M$ is a pointwise limit in $\tau$-rank-metric of inner derivations.