UCLA

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 $\tau$ on $M$. In particular, letting $M\subset \B= \B(L^2M)$, and $2\leq p < \infty$, we first consider the space $\B(p)$, obtained as the completion of $\B$ in the norm $|||T|||_p $ defined as the supremum of $|\varphi(T)|$ over all linear functionals $\varphi \in \B^*$ such that $|\varphi(xYz)| \leq \|x\|_p \|Y\| \|z\|_p$, for all $x,y \in M$ and $Y \in \B$. We continue by examining the subspace $\K(p)\subset \B(p)$, obtained as the closure in $\B(p)$ of the space of compact operators $\K(L^2M)$. We then construct the space $\K_p\subset \B$ of operators that are $||| \ |||_p$-limits of bounded sequences of operators in $\K(L^2M)$. We prove that $\K_p$ are all equal to the \textit{$\tau$-rank-completion} of $\K(L^2M)$ in $\B$, denoted as $\text{q}\K_M$, which is the subspace of all operators $K \in \B(L^2M)$ such that there exists a sequence operators $K_n \in \K(L^2M)$ and projections $p_n\in \mathcal{P}(M)$ with $\lim_n \|p_n(K-K_n)p_n\|= 0$ and $\lim_n\tau(1-p_n)=0$. We conclude by proving that any separable II$_1$ factor $M$ admits non-inner derivations into $\text{q}\K_M$, but that any derivation $\delta:M \rightarrow \text{q}\K_M$ is a pointwise limit in $\tau$-rank-topology of inner derivations. This gives us an example of a ``non-vanishing" theory of 1-cohomology for II$_1$ factors $M$.