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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.06242 |
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| _version_ | 1866929266548015104 |
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| author | Hiatt, Patrick Peterson, Jesse Popa, Sorin |
| author_facet | Hiatt, Patrick Peterson, Jesse Popa, Sorin |
| contents | 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 $M\subset \B= \B(L^2M)$, and $2\leq p < \infty$, we consider: $(1)$ the space $\B(p)$, obtained as the completion of $\B$ in the norm \[ \vertiii{T}_p := \sup \{|φ(T)| \mid φ\in \B^*, \sup\{|φ(xYz)| \mid Y\in (\B)_1, x, z \in M\cap (L^pM)_1\} \leq 1 \}; \] $(2)$ the subspace $\K(p)\subset \B(p)$, obtained as the closure in $\B(p)$ of the space of compact operators $\K(L^2M)$; $(3)$ the space $\K_p\subset \B$ of operators that are $\vertiii{ \, \cdot \, }_p$-limits of bounded sequences of operators in $\K(L^2M)$. We prove that $\K_p$ are all equal to the {\it $τ$-rank-completion} of $\K(L^2M)$ in $\B$, defined by \begin{align} \text{\rm 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 $\text{\rm q}\K_M$, but that any derivation $δ:M \rightarrow \text{\rm q}\K_M$ is a pointwise limit in $τ$-rank-metric of inner derivations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_06242 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some classes of smooth bimodules over II$_1$ factors and their associated 1-cohomology spaces Hiatt, Patrick Peterson, Jesse Popa, Sorin Operator Algebras 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 $M\subset \B= \B(L^2M)$, and $2\leq p < \infty$, we consider: $(1)$ the space $\B(p)$, obtained as the completion of $\B$ in the norm \[ \vertiii{T}_p := \sup \{|φ(T)| \mid φ\in \B^*, \sup\{|φ(xYz)| \mid Y\in (\B)_1, x, z \in M\cap (L^pM)_1\} \leq 1 \}; \] $(2)$ the subspace $\K(p)\subset \B(p)$, obtained as the closure in $\B(p)$ of the space of compact operators $\K(L^2M)$; $(3)$ the space $\K_p\subset \B$ of operators that are $\vertiii{ \, \cdot \, }_p$-limits of bounded sequences of operators in $\K(L^2M)$. We prove that $\K_p$ are all equal to the {\it $τ$-rank-completion} of $\K(L^2M)$ in $\B$, defined by \begin{align} \text{\rm 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 $\text{\rm q}\K_M$, but that any derivation $δ:M \rightarrow \text{\rm q}\K_M$ is a pointwise limit in $τ$-rank-metric of inner derivations. |
| title | Some classes of smooth bimodules over II$_1$ factors and their associated 1-cohomology spaces |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2304.06242 |