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\title{Derivations with values in noncommutative symmetric spaces}

\author{\firstname{Jinghao} \lastname{Huang}\IsCorresp}
\address{Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology, Harbin, 150001,\,China}
\email{jinghao.huang@hit.edu.cn}

\author{\firstname{Fedor} \lastname{Sukochev}}
\address{School of Mathematics and Statistics, University of New South Wales, Kensington, 2052, Australia}
\email{f.sukochev@unsw.edu.au}

\thanks{J. Huang was supported by the Natural Science Foundation of China (No. 12031004 and 12301160). F. Sukochev's research was supported by the Australian Research Council (FL170100052).}

\CDRGrant[Natural Science Foundation of China]{No.12031004}
\CDRGrant[Natural Science Foundation of China]{No.12301160}
\CDRGrant[Australian Research Council]{FL170100052}

\begin{abstract}
Let $E=E(0,\infty)$ be a symmetric function space and $E(\mathcal{M},\tau)$ be the noncommutative symmetric space corresponding to $E(0,\infty)$ associated with a von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $\delta:\mathcal{A}\to E(\mathcal{M},\tau)$ is necessarily inner for each $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.
\end{abstract}

\keywords{\kwd{derivation}
\kwd{noncommutative symmetric space}
\kwd{semifinite von Neumann algebra}}

\begin{document}
\maketitle

\section{Historical Background and Motivations}

Let $\cA $ be a $C^*$-algebra and let $J$ be an $\cA$-bimodule~\cite{Sinclair_S}. A derivation $\delta :\cA \rightarrow J$ is a linear mapping satisfying $\delta (xy) = \delta(x)y + x\delta(y)$, $x,y \in \cA $. In particular, if $a\in J$, then $\delta_a(x): = xa-ax $ is a derivation. Such derivations implemented by elements in $J$ are said to be \emph{inner}. One of the classical problems in operator algebra theory is the question whether every derivation from $\cA$ into $J$ is automatically inner.


At a conference held in 1953, Kaplansky asked Singer if he had an idea of what the derivations of $C(X)$ (the algebra of continuous functions on a compact Hausdorff space $X$) might be. A day later, Singer gave Kaplansky a short, clever argument that such derivations must map all of $C(X)$ to $0$~\cite{K00}. Kaplansky's paper~\cite{Kaplansky} and the strong interest in derivations of operator algebras grew out of Singer's result. Kaplansky showed that each derivation of a type $I$ von Neumann algebra (for example, $B(\cH)$, the algebra of all bounded operators on a Hilbert space $\cH$) into itself is inner. In the course of his argument, Kaplansky proved that each such derivation is (norm-)continuous and conjectured that automatic continuity is true for all $C^*$-algebras. This conjecture was proved a few years later by Sakai~\cite{Sakai60} and extended by Ringrose~\cite{Ringrose} to derivations of a $C^*$-algebra into a Banach bimodule. These were among the earliest automatic-continuity results. In~\cite{Kadison,Sakai66}, it was proved that each derivation of a $C^*$-algebra acting on $\cH$ extends to a derivation of the strong-operator closure of that algebra, a von Neumann algebra, and that each derivation of a von Neumann algebra is inner. However, there exist $C^*$-algebras $\cA$ with non-inner derivations $\delta:\cA\to \cA$, e.g., if $\cA$ is the $C^*$-algebra $K(\cH)$ of all compact linear operators on $\cH$, then for any $a{\color{black}\not\in} K(\cH)+\mathbb{C}\un$, $\delta_a$ is a non-inner derivation on $\cA$~\cite[Example~4.1.8]{Sakai98}. Therefore, it would be desirable to identify those $\cA$-bimodules $\cA$ such that all derivations from $\cA$ into $\cJ$ are necessarily inner, see e.g.~\cite[Section~10.11]{Johnson}, \cite{Hoover} and~\cite[p.~60]{Sinclair_S}. Recall several important results in this direction:
\begin{enumerate}
\item Derivations from a $C^*$-algebra $\cA$ into any reflexive $\cA$-bimodule are inner~\cite{Johnson};
\item Derivations from a hyperfinite von Neumann algebra $\cA$ into any dual normal\footnote{Let $\cM$ be a von Neumann algebra. An $\cM$-bimodule $X$ is said to be a dual normal $X$-bimodule if $X$ is a dual space and the maps $m\mapsto mx$ and $m\mapsto xm$ are both ultraweak-weak$^*$ continuous from $\cM$ into $X$ for each fixed element $x\in X$. } $\cA$-bimodule are inner~\cite{JKR};
\item Derivations from a nuclear $C^*$-algebra $\cA$ into a dual Banach $\cA$-module are inner~\cite{Haagerup}.
\end{enumerate}



Let $\cM$ be a semifinite von Neumann algebra acting on a Hilbert space $\cH$ equipped with a semifinite faithful normal trace $\tau$. Let $\un$ be the identity of $\cM$. Let $\cP(\cM)$ be the collection of all projections in $\cM$. A densely-defined closed operator $x$ affiliated with $\mathcal{M}$ is $\tau$-measurable (see~\cite{FK}) if and only if
\[
\tau\left(e^{|x|}(n,\infty)\right)\rightarrow 0,\quad n\to\infty,
\]
where $ e^{|x|}(n,\infty)$ is the spectral projection of $|x|$ corresponding to the interval $(n,\infty)$. The collection of all $\tau$-measurable operators with respect to $\mathcal{M}$ is denoted by $S\left(
\mathcal{M},\tau \right) $. Let $x\in S(\mathcal{M},\tau)$. The generalised singular value function $\mu(x):t\rightarrow \mu(t;x)$, $t>0$, of the operator $x$ is defined by setting
\[
\mu(t;x) =
\inf \left\{
\left\|xp\right\|_\infty :p\in \cP(\cM), \tau(\un-p)\leq t
\right\},
\]
where $\norm{\,\cdot\,}_\infty$ denotes the usual operator norm. If $x\in S(\cM,\tau)$ satisfies that $\mu(\infty;x)=0$, then $x$ is said to be a $\tau$-compact operator. The collection of all $\tau$-compact operators in $S(\cM,\tau)$ is denoted by $S_0(\cM,\tau)$. Let ${\cE}$ be a linear subset in $S({\mathcal{M}, \tau})$ equipped with a complete norm $\norm{\,\cdot\,}_{{\cE}}$. We say that ${\cE}$ is a \emph{symmetric space} if for $x \in {\cE}$, $y\in S({\mathcal{M}, \tau})$ and $\mu(y)\leq \mu(x)$ imply that $y\in {\cE} $ and $\left\|y\right\|_{{\cE}}\leq \left\|x\right\|_{\cE}$. If ${\cE}$ is a symmetric space, then the carrier projection $c_{\cE}\in \cP(\cM)$ is defined by setting
\[
c_{\cE} = \bigvee\{p:p\in \cP(\cM),~p \in {\cE} \}.
\]
We remark that, replacing the von Neumann algebra $\cM$ by the reduced von Neumann algebra $\cM_{c_{\cE}}$, it is often assumed that the carrier projection of $ {\cE}$ is equal to $\un$, see e.g.~\cite{DP2}.


If $x,y\in S(\cM,\tau)$, then $x$ is said to be \emph{submajorized} by $y$, denoted by $x\prec\prec y$, if
\begin{align*}
\int_{0}^{t} \mu(s;x) \,\ds \le \int_{0}^{t} \mu(s;y) \,\ds \quad\text{for all $t\ge 0$.}
\end{align*}
A symmetric space ${\cE} \subset S(\cM,\tau)$ is called \emph{strongly symmetric} if its norm $\norm{\,\cdot\,}_{\cE}$ has the additional property that $\left\|x\right\|_{\cE} \le \left\|y\right\|_{\cE}$ whenever $x,y \in {\cE}$ satisfy $x\prec\prec y$. In addition, if $x\in S(\cM,\tau)$, $y \in {\cE}$ and $x\prec\prec y$ imply that $x\in {\cE}$ and $\left\|x\right\|_{\cE} \le \left\|y\right\|_{\cE}$, then ${\cE}$ is called \emph{fully symmetric space} (of $\tau$-measurable operators). If ${\cE}$ is a strongly symmetric space with $c_{\cE}=\un$ (or a symmetric space affiliated with a semifinite von Neumann algebra which is either atomless or atomic with all minimal projections having equal trace), then we have~\cite{DP2,KPS}
\begin{align}\label{largerthan1infty}
(L_1\cap L_\infty)(\cM,\tau)\subset {\cE} \subset (L_1+ L_\infty) (\cM,\tau).
\end{align}


If ${\cE}$ is a symmetric space, then the norm $\norm{\,\cdot\,}_{\cE}$ is called \emph{order continuous} if $\left\|x_\alpha \right\|_{\cE} \rightarrow 0$ whenever $\{x_\alpha\}$ is a downwards directed net in ${\cE}^+$ satisfying $x_\alpha \downarrow 0.$ A symmetric space ${\cE}$ is said to have the\emph{Levi property}~\cite[Definition~7]{AA}\footnote{The Soviet school on Banach lattices used the term monotone complete norm or property (B), see also~\cite[p.~89]{AA}.}, if for every upwards directed net $\{x_\beta\}$ in ${\cE}^+$, satisfying $\sup_\beta \left\|x_\beta\right\|_{\cE} <\infty$, there exists an element $x\in {\cE}^+$ such that $x_\beta \uparrow x$ in ${\cE}$. It is well known that if a norm is Levi, then necessarily it is also weak Fatou, i.e., there exists a constant $K\ge 1$ such that
\[
0\le x_\beta \uparrow x \Longrightarrow \norm{x}_{\cE} \le K\lim_\beta \norm{x_\beta }_{\cE}.
\]
If the constant $K$ is $1$, then ${\cE}$ is said to have the \emph{Fatou property}. If ${\cE}$ has the Fatou property and order continuous norm, then it is said to be a \emph{KB}-space (or Kantorovich--Banach space)\cite{DP2}.


The so-called K\"{o}the dual is identified with an important part of the dual space. If ${\cE}$ is a symmetric space, then the K\"{o}the dual ${\cE}^\times $ of ${\cE}$ is defined by
\[
{\cE}^\times =\left\{x\in S(\cM,\tau) : \sup_{\left\|y\right\|_{\cE} \le 1, y\in {\cE}}\tau (|xy|) <\infty \right\},
\]
and for every $x\in {\cE}^\times$, we set $\left\|x\right\|_{{\cE}^\times} = \sup
\left\{\tau(|yx|) : y\in {\cE},\, \left\|y\right\|_{\cE} \le 1\right\}$~\cite{DP2}.

Reciprocally, a wide class of symmetric spaces of $\tau$-measurable operators is representable as $E(\cM, \tau)$, the noncommutative symmetric space associated with a given symmetric function space $E(0,\infty)$ and a given von Neumann algebra $\cM$ equipped with a semifinite faithful normal trace $\tau$: let $(E(0,\infty),\norm{\,\cdot\,}_{E(0,\infty)})$ be a symmetric function space on the semi-axis $(0,\infty)$. The space
\[
E(\cM,\tau)=\{x\in S(\cM,\tau):\mu(x)\in E(0,\infty)\}
\]
equipped with the norm $\left\|x\right\|_{E(\cM,\tau)}:=\left\|\mu(x)\right\|_{E(0,\infty)}$ is a symmetric operator space affiliated with $\cM$ with $c_{\cE} =\un$, see e.g.~\cite{Kalton_S}, \cite[Proposition~28]{DP2}. For convenience, we denote $\norm{\,\cdot\,}_{E(\cM,\tau)}$ by $\norm{\,\cdot\,}_E$.

Due to the rapid development of noncommutative analysis and motivated by questions due to Johnson et\,al., there are a number of papers concerning various versions of the following question~\cite{BP,BCS_2014,BGM}:

\begin{ques}\label{que:1}
Assume that $\cM$ is a von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Let ${\cE}$ be a symmetric space of $\tau$-compact operators affiliated with $\cM$. How can one identity those ${\cE}$ such that derivations from an arbitrary $C^*$-subalgebra $\cA$ of $\cM$ into ${\cE}$ are necessarily inner?
\end{ques} Experts in the operator theory are probably more familiar with symmetrically normed ideals in $B(\cH)$, which are a special case of noncommutative symmetric spaces. Various versions of Question~\ref{que:1} for derivations with values in ideals of a von Neumann algebra were asked and discussed in~\cite{Kaftal_W,BHLS,BHLS2,Johnson_P,Popa, Popa_R, Hoover}.
\section*{The Johnson--Parrot{t}--Popa Theorem and Its Semifinite Versions} Johnson and Parrott~\cite{Johnson_P} initiated the study of derivations with values in ideals of a von Neumann algebra by showing that derivations from an abelian/properly infinite von Neumann subalgebra of $B(\cH)$ into the algebra $K(\cH)$ of all compact operators on $\cH$ are inner. However, they failed to resolve the case when $\cA$ is a type $\II_1$ von Neumann algebra, which remained open until Popa's penetrating work~\cite{Popa} in 1987. This result is now known as the so-called the Johnson--Parrott--Popa theorem:
\emph{every derivation from an arbitrary von Neumann subalgebra $\cA$ of $B(\cH)$ into $K(\cH)$ is inner.} Note that the condition that $\cA$ is a von Neumann algebra can not be relaxed to the setting of a $C^*$-subalgebra of $B(\cH)$ as mentioned above.

A natural development of the Johnson--Parrott--Popa theorem is to establish a suitable semifinite version of the result. In 1985, Kaftal and Weiss~\cite{Kaftal_W} proved that if $\cA$ is an abelian (or properly infinite) von Neumann subalgebra of $\cM$ containing the center $\cZ(\cM)$ of $\cM$, then any derivation $\delta:\cA\to \cJ(\cM)$ is inner, where $\cJ(\cM)$ is an ideal of $\cM$ generated by all finite projections in a semifinite von Neumann algebra $\cM$. This result was later extended to the setting of more general von Neumann subalgebras by Popa and R\u{a}dulescu~\cite{Popa_R}. However, Popa and R\u{a}dulescu established the existence of non-inner derivations $\delta:\cA \to \cJ(\cM)$ for a specific semifinite von Neumann algebra $\cM$ and an abelian von Neumann subalgebra $\cA$ of $\cM$, which is the first nonvanishing $1$-cohomological result in the theory of von Neumann algebras.

In 1987, Christensen~\cite{Christensen87} introduced the notion of \emph{generalized compacts} associated with a von Neumann algebra and showed that derivations from a properly infinite von Neumann algebra into the generalized compacts associated with this von Neumann algebra are inner. However, the question whether derivations from a type $\II_1$ von Neumann algebra into the generalized compacts associated with this von Neumann algebra are inner was left open, which was recently answered in the affirmative in~\cite{Galatan_P}. We note that the ideals considered in~\cite{Kaftal_W,Galatan_P, Christensen87,Popa_R} are not necessarily symmetrically normed ideals in a semifinite von Neumann algebra (see e.g.~\cite[Section~2.3]{BHLS2}) and these results lie outside of the scope covered by Question~\ref{que:1}.

In our recent joint paper~\cite{BHLS2} with Ber and Levitina, we established the Johnson--Parrott--Popa theorem for another type of semifinite version of the ideal $K(\cH)$, namely the ideal $C_0(\cM,\tau)$ of $\tau$-compact operators in a semifinite von Neumann algebra $\cM$: \emph{every derivation from a von Neumann subalgebra of $\cM$ into $ C_0(\cM,\tau) $ is necessarily inner.} Even though $C_0(\cM,\tau)$ and $\cJ(\cM)$ are similar in many respects (see~\cite{BHLS2}), our result is in strong contrast with the result in~\cite{Popa_R} and our result seems to be spiritually closer to the original Johnson--Parrott--Popa theorem, since we do not impose any additional condition on the von Neumann subalgebra $\cA$.


\section*{Derivations into an Ideal of a von Neumann Algebra}

An important class of ideals in a von Neumann algebra is given by the Schatten--von Neumann $p$-classes. In~\cite{Hoover}, Hoover used the Ryll-Nardzewski fixed point theorem (as suggested by Johnson~\cite{Johnson, Johnson_P}) and the reflexivity of the Schatten--von Neumann $p$-class $C_p(\cH)$, $1<p<\infty$, to show that every derivation from a $C^*$-subalgebra of $B(\cH)$ into $C_p(\cH)$ is inner. Actually, the Ryll-Nardzewski fixed point theorem is applicable to all reflexive $\cA$-bimodules (in particular, noncommutative $L_p$-spaces when $1<p<\infty$), {see e.g.~\cite[Theorem~3.4]{Johnson}}. Hoover also resolved the special case for the trace class $C_1(\cH)$ by a $p$-convexification technique~\cite{Hoover}.

Let $\cM$ be a von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. Denote by $L_p(\cM,\tau)$, $p\ge 1$, the noncommutative $L_p$-space affiliated with $\cM$, and denote $C_p(\cM,\tau):=L_p(\cM,\tau)\cap \cM$. In general, $C_p(\cM,\tau)$ is not reflexive even for $1< p<\infty$ and therefore the Ryll-Nardzewski fixed point theorem can not be applied directly (the method used in~\cite{BGM} or~\cite{Pfitzner} is not applicable, either). In 1985, using Johnson and Parrott's trick~\cite{Johnson_P}, Kaftal and Weiss~\cite{Kaftal_W} showed that every derivation from an abelian (or properly infinite) von Neumann subalgebra of $\cM$ into the $C_p$ ideal of $\cM$ is inner when $1 \le p<\infty$. However, the case for general von Neumann subalgebras of $\cM$ was left unanswered. Using noncommutative integration techniques, it was proved in~\cite{BHLS} that derivations from a $C^*$-subalgebra of $\cM$ into any symmetric \emph{KB}-ideal of $\cM$ are inner, which, in particular, fully resolves the untreated cases for derivations with values in $ C_p(\cM,\tau)$ in the paper~\cite{Kaftal_W} by Kaftal and Weiss. For the special case when $\cA$ is a von Neumann subalgebra of $\cM$, it is shown in~\cite{BHLS2} that derivations from $\cA$ into any ideal of $\tau$-compact operators, generated by a noncommutative strongly symmetric space ${\cE}$ having the Fatou property, are inner; it was shown in~\cite{Ber_S_1,Ber_S_2} that derivations from a (not necessarily semifinite) von Neumann algebra $\cM$ into any ideal of $\cM$ are inner.

\section{Derivations into Symmetric Spaces of Possibly Unbounded Operators} An important class of $\cA$-bimodules is given by the noncommutative $L_p$-spaces. Since a noncommutative $L_p$-space is reflexive if $1<p<\infty$, it follows immediately from Johnson's result that derivations into a noncommutative $L_p$-space, $1<p<\infty$, are inner. Let $\cU(\cA)$ be the unitary group of a unital $C^*$-algebra $\cA$. Consider a derivation $\delta$ from $\cA$ into a reflexive $\cA$-bimodule $\cJ$. Since the unit ball of a reflexive Banach space is weakly compact, it follows that the set
\[
K_\delta:= \overline{\conv\{\delta(u)u^* \mid u\in \cU(A)\}}^{\norm{\,\cdot\,}_{\cJ }} \footnote{\color{black}This convex set $ K_\delta$ was first considered by Kadison, in connection with innerness of derivations on finite von Neumann algebras, and using a fixed point method.}
\]
is weakly compact in $\cJ$. Therefore, one can easily apply the Ryll-Nardzewski fixed point theorem to prove the innerness of $\delta$. Recall, also, that every derivation from a hyperfinite von Neumann algebra $\cA$ into a dual normal $\cA$-bimodule is inner. However, the predual (i.e., a noncommutative $L_1$-space) of a semifinite von Neumann algebra $\cM$ is not reflexive unless the underlying von Neumann algebra is finite-dimensional, which is the main obstacle to resolve derivation problem for noncommutative $L_1$-spaces; moreover, in general, a noncommutative $L_1$-space does not even have a predual (i.e., it does not have the ``dual normal'' property), which makes this problem even harder.


Hoover~\cite{Hoover} resolved the special case for the trace class. The case for derivations from a $C^*$-subalgebra of a finite von Neumann algebra $\cM$ into the predual of $\cM$ was resolved by Bunce and Paschke in~\cite{BP}, where they also proved the derivations from a semifinite von Neumann algebra $\cM$ into its predual $\cM_*$ are automatically inner (see also~\cite{Haagerup} for the case of type $\III$ von Neumann algebras). However, it was a long-standing open question whether every derivation a $C^*$-subalgebra of $\cM$ into $\cM_*$ must be inner~\cite[p.~247]{BP}, which was resolved completely by Bader, Gelander and Monod in 2012~\cite{BGM} (see also~\cite{Pfitzner} for a slightly different proof due to Pfitzner). Bader, Gelander and Monod considered the so-called Chebyshev center (which is weakly compact) in the so-called $L$-embedded Banach spaces\footnote{That is, a Banach space $V$ whose bidual can be decomposed as $V ^{** }= V \oplus_1 V_0$ for some $V_0 \subset V^{**} $ (and $\oplus _1$ indicates that the norm is the sum of the norms on $V$ and $V_0$).} (e.g. the predual of a von Neumann algebra), where the Ryll-Nardzewski fixed point theorem is applicable. Hereby, they provided a beautiful and short resolution to the derivation problem for noncommutative $L_1$-spaces. In fixed point theory, it is natural to claim only convex compactness instead of compactness~\cite[Remark~8]{Pfitzner}. Therefore, the idea used by Pfitzner in~\cite{Pfitzner} seems to be more natural, where he introduced a new topology for general $L$-embedded spaces (in particular, for the predual of von Neumann algebras) and apply a variant of the Ryll-Nardzewski fixed point theorem to a weakly compact convex set. This direction of thought has been completed in~\cite{HLS}, which shows that derivations into $L$-embedded symmetric spaces are inner. It is worth mentioning that derivations from a semifinite von Neumann algebra into a symmetric space affiliated with itself are necesarily inner~\cite{BCS_2014}.



\section{Main Results and Methods}

$L$-embedded Banach spaces are very special in the field of Banach spaces, and the method used in~\cite{BGM} (or~\cite{Pfitzner}) does not have any chance to deliver the full answer on Question~\ref{que:1}. This fact was emphaiszed in~\cite[Section~3]{BGM}, where the following points were raised:\medskip

\begin{adjustwidth}{\parindent}{\parindent}
\begin{enumerate}\alphenumi
\item \label{a} In marked contrast to the classical fixed point theorems, there is no hope to find a fixed point inside a general bounded closed convex subset of $L^1$ $\cdots$ the weak compactness $\cdots$ seems almost unavoidable $\cdots$
\item \label{b} $\cdots$ a canonical norm one projection~$V^{**}\to V$ is not enough.
\item \label{c} It would be interesting to find a purely geometric version of the proposition $\cdots$
\end{enumerate}
\end{adjustwidth}\medskip

\noindent
The fact that the ``fixed point'' obtained in~\cite{BGM} is not inside a general bounded closed convex subset of $L^1$ leads to extra difficulties in the general case. Our principal result provides a complete answer to Question~\ref{que:1} above.

Recall the noncommutative version of Grothendieck's theorem established in~\cite{ADG} stating that an arbitrary bounded linear map from a $C^*$-algebra into a weakly sequentially complete Banach space is weakly compact. Moreover, a symmetric $KB$-function space $F(0,\infty)$ generates a weakly sequentailly complete noncommutative symmetric space $F(\cM,\tau)$~\cite{DP2}. We obtain that derivations into $F(\cM,\tau)$ must be weakly compact. However, it is not enough to show that the convex set $K_\delta$ is weakly compact in $F(\cM,\tau)$, {see e.g.,} the case when $F(0,\infty)=L_1(0,\infty)$ and $\cM$ is a non-finite semifinite von Neumann algebra~\cite[Theorem~III.5.4]{Tak}. We need the following result to guarantee the weak compactness of $K_\delta$, whose proof relies on the weak compactness criteria in symmetric spaces obtained in~\cite{DDP,DSS}.

For a Banach space $X$, we denote by $B_X$ the unit ball of $X$.

\begin{prop}\label{proprwc}
Let $\cM$ be a von Neumann algebra equipped with a semifinite
faithful normal trace $\tau$. Assume that ${\cE}$ is a strongly symmetric $KB$-space such that ${\cE}^\times \subset S_0(\cM,\tau)$. Let $\cA$ be a $C^*$-subalgebra $ \cM$ and let $T$ be a bounded linear operator from $\cA$ into ${\cE}$. Then, the set
\[
B_\cM T(B_\cA)B_\cM :=\left\{aT(x)b : a,b\in B_\cM,x\in B_\cA \right \}
\]
is relatively weakly compact in ${\cE}$.
\end{prop}

For derivations into noncommutative symmetric spaces, we have the following consequence.

\begin{prop}\label{prop:rwc}
Let $\cM$ be a von Neumann algebra equipped with a semifinite
faithful normal trace $\tau$ and let $\cA$ be a unital $C^*$-subalgebra $ \cM$. Assume that ${\cE}$ is a strongly symmetric KB-space such that ${\cE}^\times \subset S_0(\cM,\tau)$. Let $\delta:\cA\to {\cE}$ be a derivation. Then,
\[
\{\delta(u)u^* \mid u\in \cU(\cA) \}
\]
is relatively weakly compact in ${\cE}$. Consequently, the closure $\overline{\conv\{\delta(u)u^* \mid u\in \cU(\cA) \}}^{\norm{\,\cdot\,}_{{\cE}} }$ of the convex hull is weakly compact.
\end{prop}

Assume that ${\cE}$ is a strongly symmetric space with $c_{\cE} =\un$. Note that, if $\tau(\un)<\infty$, then the condition ${\cE}^\times \subset S_0(\cM,\tau)$ holds for any symmetric space ${\cE}$. If $\tau(\un)=\infty$, then ${\cE}^\times \subset S_0(\cM,\tau)$ if and only if ${\cE}\cap \cM \supsetneqq L_1(\cM,\tau)\cap \cM$. Indeed, assume that ${\cE} \cap \cM \supsetneqq L_1 (\cM,\tau)\cap \cM $. Then, there exists an element $0\le z\in {\cE} \cap \cM $ but $z\notin L_1(\cM,\tau)$. In particular, we have $\tau(z)=\infty$. By the definition of K\"{o}the duals, we infer that $\un\notin {\cE}^\times$, which, in turn, implies that ${\cE}^\times \subset S_0(\cM,\tau)$. On the other hand, assume by contradiction that $L_1(\cM,\tau)\cap \cM$ is not a proper subspace of ${\cE} \cap \cM $. By~\eqref{largerthan1infty}, we have $L_1(\cM,\tau)\cap \cM\subset {\cE} \cap \cM $. Therefore, we obtain that ${\cE}\cap \cM =L_1(\cM,\tau)\cap \cM $. By the fact that ${\cE}\stackrel{\eqref{largerthan1infty}}{\subset} (L_1+L_\infty)(\cM,\tau)$, we obtain that for any element $x\in {\cE}$, $\mu(x)\chi_{(0,1)}\in L_1(0,1)$. Hence, all elements in ${\cE}$ belong to $L_1(\cM,\tau)$. Therefore, ${\cE}^\times \supset L_1(\cM,\tau)^\times =\cM$~\cite[Definition~5.1]{DDP93}. That is, ${\cE}^\times \not\subset S_0(\cM,\tau)$, which completes the proof.



Now, one may apply the Ryll-Nardzewski fixed point theorem to obtain the following result for a wide class of noncommutative symmetric spaces.

\begin{lemm}\label{theorem:L_1+E}
Let $\cM$ be a von Neumann algebra with a faithful normal semifinite trace $\tau$ and let $\cA$ be a unital $C^*$-subalgebra of $\cM$. Let ${\cE}$ a strongly symmetric $KB$-space such that ${\cE}^\times \subset S_0(\cM,\tau)$. For every derivation $\delta:\cA \rightarrow {\cE}$, there exists an element $a \in \overline{\conv\{\delta(u)u^* \mid u\in \cU(A)\}}^{\norm{\,\cdot\,}_{{\cE} }} $ such that $\delta=\delta_a$ on $\cA$. In particular, $\left\|a\right\|_{{\cE}} \le \left\|\delta\right\|_{\cA \rightarrow {\cE}}$.
\end{lemm}

For the case of general noncommutative symmetric spaces (in particular, $L_1(\cM,\tau)$), we need the following noncommutative version of the ``reflexive gate type'' result. Here, we recall the reflexive gate type result~\cite[Corollary~3.2.3]{Novikov} {since it seems not to be well-known} but plays a significant role in our approach: if $E(0,1)\ne L_1(0,1) $ and $E(0,1)\ne L_\infty (0,1)$, then there exist two reflexive symmetric spaces $F_1(0,1)$ and $ F_2(0,1)$ such that $F_2(0,1)\subset E(0,1)\subset F_1(0,1)$. {\color{black}The main ingredients of the proof of~\cite[Corollary~3.2.3]{Novikov} are the construction of a Lorentz space which embeds into $E(0,1)$ continuously, and a $p$-convexification technique. For the case when the measure/trace is infinite, we need the famous Davis--Figiel--Johnson--Pe{\l}czy\'{n}ski construction for reflexive spaces\cite{DFJP}. }

\begin{theo}\label{prop:ncembe}
Let $\cM $ be a von Neumann algebra equipped with a semifinite faithful normal trace $\tau$. Let ${\cE}$ be a strongly symmetric space such that ${\cE}^{\times \times}\subset S_0(\cM,\tau)$. Then, there exists a symmetric $KB$-function space $F(0,\infty)$ such that
\[
{\cE}\subset F(\cM,\tau).
\]
\end{theo}


It is important to emphasize that the Fatou/Levi property was hatched in the theory of Banach lattices~\cite{BVG,AA}, and was even included into the original definition of Banach function spaces over $\sigma$-finite measure spaces. The property is somewhat analogous to the so-called ``dual normal'' property. The importance of the Fatou/Levi property in the theory of Banach function spaces and symmetric operator spaces is hard to overestimate~\cite{DDP93,DP2012,DDST}. It seems appropriate to recall here that every derivation from a hyperfinite von Neumann algebra $\cA$ into a dual normal $\cA$-bimodule is inner. Recall also, that derivations from a nuclear $C^*$-algebra $\cA$ into a dual Banach $\cA$-module are inner. However, Theorem~\ref{main} below holds for arbitrary $C^*$-subalgebras $\cA$ of $\cM$ and for symmetric spaces which may not have a predual.

\begin{theo}\label{main}
Let $\cM$ be a von Neumann algebra with a faithful normal semifinite trace $\tau$ and let $\cA$ be a $C^*$-subalgebra of $\cM$. If ${\cE}$ is a strongly symmetric space of $\tau$-compact operators (i.e., ${\cE}\subset S_0(\cM,\tau)$\footnote{If $\tau(\un)<\infty$, then ${\cE}\subset S_0(\cM,\tau)$ holds for any symmetric space ${\cE}$ affiliated with $\cM$. }) having the Fatou property (resp., the Levi property), then every derivation $\delta:\cA \rightarrow {\cE}$ is inner. That is, there exists an element $a \in \overline{\conv
\left\{\delta(u)u^* \mid u\in \cU(\cA)\right\}}^{t_\tau} \subset{\cE}$ with $\|a \|_{\cE}\le \|\delta\|_{\cA\rightarrow {\cE}}$ (resp., $\left\|a \right\|_{\cE} \le c\left\|\delta\right \|_{\cA\rightarrow {\cE}}$ for some constant $c$ depending on ${\cE}$ only) such that $\delta=\delta_a $ on $\cA$.
\end{theo}

\begin{proof}
Without loss of generality, we may assume that the carrier projection {$c_{\cE} =\un$, $\cA$ is unital and $\cE$ has the Fatou property.}

Since ${\cE}$ has the Fatou property and ${\cE}\subset S_0(\cM,\tau)$, it follows that ${\cE}^{\times \times }\subset S_0(\cM,\tau)$. By Theorem~\ref{prop:ncembe}, there exists a symmetric $KB$-function space $F(0,\infty)\subsetneqq (L_1+C_0)(0,\infty) $ such that
\[
{\cE} \subset F(\cM,\tau)\subset S_0(\cM,\tau).
\]
Without loss of generality, we may assume that $L_2(0,\infty)\subset F(0,\infty)$ by replacing $F(0,\infty)$ with $L_2(0,\infty)+F(0,\infty)$. In particular, $F(0,\infty)\cap L_\infty (0,\infty)\supsetneqq L_1(0,\infty)\cap L_\infty (0,\infty)$. In particular, $F(0,\infty)^{\times }\subset S_0(0,\infty)$, and therefore, $F(\cM,\tau)^{\times }\subset S_0(\cM,\tau)$.


Note that $\delta(\cA)\subset {\cE} \subset F(\cM,\tau)$.
By Lemma~\ref{theorem:L_1+E}, there is an element
\[
a\in \overline{\conv
\left\{\delta(u)u^* \mid u\in \cU(\cA)\right\}}^{\norm{\,\cdot\,}_F}
\]
such that $\delta=\delta_a$ on $\cA$.
Hence, there exists a sequence
\[
\left(x_n\right)_{n=1}^\infty\subset \conv \left\{\delta(u)u^* \mid u\in \cU(\cA)\right\}
\]
such that $\left\|x_n -a\right\|_F\rightarrow_n 0$.
Since $F(\cM,\tau)$ is a symmetric space, it follows from~\cite[Proposition~20]{DP2} that $x_n\rightarrow_{t_\tau} a $ as $n\to \infty$\footnote{For every $\varepsilon,\delta>0,$ we define the set
\[
V(\varepsilon,\delta)=
\left\{x\in S(\mathcal{M},\tau):\ \exists p\in \cP\left(\mathcal{M}\right)\text{ such that } \left\|x(\un-p)\right\|_\infty \leq\varepsilon,\ \tau(p)\leq\delta
\right\}.
\]
The topology generated by the sets $V(\varepsilon,\delta)$, $\varepsilon,\delta>0,$ is called the \emph{measure topology} $t_\tau$ on $S(\cM,\tau)$~\cite{FK}. It is well known that the algebra $S(\cM,\tau)$ equipped with the measure topology is a complete metrizable topological algebra.}.

By Ringrose's theorem~\cite{Ringrose}, we have that $\delta: (\cA,\norm{\,\cdot\,}_\infty) \rightarrow \left({\cE},\norm{\,\cdot\,}_{\cE} \right)$ is a bounded mapping. Since ${\cE}$ has the Fatou property, it follows that the closed ball $\left({\cE},\norm{\,\cdot\,}_{\cE} \right)$ with radius $\left\| \delta\right\|_{\cA\rightarrow {\cE}}$ is closed in $S(\cM,\tau)$ with respect to the measure topology~\cite{DP2,DDP93}. Noticing that every element $x_n$, $n\ge 1$, belongs to the ball of radius $\left\|\delta\right\|_{\cA \rightarrow {\cE}}$ in ${\cE}$ and $x_n\rightarrow a$ in the measure topology, we conclude that $a\in {\cE}$ with $\left\|a\right\|_{\cE} \le \left\|\delta\right\|_{\cA\rightarrow {\cE}}$.
\end{proof}

The new approach devised in this paper answers to points (a), (b) and (c) above raised in~\cite{BGM}. In particular, it provides an alternative proof for the resolution of the question raised by Bunce and Paschke\cite{BP}, without involving weak compactness of a subset in a $L$-embedded space as~\cite{BGM} and~\cite{Pfitzner} did.
\begin{enumerate}
\item This enables us to find a ``fixed point'' (implementing the derivation) from a not necessarily weakly compact closed convex subset of a noncommutative symmetric space.
\item The Levi property of a symmetric space ${\cE}$ (not necessarily an $L$-embedded Banach space) is equivalent to the existence of a canonical norm one projection from the bidual ${\cE}^{**}$ onto ${\cE}$ \cite{DP2012,DDST}\footnote{Indeed, Theorem~\ref{main} holds for the case when the projection constant is not necessarily 1.}), which is enough to prove the existence of a fixed point, see Theorem~\ref{main}.
\item On the other hand, the Levi property of the space ${\cE}$ means that ${\cE}$ coincides with its second K\"{o}the dual and this geometrical condition is the only one required in Theorem~\ref{main} thus delivering (at least spiritually) an answer to the question suggested in~\cite[Comment c]{BGM} above.
\end{enumerate}
We believe that the method developed in this work is of interest in its own right.

We also show that whenever $E(0,\infty)$ does not have the Levi property, there exist non-inner derivations $\delta:\cA\to E(\cM,\tau)$ for some semifinite von Neumann algebra $\cM$ and a $C^*$-subalgebra $\cA$ of $\cM$.

\begin{coro}
For a given symmetric function space $E(0,\infty)\subset S_0(0,\infty)$, the following two statements are equivalent:
\begin{enumerate}
\item for any von Neumann algebra $\cM$ equipped with a semifinite faithful normal trace $\tau$ and any $C^*$-subalgebra $\cA$ of $\cM$, derivations $\delta:\cA\to E(\cM,\tau)$ are necessarily inner;
\item $E(0,\infty)$ has the Levi property.
\end{enumerate}
\end{coro}

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