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Let $M=C^*_\lambda(G)''$ be group von Neumann algebra of $G$. The the condition above implies:

$d(T, M')\leq C ||ad(T)|_{M}||$ for every $T\in B(l^2 G)$. The last inequality is equivalent to saying that every derivation of $M$ into $B(l^2 G)$ is inner.

Edit: the above inequality is satisfied automatically (was clarified to me by Stuart White).

It is known that if $M\subset B(H)$ has a cyclic vector, then every bounded derivation from $M$ into $B(H)$ is inner [E. Christensen, Extensions of derivations II, Math. Scand., 1982]. Thus [Christensen, op cit, Cor 5.4] we have

$d(T,M')\leq 3/2\|(\mathrm{ad}| T)_{M}\|$

for every $T\in B(l^2 G)$.

Let $M=C^*_\lambda(G)''$ be group von Neumann algebra of $G$. The the condition above implies:
$d(T, M')\leq C ||ad(T)|_{M}||$ for every $T\in B(l^2 G)$.
It is known that if $M$. NamelyM\subset B(H)$has a cyclic vector, then every bounded derivation of from$M$into$B(H)$is inner [E. Christensen, Extensions of derivations II, Math. Scand., 1982]. Thus [Christensen, op cit, Cor 5.4] we have$d(T,M')\leq 3/2\|(\mathrm{ad}| T)_{M}\|$for every$T\in B(l^2 G)$. 1 Let$M=C^*_\lambda(G)''$be group von Neumann algebra of$G$. The the condition above implies:$d(T, M')\leq C ||ad(T)|_{M}||$for every$T\in B(l^2 G)$. The last condition is known to be equivalent to the derivation problem for$M$. Namely, every derivation of$M$into$B(H)\$ is inner.