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Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$

$$\inf \lbrace\|T-S\| \mid S \in \lambda(G)' \rbrace \leq C \cdot \sup\lbrace \|\lambda(g)T- T \lambda(g) \| \mid g \in G\rbrace.$$

(Here $\lambda(G)'$ denotes the commutant of $G$ in $B(\ell^2 G)$.)

Question: Is $G$ amenable?

It is fairly easy to see that amenability of $G$ implies the existence of such a constant. Indeed, one may take $S$ to be some fixed point for the conjugation action on $\overline{\rm conv}\lbrace \lambda(g)T\lambda(g)^* \mid g \in G\rbrace$. I am asking for the converse of this statement.

EDIT: Since the derivation problem came up in Kate's comment, I want to clarify to what version of it my question is related. The inequality above holds for some $C$ if and only if the first bounded cohomology of $G$ with coefficients in $B(\ell^2 G)$ (with the conjugation action induced by $\lambda$) is reduced. This is a straightforward application of the open mapping theorem. Now, two things are unclear:

Question: Can $H^1_b(G,B(\ell^2 G))$ be reduced without being zero?

and

Question: Can $H^1_b(G,B(\ell^2 G))$ be zero without $G$ being amenable?

I am essentially stuck with both of the questions. My intension was to look for entirely different approaches to the problem, which relate it to other questions.
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Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$

$$\inf \lbrace\|T-S\| \mid S \in \lambda(G)' \rbrace \leq C \cdot \sup\lbrace \|\lambda(g)T- T \lambda(g) \| \mid g \in G\rbrace.$$

(Here $\lambda(G)'$ denotes the commutant of $G$ in $B(\ell^2 G)$.)

Question: Is $G$ amenable?

It is fairly easy to see that amenability of $G$ implies the existence of such a constant. Indeed, one may take $S$ to be some fixed point for the conjugation action on $\overline{\rm conv}\lbrace \lambda(g)T\lambda(g)^* \mid g \in G\rbrace$. I am asking for the converse of this statement.

EDIT: Since the derivation problem came up in Kate's comment, I want to clarify to what version of it my question is related. The inequality above holds for some $C$ if and only if the first bounded cohomology of $G$ with coefficients in $B(\ell^2 G)$ (with the conjugation action induced by $\lambda$) is reduced. This is a straightforward application of the open mapping theorem. Now, two things are unclear:

Question: Can $H^1_b(G,B(\ell^2 G))$ be reduced without being zero?

and

Question: Can $H^1_b(G,B(\ell^2 G))$ be zero without $G$ being amenable?

I am essentially stuck with both of the questionquestions. My intension was to look for entirely different approaches to the problem, which relate it to other questions.
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Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$

$$\inf \lbrace\|T-S\| \mid S \in \lambda(G)' \rbrace \leq C \cdot \sup\lbrace \|\lambda(g)T- T \lambda(g) \| \mid g \in G\rbrace.$$

(Here $\lambda(G)'$ denotes the commutant of $G$ in $B(\ell^2 G)$.)

Question: Is $G$ amenable?

It is fairly easy to see that amenability of $G$ implies the existence of such a constant. Indeed, one may take $S$ to be some fixed point for the conjugation action on $\overline{\rm conv}\lbrace \lambda(g)T\lambda(g)^* \mid g \in G\rbrace$. I am asking for the converse of this statement.

EDIT: Since the derivation problem came up in Kate's comment, I want to clarify to what version of it my question is related. The inequality above holds for some $C$ if and only if the first bounded cohomology of $G$ with coefficients in $B(\ell^2 G)$ (with the conjugation action induced by $\lambda$) is reduced. This is a straightforward application of the open mapping theorem. Now, two things are unclear:

Question: Can $H^1_b(G,B(\ell^2 G))$ be reduced without being zero?

and

Question: Can $H^1_b(G,B(\ell^2 G))$ be zero without $G$ being amenable?

I am essentially stuck with both of the question. My intension was to look for entirely different approaches to the problem, which relate it to other questions.
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