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Open Problems in Linear Analysis and Probability

The problems here were either submitted specifically for the purpose of inclusion in this page, or were taken from talks given during the Workshop in Linear Analysis and Probability.

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Problem #1 (Submitted by Leonid Kovalev)

A continuous map $T:X\rightarrow X$ is called "low-rotational" if \, $\langle Tx-Ty\,,x-y\rangle _{-}\,\, \geq \delta\|Tx-Ty\|\|x-y\|$\,\, for some $\delta >0$.
Now suppose $T:X\rightarrow X$ is "low-rotational", $T\not = const$ and $\dim X>1$.
\begin{itemize}
\item Is $T$ injective?
\item Is $T$ surjective?
\item Is $T$ quasisymmetric?
\item Are there $p_1, p_2$ such that $\|x\|^{p_1}\leq\|Tx\|\leq\|x\|^{p_2}$ ?
\item If one uses $\langle Tx-Ty,x-y\rangle _{+}$ instead, would this change anything?
\end{itemize}

Problem #2 (Submitted by Gilles Pisier)

Let $G$ be a locally compact group. Does $G$ unitarizible implies $G$ amenable?\\
The answer to this question is known to be "YES" in the cases:
\begin{itemize}
\item $G$ - locally compact connected group
\item $G$ - discrete group containing copy of $\mathbb{F}_2$ (free group with 2 generators)
\end{itemize}
and the answer is also known to be positive if we assume some additional conditions.\\

Problem #3 (Submitted by Gilles Pisier)

Does $l(A,B)<\infty$ imply $\displaystyle A\otimes_{min}B=A\otimes_{max}B$ ?\\
Here $l(A,B)$ is the length corresponding to the way that the pair $(A,B)$ generates $\displaystyle A\otimes_{max}B$. Another way to look at $l(A,B)$ is as the smallest $d$ such that for any pair $u: A\to B(H),\,\, v: B\to B(H)$ of c.b homomorphisms with commuting ranges, the homomorphism $u.v: a\otimes b\to u(a)v(b)$ is c.b on $A\otimes_{max}B$, with
$$ \|u.v\|_{cb}\leq C\left ( \max\lbrace\|u\|_{cb}, \|v\|_{cb}\rbrace\right )^d . $$

Problem #4 (Submitted by Roger Smith)

Can an infinite subset of $\mathbb{N}$ be a value of $\,\,\, \textrm{Puk}_{L(\mathbb{F}_k)}(A)$ for some masa $A$.\\

Problem #5 (Submitted by Deguang Han)

For $d>1$, is it true that there exists always a compactly supported $g$ such that $\mathbf{G}(g,\mathcal{L,K})$ is a Parseval frame for $L_2(\mathbb{R}^d)$.

Problem #6 (Submitted by Michal Johanis)

Let $(B, \|\cdot\|)$ be a reflexive Banach space. Is there a renorming of $B$, such that the new norm is 2-rotund?
The answer to this question is known to be yes if $B$ is separable space.