I have heard it said that if a Banach space has a unique greedy basis, then that basis must be symmetric. I would like to see the proof of this. Is there a reference for this fact?
Definitions
Fix a basis $(e_j)$ for $X$ with biorthogonal functionals $(e_j^*)$. Let $(\mathcal{G}_m)$ be a sequence of maps such that $\mathcal{G}_m(x)=\sum_{j\in B}e_j^*(x)e_j$, where $B\subset\mathbb{N}$ is chosen so that $\#B=m$, and $|e_j^*(x)|\geq|e_k^*(x)|$ whenever $j\in B$ and $k\notin B$. We say that $(e_j)$ is greedy if there is $C\geq 1$ such that for any $x\in X$ and $m\in\mathbb{N}$ we have $\|x-\mathcal{G}_m(x)\|\leq C\inf\|x-y\|$, where the "inf" is taken over all $y\in X$ which can be expressed as the linear combination of $m$ elements of $(e_j)$.