Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis which is not another. For example, given that \begin{equation}\text{symmetric basis }\Rightarrow\text{ subsymmetric basis }\Rightarrow\text{ unconditional basis }\Rightarrow\text{ basis}\end{equation} we want to prove that the reverse implications do not hold, i.e. that \begin{equation}\tag{$*$}\text{basis }\nRightarrow\text{ unconditional basis }\nRightarrow\text{ subsymmetric basis }\nRightarrow\text{ symmetric basis}.\end{equation} The above nonimplications are well-known.
However, for every type of basis, there is a corresponding property of Banach spaces for admitting that type of basis. To continue the above example, we have: \begin{equation}\begin{array}{c}\text{admits a}\\\text{symmetric basis}\end{array}\Rightarrow\begin{array}{c}\text{admits a}\\\text{subsymmetric basis}\end{array}\Rightarrow\begin{array}{c}\text{admits an}\\\text{unconditional basis}\end{array}\Rightarrow\begin{array}{c}\text{admits a}\\\text{basis}\end{array}\end{equation} Let us now consider the reverse nonimplications. \begin{equation}\tag{$**$}\begin{array}{c}\text{admits a}\\\text{basis}\end{array}\nRightarrow\begin{array}{c}\text{admits an}\\\text{unconditional basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{subsymmetric basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{symmetric basis}\end{array}\end{equation} Proving ($**$) is more difficult than proving ($*$). However, the first two nonimplications in ($**$) are already known. For instance, $L_1[0,1]$ admits a basis but not an unconditional one. $L_p[0,1]$, $p\in(1,2)\cup(2,\infty)$, admits an unconditional basis but not a subsymmetric one.
Garling (1968) exhibited an example of a basis which is subsymmetric but not symmetric. His basis was a modification of the canonical basis of the Lorentz sequence space $d(w,p)$ with $w=(n^{-1/2})_{n=1}^\infty$ and $p=1$. In a Lorentz sequence space, the norm of a vector is taken by considering the supremum over all permutations of an $\ell_p$ basis, with weights applied in a certain way. If instead you consider the supremum over all subsequences of an $\ell_p$ basis (with weights similarly applied), you get a Garling-like basis.
It is known that $d(w,p)$ admits a unique symmetric basis. Using similar arguments, we can show that Garling's space (and indeed a wider class of spaces based on Garling's construction) admits a unique subsymmetric basis. Consequently, we have examples of spaces which admit a subsymmetric basis but not a symmetric one. This proves the third nonimplication in ($**$), and brings me to my first question.
Question 1. Was the third nonimplication in ($**$) already known? In other words, was there already known an example of a Banach space which admits a subsymmetric basis but not a symmetric one?
I am hoping the answer is no, as it feels good to contribute something new to the math community : )
The next natural question is to expand ($**$) to include other types of bases. For example, we have the following: \begin{equation}\text{unconditional basis }\nRightarrow\text{ greedy basis }\nRightarrow\text{ subsymmetric basis}\end{equation} Accordingly, we can show \begin{equation}\tag{$***$}\begin{array}{c}\text{admits an}\\\text{unconditional basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{greedy basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{subsymmetric basis}\end{array}\end{equation} Dilworth/Kalton/Kutzarova (2003) proved that $c_0$ is the unique $\mathcal{L}_\infty$-space admitting a quasigreedy basis, and hence the unique $\mathcal{L}_\infty$-space admitting a greedy basis. So, if we consider any $\mathcal{L}_\infty$ space with an unconditional basis but which is not isomorphic to $c_0$ then we prove the first nonimplication in ($***$). The Haar basis for $L_p[0,1]$, $p\in(1,2)\cup(2,\infty)$, is greedy, which proves the second nonimplication in ($***$).
We can further subdivide greedy bases as follows. \begin{equation}\text{greedy basis }\Rightarrow\text{ almost greedy basis }\Rightarrow\text{ quasigreedy basis }\Rightarrow\text{ basis}\end{equation} and \begin{equation}\text{basis }\nRightarrow\text{ quasigreedy basis }\nRightarrow\text{ almost greedy basis }\nRightarrow\text{ greedy basis}\end{equation} leading to the conjecture \begin{equation}\tag{$****$}\begin{array}{c}\text{admits a}\\\text{basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{quasigreedy basis}\end{array}\nRightarrow\begin{array}{c}\text{admits an}\\\text{almost greedy basis}\end{array}\nRightarrow\begin{array}{c}\text{admits a}\\\text{greedy basis}\end{array}\end{equation} The $(\oplus_{n=1}^\infty\ell_\infty^n)_{\ell_1}$ counterexample proves the first nonimplication in ($****$), and Gogyan's construction (2010) of an almost greedy basis for $L_1[0,1]$ proves the third. However, we have the following question
Question 2. Does the second nonimplication in ($****$) hold? In other words, can we find an example of a Banach space admitting a quasigreedy basis but not an almost greedy basis?
I suspect that the answer to Questions #2 may be "no," as the connections between almost/quasi/fully greedy bases seem to be closer than between other types of bases. For example, Albiac/Ansorena (2015) showed that if a basis is both 1-quasigreedy and 1-almost-greedy then it is 1-greedy. Meanwhile, Dilworth/Kalton/Kutzarova (2003) proved that $c_0$ is the only infinite-dimensional Banach space admitting a unique quasigreedy basis, which ruins any chance of using on ($****$) the method I used in proving the third nonimplication in ($**$).
Finally:
Question 3. What are some other nonimplications of the form of ($**$)/($***$)/($****$) which are known, or unknown?
Question 4. Are Questions #1-3 interesting?
Thanks!
