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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!

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    $\begingroup$ Answer to Q1 is Yes. A version of Tzafriri spaces (called Trilman spaces in Casazza-Shura's book) have subsymetric bases but do not even contain symmetric basic sequences. See impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/… $\endgroup$ – Bunyamin Sari Mar 16 '16 at 18:10
  • $\begingroup$ Thanks! I am a bit disappointed that this is not new, but it's still good to know. $\endgroup$ – Ben W Mar 16 '16 at 21:57
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    $\begingroup$ As I understand, such people as Dilworth, Kutzarova, and Schlumprecht do not participate in MO. Maybe, to get an answer to your question, it is worthwhile to e-mail them. $\endgroup$ – Mikhail Ostrovskii Mar 18 '16 at 19:58
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    $\begingroup$ So, it looks as if Q1 and Q2 are answered affirmatively. Thank you to Bunyamin Sari and Anso for those. With regard to Q3, it occurs to me that the James space has a spreading basis but not an unconditional one, and that $L_p[0,1]$, $p\in(1,2)\cup(2,\infty)$, has an unconditional basis but not a spreading one. Still no word on Q4... $\endgroup$ – Ben W Mar 24 '16 at 19:52
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For Q2, consider the original Tsirelson space. It has an unconditional, then quasi greedy, basis. However, it does not contain a democratic basis (see Remark 5.8 in DKK2003).

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