What are good examples of Banach spaces which are and aren't super-reflexive? Whenever properties of Banach spaces like super-reflexivity, uniform convexity etc. are discussed, my impression is that almost always the only examples used are $l^p$, $L_{p}$ and $c_{0}$ (at least in places I looked at), but there must be many more interesting Banach spaces to look at - in particular, are there natural (i.e. encountered "in nature", not artificially constructed as counterexamples) examples of reflexive spaces which are not super-reflexive?
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Probably the most natural examples of reflexive spaces that are not super-reflexive are the spaces $(\bigoplus_{n=1}^\infty\ell_q^n)_{\ell_p}$, where $q\in\{1,\infty\}$ and $1<p<\infty$. They are reflexive as the $\ell_p$-direct sum of a sequence of finite dimensional spaces, but not super-reflexive since they contain isometric copies of $\ell_q^n$ for all $n$.
Tsirelson's space is also an example of a reflexive space that is not super-reflexive, but obviously it is not quite so natural since it was invented to be a Banach space having no subspace isomorphic to $\ell_p$ or $c_0$. Figiel and Johnson later constructed a super-reflexive space having no subspace isomorphic to $\ell_p$ or $c_0$.
answered Oct 11, 2013 at 2:58
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$\begingroup$ I would like to mention that the first example given by Philip appears "in nature" as the following statement in [S. J. Dilworth et al., Constr. Approx. 34 (2011), no. 2, 281–296] shows: "We prove that the Banach space $(\oplus^\infty_{n=1}\ell^n_p)_{\ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\le p\le\infty$ and $1<q<\infty$". $\endgroup$ Commented Feb 28, 2016 at 5:01