Let $a,b\in\mathbb C$ be suc that $\max\{|a+b|,|a-b|\}\leq 1$ but $|a|+|b|>1.$ According to this paper by Arias, Figiel, Johnson and Schechtman <https://www.jstor.org/stable/2155206?origin=crossref#metadata_info_tab_contents>, the two dimensional complex subspace spanned by $(1,0,a)$ and $(0,1,b)$ in complex $\ell\_\infty^3$ is not isometric to complex $\ell\_\infty^2$. Can anyone prove that?



Subspaces of $\ell_\infty^3$
A funny isometry invariant to distinguish these normed spaces is: *The space of spheres of radius $2$ in the unit sphere of $(X,\|\cdot\|\_X)$ which are maximal by inclusion*, as described below. It turns out that for $\ell\_\infty^2(\mathbb C)$ it is a torus $\mathbb S^1\times \mathbb S^1$, and for the space $Y:=\text{span}\big((1,0,a),(0,1,b)\big)\subset \mathbb C^3$ as defined above it is a disjoint union of two circles, $\mathbb S^1\times \mathbb S^0$.  

Another one is: *The length of the poset of all spheres of radius $2$ in the unit sphere of $(X,\|\cdot\|\_X)$ ordered by inclusion.* One gets respectively $2$ and $3$.


Consider the unit sphere $S\_X:=\{x\in X:\|x\|\_X=1\}$ of the normed space $(X,\|\cdot\|\_X)$ as a metric space, and the set $\Sigma\_2(S\_X)$ of all spheres of radius equal to the diameter $2$ of $S\_X$. This is both a metric space with the Hausdorff distance, and a POS with the inclusion relation, and these structures are clearly an isometry invariant of $X$. We can further consider the subspace $\Sigma^\*\_2(S\_X)$ of all maximal elements of $\Sigma\_2(S\_X)$ w.r.to inclusion. For instance, in the case of $X:=\ell^2\_\infty(\mathbb C)$, denoting by $\Delta$ the closed unit disk of $\mathbb C$, the sphere of radius $2$ and center $(x,y)\in S\_X$ is either $\{-x\}\times\Delta$, or $\Delta\times\{-y\}$, or $\big(\{-x\}\times\Delta\big)\cup\big(\Delta\times\{-y\}\big)$ according whether $|y|<|x|=1$, or $|x|<|y|=1$, or respectively $|x|=|y|=1$. In particular, maximal spheres of radius $2$ are exactly those of the third type, that is with center $(x,y)\in\partial\Delta\times\partial\Delta$. Since these spheres are homeomorphically parametrised by their centers, we conclude that in this case the space $\Sigma^\*\_2(S\_X)$ is topologically a torus $\mathbb S^1\times\mathbb S^1$. Note that this easily generalises for all $\ell\_\infty^d(\mathbb C)$, in particular for $d=3$: now the maximal spheres of radius $2$ in $S\_{\ell\_\infty^3(\mathbb C)}$ are exactly those of center $(x,y,z)\in\partial\Delta\times\partial\Delta\times\partial\Delta$.


With some more computations, yet by elementary arguments, it is also easy to compute this object for the given complex space $Y:=\text{span}\big((1,0,a),(0,1,b)\big)\subset \mathbb C^3$ as defined above. We just have to look at the traces on $Y$ of the inclusion of spheres of radius $2$ in the unit sphere of $S\_{\ell\_\infty^d(\mathbb C)}$: it follows from the assumptions on $a$ and $b$ that the maximal spheres of radius $2$ of $S\_Y$ are exactly those whose center $(x,y,z)\in Y$ has $|x|=|y|=|z|=1$. Moreover, for any $x\in\partial\Delta$ there are exactly $2$ distinct values of $y$ for which $(x,y,ax+by)\in Y$, which are of the form $y=\theta\_1 x$ and $y=\theta\_2 x$, for some complex numbers $\theta\_1\neq\theta\_2$ of unit modulus (reason: by the assumptions one has $|a+\theta b|\le 1$ for $\theta=1$ and $|a+\theta b|=|a|+|b|> 1$ for $\theta=\text{sgn}b /\text{sgn}a$, so the circle of center $a$ and radius $|b|$ has exactly $2$ intersections with the unit circle $\partial \Delta$, namely $|a+\theta\_1b|=|a+\theta\_2b|=1$ for $\theta\_1|=|\theta\_2|=1$, so that also $|ax+(\theta\_1x)b|=|a+(\theta\_2x)b|=1$ for any $|x|=1$). By similar arguments one checks that a strict inclusion of spheres of radius $2$ in $S\_{\ell\_\infty^3(\mathbb C)}$ gives a strict inclusion on the trace on the space $Y$, so that there are no other maximal spheres in $S\_Y$.


So the set of the maximal spheres now is a space homeomorphic to the union of the graphs of $\partial\Delta\ni x\mapsto \big(\theta\_j x, (a +b\theta\_j)x\big)$, ($j=1,2$),thus to $\mathbb S^1\times \mathbb S^0$.


**Rmk** As a variant, we may consider the length of $\Sigma\_2(S\_X)$ as a partially ordered set (the maximum cardinality of its chains). Then, by the above computations, one gets length $d$ for $\ell^d\_\infty(\mathbb C)$, and length $3$ for the space $Y$.


