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We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its Banach space structure.

Clearly, finite-dimensionality and other cardinality properties can be detected. Thus, to avoid these cases, let us restrict to separable, infinite-dimensional $C^*$-algebras.

For commutative $C^*$-algebras the problem has been solved by Milutin. In particular, $C(X)$ and $C(Y)$ are isomorphic as Banach spaces for any uncountable, compact, metric spaces $X$ and $Y$.

In the paper
Hamana. On linear topological properties of some $C^*$-algebras, Tohoku Math. J., II. Ser. 29, 157-163 (1977).
it is shown that the Banach space structure also reflects if all irreducible representations of the $C^*$-algebra are finite-dimensional.

Let us ask what happens at the 'opposite' end of the scale:

Question: Are all simple, separable, infinite-dimensional $C^*$-algebras isomorphic as Banach spaces?

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    $\begingroup$ Hi Hannes, Kirchberg showed in his subalgebras of CAR algebra paper that all separable nuclear, non Type I C*-algebras are isomorphic as operator spaces (much stronger than just isomorphic as Banach spaces). I think Christensen and others also did some things in this direction for von Neumann algebras: those results are in Pisier's operator space book, but my copy isn't with me. $\endgroup$
    – Caleb Eckhardt
    Commented Feb 4, 2014 at 3:52
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    $\begingroup$ A complement to Caleb's remark: Being type I is invariant under a Banach space isomorphism (at least in the separable case). This follows from Haagerup--Rosenthal--Sukochev's classification of noncommutative $L^1$-spaces up to Banach isomorphism. $\endgroup$
    – Narutaka OZAWA
    Commented Feb 4, 2014 at 8:49
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    $\begingroup$ By Szankowski (Acta Math 1981), there is a separable (simple, unital, etc.) $C^*$-algebra which does not have the approximation property. Such a $C^*$-algebra cannot be Banach isomorphic to a nuclear one. $\endgroup$
    – Narutaka OZAWA
    Commented Feb 4, 2014 at 8:57
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    $\begingroup$ Hannes, this ma.utexas.edu/users/rosenthl/pdf-papers/93.pdf survey article of Rosenthal gives a nice overview of the Banach/operator-space structure of C*-algebras. $\endgroup$
    – Tomasz Kania
    Commented Feb 5, 2014 at 15:32
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    $\begingroup$ It is an open problem whether $C^∗_r(F_2)$ is Banach isomorphic to a nuclear $C^*$-algebra (say the CAR algebra $B$ as a model). It is known that (1) the dual of $B$ has the bounded approximation property (because it's AFD), (2) $B$ has a Schauder basis (Junge--Nielsen--Ruan--Xu, Adv. Math. 2004), and (3) $B\cong B\otimes_\alpha B$ for some reasonable tensor product (in this case the spatial tensor product). None of these properties is known for $C^∗_r(F_2)$. $\endgroup$
    – Narutaka OZAWA
    Commented Feb 6, 2014 at 8:33

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