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$$C^\ast$$-algebras isomorphically representable on $$l^p$$. (English) Zbl 07324213
A Banach algebra $$A$$ is (isometrically) representable on some Banach space $$X$$ if there exists an (isometric) isomorphism between $$A$$ and a closed subalgebra of the bounded, linear operators on $$X$$.
It is well known that every C*-algebra is isometrically representable on some Hilbert space. On the other hand, it was shown in [E. Gardella and H. Thiel, Math. Z. 294, No. 3–4, 1341–1354 (2020; Zbl 1456.46042)] that a C*-algebra is isometrically representable on some $$L_p$$-space for $$p\in[1,\infty)\setminus\{2\}$$ if and only if the C*-algebra is commutative.
Turning to isomorphic representability on $$L_p$$-spaces, using that the Hilbert space $$\ell_2$$ is isomorphic to a complemented subspace of $$L_p=L_p([0,1])$$, we see that every C*-algebra that is representable on $$\ell_2$$ (in particular, every separable C*-algebra) is also representable on $$L_p$$, for every $$p\in[1,\infty)$$.
This paper complements these results by showing that a C*-algebra is representable on the $$L_p$$-space $$\ell^p(J)$$ for $$p\in(1,\infty)\setminus\{2\}$$ and some set $$J$$ if and only if the C*-algebra is residually finite-dimensional. It remains open if the result also holds for $$p=1$$.
##### MSC:
 46L05 General theory of $$C^*$$-algebras 46H20 Structure, classification of topological algebras 46H15 Representations of topological algebras
##### Keywords:
$$l^p$$ space; $$C^\ast$$-algebra
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