\(C^\ast\)-algebras isomorphically representable on \(l^p\) (Q2657024)

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 [\textit{E. Gardella} and \textit{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 \(p=1\).