\(C^\ast\)-algebras isomorphically representable on \(l^p\). (English) Zbl 1473.46057

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 \(p=1\).


46L05 General theory of \(C^*\)-algebras
46H20 Structure, classification of topological algebras
46H15 Representations of topological algebras


Zbl 1456.46042
Full Text: DOI arXiv


[1] 10.2140/pjm.2019.303.401 · Zbl 1503.47120
[2] 10.2307/1989630 · Zbl 0015.35604
[3] 10.1007/s00209-019-02315-8 · Zbl 1456.46042
[4] 10.1007/978-3-319-49847-8 · Zbl 1396.37001
[5] 10.2140/pjm.1958.8.459 · Zbl 0085.09702
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[7] 10.4064/sm-19-2-209-228 · Zbl 0104.08503
[8] 10.1215/S0012-7094-66-03346-1 · Zbl 0171.11503
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