\(C^\ast\)-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 07179013
[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
[6] ; Lindenstrauss, Classical Banach spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, 92 (1977)
[7] 10.4064/sm-19-2-209-228 · Zbl 0104.08503
[8] 10.1215/S0012-7094-66-03346-1 · Zbl 0171.11503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.