## $$C^\ast$$-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 $$p=1$$.

### 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

Zbl 1456.46042
Full Text:

### References:

 [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
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