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There are many Banach algebras which, as Banach spaces, are reflexive. Of course, unitisation is just adding one dimension so this operation preserves reflexivity, hence there are many reflexive, unital Banach algebras. On the other hand, it is an open question of whether reflexive, amenable Banach algebras are finite-dimensional. Hence my question,

Is there an infinite-dimensional, unital, reflexive Banach algebra which is moreover simple (i.e., it does not have non-trivial ideals)?

If there is one, most likely it will be non-amenable as otherwise one would have solved an old problem.

Update 25.04.2021: I have constructed a whole family parametrised by $p\in (1,\infty)$ of non-separable super-reflexive simple unital Banach algebras. Using some model-theoretic machinery, I can manufacture separable examples too. Once I have more time, I will post the construction here.

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    $\begingroup$ What is the reference for the old problem? $\endgroup$
    – Andreas Thom
    Commented Aug 9, 2016 at 18:08
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    $\begingroup$ @AndreasThom, it was raised in J. E. Gale, T. J. Ransford, and M. C. White, Weakly compact homomorphisms. Trans. Amer. Math. Soc. 331 (1992), 815–824. $\endgroup$
    – Tomasz Kania
    Commented Aug 9, 2016 at 18:09
  • $\begingroup$ Tomek, I guess you require no non-trivial ideals, and not merely no non-trivial closed ideals? $\endgroup$
    – Yemon Choi
    Commented Aug 9, 2016 at 18:14
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    $\begingroup$ @YemonChoi, proper ideals of unital Banach algebras have proper closures. So this is the same question. $\endgroup$
    – Tomasz Kania
    Commented Aug 9, 2016 at 18:15
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    $\begingroup$ Oops - I forgot the unital bit. Thanks $\endgroup$
    – Yemon Choi
    Commented Aug 9, 2016 at 18:16

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