Let us consider the class of Banach algebras with homomorphisms that are bounded below but not necessarily isometric.

  1. Is there a separable Banach algebra that contains isomorphic images of all separable Banach algebras?

  2. Is there a commutative separable Banach algebra that contains copies of all commutative separable Banach algebras?

The trick with bounding the distance between commuting projections (of arbitrary norm) from below by 1 does not work in either case.

  • 1
    $\begingroup$ In math.yorku.ca/~ifarah/Ftp/2008a11-problems.pdf Ilijas Farah remarks that a result of Junge and Pisier implies that the $C^*$ version of your question (1) has a negative answer. OTOH, the the $C^*$ version of your question (2) has an affirmative answer. (The continuous functions on the Cantor set is universal since every compact metric space is a continuous image of the Cantor set.) $\endgroup$ – Bill Johnson Apr 19 at 22:01

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