In this question Yemon Choi asked whether there is a standard term for Banach algebras for which the submultiplicative law ($\|ab\| \leq \|a\| \|b\|$) is weakened to merely requiring the product to be continuous. When looking for an answer I discovered that Gelfand originally used the terms "normed ring" and "Banach algebra" in this weaker sense.

I'm working on a second edition of my book on Lipschitz algebras, and I still find that the "max" norm $\max(\|f\|_\infty, L(f))$ is the most appropriate one, despite failing to be submultiplicative.

It has been noted that the term "Banach algebra" is unfair to Gelfand, who is uncontroversially credited with developing their basic theory. How about using "Gelfand algebra" as a name for the above mentioned weakening of the (now standard) definition of Banach algebras? Are there other good options I'm not aware of?

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    $\begingroup$ As a side comment, to "assign blame": I always understood the term Banach algebra was first used in a 1945 paper by Ambrose (dx.doi.org/10.1090/S0002-9947-1945-0013235-8). Is that correct? $\endgroup$ Oct 15, 2015 at 14:01
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    $\begingroup$ @ChrisHeunen: it looks like you are right. I didn't know that. I have to say that my sense of injustice is inflamed by Ambrose's obvious failure to recognize the significance of Gelfand's monumental contribution. From a modern perspective the passage comparing and contrasting his obscure structure theorem with the GNS construction appears fairly ridiculous ... $\endgroup$
    – Nik Weaver
    Oct 15, 2015 at 14:32
  • $\begingroup$ Don't forget to apply Stigler's law of eponymy! $\endgroup$
    – BWW
    Oct 15, 2015 at 16:50


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