## Terminology: Banach spaces equipped with continuous associative product?

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This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach algebra/functional analysis books on my shelves and in the library. However, it strikes me that this is easily crowd-sourceable and that people may know of texts I am less familiar with. My reason for asking on MO rather than MSE is that I think it will get better answers here.

So: the usual definition of a Banach algebra is that it is a (complex) algebra equipped with a complete vector-space norm, such that $\Vert ab\Vert\leq \Vert a\Vert \Vert b\Vert$ for all elements $a,b$.

Now suppose we have a (complex) algebra $A$ equipped with a complete vector-space norm $\Vert\cdot\Vert$ and a constant $K>0$ such that $\Vert ab\Vert\leq K\Vert a\Vert \Vert b\Vert$ for all $a,b$. These are much rarer in the literature, most likely for the following reason: a standard exercise doled out to students is to show that there is an equivalent norm on $A$ for which multiplication is contractive, i.e. rendering $A$ (in this new norm) a Banach algebra in the usual sense. In this sense "one has nothing new".

However, in some joint work I am writing up, I am toying with the idea of working in this greater generality, in order to let certain technical functorial constructions have more natural formulations. (In a bit more detail, it is to do with certain homologically flavoured constructions for Banach algebras and Banach bimodules more naturally living in a world where multiplication need not be contractive.)

So my question is this:

do these kinds of algebra have a standard name, and where are the established sources for such terminology?

I have a dim recollection that they are given a name of their own in Zelazko's old book on Banach algebras, but I don't recall what the name was, and I can't find anything in Bonsall & Duncan.

Note: I am not after arguments as to what terminology should or should not be, or observations about one definition being a "semigroup object in Ban$_1$" while the other is a "semigroup object in Ban". Rather, I need some idea of whether one choice of terminology is standard, and hence least likely to cause confusion/irritation to the intended audience, should I decide to pursue this course.

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## 2 Answers

To my knowledge, something in the same vein was first (?) considered in: C. Le Page (1967), Sur quelques conditions impliquant la commutativité dans les algèbres de Banach, C.R.A.S. Paris, Ser. A-B, 265, A235-A237 (click). In any event, if really necessary, I would refer to a norm (of ring-like structures) with that property as a quasi-norm wrt multiplication. Indeed, the kind of algebras that you're considering are somehow related to quasi-normed algebras (cf. Wiki). But this is not the real answer that I would have liked to give.

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It sounded to me like Yemon's constant only appears in the product inequality, not in the triangle inequality, so "quasi-normed algebra" isn't exactly what he wants ... – Nik Weaver Jun 14 at 15:20
Sorry, I was a little bit sloppy. I'd better say that the kind of algebras considered by Yemon are reminiscent of quasi-normed algebras. Let me fix it. – Salvo Tringali Jun 14 at 15:44

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Yemon, I have used the term "weak Banach algebra" for such things. I don't think there is a standard term, though. I vaguely recall seeing people simply call them Banach algebras (probably in some older papers when the terminology in the subject hadn't really stabilized).

(I ran into this issue when dealing with the Lipschitz algebra $Lip_0(X)$ for $X$ a complete finite diameter metric space. You really want to use Lipschitz number as the norm, even though this only makes it a weak Banach algebra. There's no real penalty for doing this, and the advantage is that it allows you to identify $X$ isometrically with the normal spectrum of $Lip_0(X)$.)

Edit: I've just realized that this is what Gelfand meant by "normed ring". E.g., on the first page of his book Commutative Normed Rings (1960) he writes:

"A normed ring is a complex Banach space in which an associative multiplication is defined that is permutable with the multiplication by complex numbers, distributive with respect to addition, and continuous in each factor."

and there is a footnote which says "In another terminology, a Banach algebra."

A few pages in he proves that you can always achieve $\|xy\| \leq \|x\|\|y\|$ by renorming.

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Thanks for the update, Nik! Interesting how "working definitions" become fossilized as new axioms – Yemon Choi Jan 11 at 19:00
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