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.