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… need not be $1$, but it shouldn't be larger; remarks about this are now at Banach algebra (and also at JB-algebra).
I’m afraid this is exactly the wrong way round! A Banach algebra, perhaps obtained by some renorming process, or by taking a certain ideal, can easily have an identity element of norm greater than 1. (Just take an idempotent 2 by 2 matrix which is not self-adjoint.) In fact, once you get onto the subject of bounded approximate identities, it can be quite natural to consider such things which may have norm 2, for instance (maximal ideals inside unital commutative Banach algebras sometimes have this property).
Toby’s calculation should have observed that the norm of 1 is less than or equal to the square of the norm of 1. This shows that the norm of the identity element is either zero or is greater than or equal to one.
Yes, one can calculate that the norm of $1$ is (if nonzero) at least $1$. And one should (?) require that the norm of one is at most $1$. The conclusion is that the norm of $1$ is (if nonzero) exactly $1$.
The requirement that ${\|1\|} \leq 1$ is a direct analogue of the requirement that ${\|x y\|} \leq {\|x\|}\,{\|y\|}$. The latter requirement ensures that one has a semigroup object in the closed monoidal category $Ban$ of Banach spaces with short linear maps and the usual hom-space of bounded linear maps (and therefore with the projective tensor product); the former requirement additionally ensures that one has a monoid object.
It is therefore interesting that there are natural Banach algebras (semigroups in $Ban$) that are unital as algebras (so monoids in $Vect$) but violate ${\|1\|} \leq 1$ (so are not monoids in $Ban$). I know that if ${\|x y\|} \leq {\|x\|}\,{\|y\|}$ is violated (but only in a bounded way), then one sometimes fixes this by rescaling the norm; does one ever do rescaling to ensure ${\|1\|} \leq 1$? (Of course, these two rescalings may ruin each other!)
Thanks, Toby; on re-reading, I see I misunderstood what you were getting at. Sorry!
Yes, rescaling to ensure $\Vert 1 \Vert\leq 1$ is very standard, and it can also be done while rescaling to ensure the norm is submultiplicative. I think the standard way to do it would be to let $A$ act on itself, and then check that it is topo-algebraically isomorphic to its image in $B(A)$. What this might do is to mess up other structure, for instance if the original algebra had underlying Banach space $\ell^1$ then this is such a nice norm in many ways that one doesn’t always want to distort or smooth it. Or, if your algebra arises as an ideal in something with a very canonical norm, one doesn’t always want to renorm the ideal without renorming the parent algebra.
I have added a hastily typed example to the Banach algebra page, to illustrate the phenomenon.
It is true that in most books one reduces to the case of a “genuine” monoid object in $Ban$, because then all the basic theory can be developed and stated in a much cleaner way. (One exception, if I recall correctly, is an old book of Zelazko where he takes some pains to give proofs that work without these restrictions.) I guess this comes back to the valuable point, which I know people on the nLab have made before, that really “Banach algebra people” are studying “topological algebras which happen to be isomorphic qua top. algebras to Banach algebras”, but the elision is so convenient and ingrained for analysts that we ignore the distinction.
Support for your terminology: in the sources I was taught from, the phrase “unital Banach algebra” expressly means that the norm is submultiplicative and the algebra is not zero and the norm of the identity element is exactly $1$. I don’t know if this is still the most usual use of the term, though.
Heh, well I would say that they are wrong to require that the algebra be non-zero (and hence wrong to require the norm to be exactly $1$ instead of $\leq 1$). But it is also what I was taught.
the elision is so convenient and ingrained for analysts that
Something is missing here.
But I would rather say that Banach algebra people should be studying Banach algebras as such (that is monoids in Ban, or semigroups if non-unital), and if they're not then they're doing something wrong. But that may not be true!
BTW, I'm perfectly happy with your edit about Banach coalgebras. (You might want to edit Banach coalgebra too!)
I will try to get round to editing the Banach coalgebra entry, but then I’ve been saying “I will try to get round to…” for several years now :)
While in the mode of general chat: as for what Banach algebra people should be studying… I think that this would be unduly prescriptive. We start with “monoids in Ban” (or semigroups) but then we are very interested – perhaps even more interested – in how these objects interact when viewed as “monoids in TVS”.
I think looking at monoids in the category of Banach spaces and bounded linear maps does, somehow, have some value beyond looking at certain monoids in the category of LCTVS and continuous linear maps, but it’s hard for me to pin down why choosing a norm feels useful. It could just be a crutch, like choosing a basis for a vector space, but I am not sure it is merely that.
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