What is an Isomorphism Between Banach Algebras?

[.....]

I can't seem to find any sort of concrete definition anywhere... it always seems a bit hand wavy :(

In particular I want to know what is an isomorphism between two banach algebras?

Thanks.

 

[radou]

There should be very exact definitions to find on the internet. Roughly speaking, an isomorphism is a bijective mapping between algebraic structures which preserves the operations defined on the structures of interest in a way.

 

[Fredrik]

Disclaimer: I don't really know what I'm talking about here (but I still think I'm right).

It's easy to define isomorphisms in terms of homomorphisms:

A homomorphism $f:X\rightarrow Y$ is said to be an isomorphism if there exists a homomorphism $g:Y\rightarrow X$ such that $f\circ g=\mbox{id}_Y$ and $g\circ f=\mbox{id}_X$.

I don't think there's a way to define homomorphisms in general, i.e. in a way that works for all algebraic structures. Instead, the definition of an algebraic structure, along with a choice of what functions to call "homomorphisms", define a category. So the class of Banach algebras with one choice of "homomorphisms" would simply be a different category than the class of Banach algebras with a different choice of "homomorphisms".

This is of course very formal, and probably not very useful. I have e.g. never seen anyone define vector space homomorphisms as anything other than linear functions. There seems to be a standard choice of homomorphisms for each kind of mathematical structure, probably determined by a consensus about which choice is the most interesting and useful. In the case of Banach algebras, I think that choice should be the same as for normed algebras, and I think that choice should be "linear functions that also preserve products and norms of vectors".

 

[HallsofIvy]

An isomorphism is a one-to-one function from one algebraic structure to another that "preserves" the operations. That is, if u+v is an operation in one structure and x*y is the corresponding operation in the other, then f(u+ v)= f(u)*f(v).

Another way of looking at it is that two systems are "isomorphic" if and only if one is just the other "relabled". That is, if you have two stuctures and an isomorphism such that f(u)= x, the just changing the "name" 'x' to 'u' (and similarly for all other elements of the structure) would make the second structure identical to the first.

For more detail than that you will have to specify what kind of "isomorphism" you are talking about isomorphism of groups, or rings, or field, or what?

 

[Landau]

For more detail than that you will have to specify what kind of "isomorphism" you are talking about isomorphism of groups, or rings, or field, or what?

He did, namely a Banach Algebra.

A Banach algebra is an associative algebra and a Banach space. So you would probably expect a (homo)morphism to preserve addition, multiplication, ring multiplication and the norm (and perhaps preserve 1); an isomorphism would then be an invertible homomorphism.

But I am not sure whether this is the right notion. In the category of Banach Spaces, the morphisms are the bounded linear operators. The isomorphisms are then the surjective linear isometries. For a Banach Algebra, you'd expect that the multiplication has to be preserved too. I agree that a precise definition is hard to find on the internet.

 

[Fredrik]

I don't think there's a way to define homomorphisms in general, i.e. in a way that works for all algebraic structures. Instead, the definition of an algebraic structure, along with a choice of what functions to call "homomorphisms", define a category. So the class of Banach algebras with one choice of "homomorphisms" would simply be a different category than the class of Banach algebras with a different choice of "homomorphisms".

Can someone confirm that I got this part right, or explain how I got it wrong? To put it differently, is there a definition of "(homo)morphism" and "mathematical structure" (or whatever term is preferred), such that if I write down the definition of a mathematical structure that you've never seen defined before, you will be able to use the definition of "homomorphism" to tell me which functions I should call homomorphisms?

By "mathematical structure", I mean of course such things as groups, algebras, topological spaces, fiber bundles and so on, i.e. things that consist of a set or several sets, and a selection of something else associated with the set(s), like some functions, a set of subsets, or whatever.

I imagine that it would be quite difficult to define "structure preserving" in a general way. Consider e.g. fiber bundles. There are several sets involved, and the functions we want to think of as "structure preserving" are maps between the total spaces of two bundles that take fibers to fibers. This choice of what functions to call homomorphisms (actually the book I studied just called them "bundle maps") is pretty natural if we understand the reason why we define fiber bundles the way we do, but it doesn't seem to be forced upon us by the definition of fiber bundle.

 

[Landau]

Hi Fredrik,

To put it differently, is there a definition of "(homo)morphism" and "mathematical structure" (or whatever term is preferred), such that if I write down the definition of a mathematical structure that you've never seen defined before, you will be able to use the definition of "homomorphism" to tell me which functions I should call homomorphisms?

Yes, there is. In model theory one studies 'structures' in a precise way, and homomorphisms between them.

 

[dx]

Can someone confirm that I got this part right, or explain how I got it wrong? To put it differently, is there a definition of "(homo)morphism" and "mathematical structure" (or whatever term is preferred), such that if I write down the definition of a mathematical structure that you've never seen defined before, you will be able to use the definition of "homomorphism" to tell me which functions I should call homomorphisms?

By "mathematical structure", I mean of course such things as groups, algebras, topological spaces, fiber bundles and so on, i.e. things that consist of a set or several sets, and a selection of something else associated with the set(s), like some functions, a set of subsets, or whatever.

I imagine that it would be quite difficult to define "structure preserving" in a general way. Consider e.g. fiber bundles. There are several sets involved, and the functions we want to think of as "structure preserving" are maps between the total spaces of two bundles that take fibers to fibers. This choice of what functions to call homomorphisms (actually the book I studied just called them "bundle maps") is pretty natural if we understand the reason why we define fiber bundles the way we do, but it doesn't seem to be forced upon us by the definition of fiber bundle.

I think you're right, at least in the case of category theory. When you define a category, you must specify what the morphisms are, i.e. you must define what a morphism is in this specific category and also how morphisms must be composed.

If we take the objects in the category to always be sets with some kind of algebra or other structure defined on it, then it seems that a definiton of homomorphism can be constructued that applies for all such objects, as Landau pointed out in the case of model theory, which studies sturctures that are "sets along with a collection of finitary functions and relations which are defined on it".

 

[Fredrik]

Thanks guys. Landau, that link was pretty enlightening. It seems easy to apply the definition to e.g. groups and modules, but does it also work for topological spaces? The "structure" of a topological space isn't defined by a few functions and relations, but by a choice of which subsets to call "open". Hm, an idea appears in my mind as I'm writing this...maybe the solution is just to view each of those subsets as a 1-ary relation.

 

[dx]

The "structure" of a topological space isn't defined by a few functions and relations, but by a choice of which subsets to call "open"

How about this: we simply construct the auxiliary set 2^X, and require a ring like algebraic structure on the designated points of this set called open sets, where union and intersection take over the role of '+' and '-'.

 

[Fredrik]

I've been looking into this terminology issue. It seems that there's a difference between the terms "mathematical structure" and "algebraic structure". Let X be a set. An n-ary relation on X is a subset of Xⁿ. An n-ary operation on X is a function from Xⁿ into X. A constant in X is a member of X. Constants don't have to be mentioned separately, because we can think of them as nullary (0-ary) functions. (I think we can also think of the n-ary functions as n+1-ary relations, but no one seems to do that). A choice of a set X, along with a bunch of relations and functions of the type mentioned above (they can have different values of n), defines a mathematical structure. An algebraic structure is a mathematical structure without any relations. (I suppose we should add "...that aren't also functions").

A topological space is a mathematical structure, but not an algebraic structure, because subsets are unary (1-ary) relations.

What's bugging me now is the definition of "metric space". I expected that to fit into all of this, but it doesn't, since it involves a function $d:X\times X\rightarrow\mathbb R$. Is there some other concept that generalizes "mathematical structure" and includes metric spaces? I could easily define one, but I'm wondering if there's a term for the thing I would be defining.

 

[quasar987]

It looks like the Bourbaki's have had fun thinking about how to rigorously notions like mathematical structure and isomorphism of structures. In chapter IV of their set theory book, from what i can gather, they define those things based on the theory of ordered sets.

The google books preview has some pages missing but here it is:

http://books.google.com/books?id=IL...&resnum=1&ved=0CCYQ6wEwAA#v=onepage&q&f=false

The general notion of isomorphism is defined on page 265.