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How useful is it that the Gelfand representation is a left adjoint?

30 January, 2015

(The question won’t get answered in this post, but it has been bugging me sufficiently that I may as well throw the question online, admit my ignorance, and see if anyone has any suggestions or critiques.)

Recently, on MathOverflow, I offered the following example of an adjunction that comes up in the theory of commutative unital Banach algebras. Let CHff be the category of compact Hausdorff spaces and continuous maps between them; and let unCBA be the category of unital commutative Banach algebras, with the morphisms being the continuous unital algebra homomorphisms between the objects.

There is a functor C from CHffop to unCBA, defined on objects by taking C(X) to be the usual algebra of continuous complex-valued functions on X, and defined on morphisms in the obvious way. Years ago I remembered convincing myself that not only does the functor C have a left adjoint, but one can define/describe the left adjoint as being the functor \Phi: \hbox{unCBA} \to \hbox{CHff}^{\rm op} which assigns to a unital commutative Banach algebra A its character space \Phi_A. Here \Phi_A is defined to be the set of characters (=non-zero multiplicative functionals from A to the ground field \bf C), equipped with the relative weak-star topology that this set inherits from the dual Banach space A^*.

What started to nag at me, after mentioning this example on MathOverflow, is that nothing in this description seems specific to the choice of complex scalars; in other words, it looks like one would obtain the same corresponding adjunction if one worked with unital commutative Banach algebras over \bf R rather than over \bf C. The choice of complex scalars is important because without it one does not get the Gelfand-Mazur theorem, and without that one does not get the fact that all maximal ideals in unital commutative Banach algebras have codimension one, and without that one does not get the following key feature of the Gelfand representation {\cal G}_A: A \to C(\Phi_A):

if a\in A and ${\cal G}_A(a)$ is invertible in C(\Phi_A), then a is invertible in A.

So the question arises: just what does one get from knowing the Gelfand representation arises as a left adjoint? What traction does it give us on the well-known examples and theorems in the theory of commutative Banach algebras? (This has been on my mind on and off for several years, because there are various possible generalizations and extensions of the Gelfand representation, either by passing to the noncommutative world or by looking at more general classes of ideals, not just the maximal ones; and I had hoped that the “left adjoint” perspective could be used as a guide when examining which of these versions is going to lead to a good theory. But if the categorical perspective I’ve outlined above can’t lead us to Gelfand-Mazur, then perhaps a rethink is needed.)

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