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Basic properties assuming the spectrum is non-empty
The following is fundamental.
Black Box #1: The spectrum is non-empty.
The idea of the proof is that if were empty, then the resolvent would be a non-constant “bounded holomorphic function,” which contradicts Liouville’s theorem. This proof actually works with no modifications for complex matrices and shows that they have eigenvalues without going through either characteristic polynomials or the fundamental theorem of algebra, and in fact this is one way to prove the fundamental theorem of algebra. In general we need to develop some complex analysis for Banach space-valued holomorphic functions, which we defer to a later post.
For now, it it is more important to see what can be done with this result.
Corollary (Gelfand-Mazur): Any Banach algebra which is a division algebra is isometrically isomorphic to .
Proof. Let be such a Banach algebra and let be nonzero. is non-empty, so let be such that is not invertible. Since is nonzero, it must be invertible, so , hence is not invertible. But then it must be equal to , so .
Corollary: Let be a commutative Banach algebra. Then if and only if there exists a morphism such that .
Proof. : if there exists such an , then lies in a maximal ideal, so cannot be invertible. (This argument works in any algebra over .)
: if , then is not invertible, hence is contained in a maximal ideal , which is already closed. Since by Gelfand-Mazur, the quotient map is the desired .
(Note that we make essential use of the axiom of choice here.)
A digression into quantum mechanics
The non-emptiness of the spectrum, besides leading to the above results, also has the following corollary, which is apparently due to Wielandt.
Proposition: The Weyl algebra does not embed into a Banach algebra.
Proof. Suppose to the contrary that are elements of a Banach algebra satisfying . We make the elementary observation that , from which it follows that
But we know that , so must be invariant under addition by . Since is non-empty, it must contain arbitrarily large elements, and this contradicts compactness.
This result is not needed for what follows, but it does imply that it is not possible to describe the position and momentum operators as elements of a Banach algebra. Equivalently, it is not possible to represent position and momentum as bounded operators on a Banach space. In their usual representation on , both position and momentum are unbounded (discontinuous) and densely defined (can only be applied to certain dense subspaces of ).
Fortunately, there is a way out: instead of dealing with position and momentum directly, we can deal with the strongly continuous one-parameter unitary groups they generate via Stone’s theorem (see also the Stone-von Neumann theorem). Indeed, these unitary groups are defined on by
for position and (up to normalization)
for momentum, both of which are defined everywhere on .
The Gelfand representation
Recall that for any commutative ring we can define a homomorphism
where the product runs over all maximal ideals of and is the image of in . This represents as a “ring of functions” on , although ideally all of the residue fields should be the same for this interpretation to really make sense, and the kernel of this homomorphism is the Jacobson radical .
We now know that if is a Banach algebra, we always have , so the above homomorphism lands in the space of functions . It’s natural to want to equip , the spectrum, Gelfand spectrum, or Gelfand space of , with a topology such that the corresponding functions are all continuous. This is usually done by identifying an element of with the corresponding map , which is in particular a continuous linear functional, hence naturally sits as a subspace of . The initial topology on making all evaluations against elements of continuous is the weak-* topology, and the topology we want on is the subspace topology with respect to this topology.
(An equivalent way to define the weak-* topology, which makes sense for replaced by any topological vector space , is to think of as a subspace of the 