# Relationship between weak topology and Gelfand topology (Banach space theory)

I don't have much background in functional analysis, so wanted to check whether my thinking about the weak topology and Gelfand topology is correct when it comes to C* algebras/Banach spaces. My understanding is that:

• The weak topology is defined on a Banach space $$X$$, and is the weakest topology such that every element of the dual space $$f\in X^*$$ is continuous.
• On the other hand the Gelfand topology is defined on commutative C* algebras (which are Banach spaces)- say the algebra A- and is such that the subspace of the double dual A** corresponding to the bounded linear maps $$\hat{x}:S(A)\rightarrow \mathbb{C}$$ where $$S(A)$$ is the space of all continuous homomorphisms of A, $$\phi : A \rightarrow C$$, $$\phi(ab)=\phi(a)\phi(b)$$, with the maps defined using the canonical isomorphism $$\hat{x}(\phi)=\phi(x)$$, every such map $$\hat{x}$$ in this subspace of A** is continuous.

Thus, it seems to me that the Gelfand topology is contained in the weak topology, and as such is weaker. Because it only requires a subspace of the double dual A** to be continuous, when acting on a subspace of A*. Namely the subspace of linear homomorphisms of A, and not merely the linear functionals on A.

These two things do not really mix much. The weak topology is defined on a topological vector space: you need a vector space to be able to define linear functionals. The weak$$^*$$ topology makes sense only on duals.
The Gelfand topology is used to give a topology to the set of characters of an abelian C$$^*$$-algebra $$A$$ (it can be done for a Banach algebra, too). What you do is you consider the characters as a subset $$\Sigma$$ (not a subspace!) of the dual $$A^*$$ of $$A$$, and you endow $$\Sigma$$ with the relative weak$$^*$$-topology (which is simply pointwise convergence).
The weak$$^*$$ topology is nice because it makes closed balls compact, which is often very useful. In particular it makes $$\Sigma$$ above compact when it's closed (which is precisely when $$A$$ is unital), and locally compact in general.