I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces.
For a Banach space $X$, we investigate its dual $X'$ and double dual $X''$. Some times these spaces give us information about $X$, eg. the embedding of $X$ in $X''$ gives notions such as reflexivity.
In Gelfand theory, we impose more algebraic structure on the space, namely, the object is algebras $A$. Now the dual space become the spectrum $\sigma(A)$ and the embedding becomes the Gelfand transformation $\Gamma:A\to C(\sigma(A))$.
Thus I wonder whether there is similar notion like reflexivity, that is, $\Gamma(A)=C(\sigma(A))$ and whether this gives some information about the algebra.
I have not gotten time to look into this problem myself. But intuitively such case should be rare for each step in $A\to\sigma(A)\to\Gamma(A)$ we lose some thing more than in the case for Banach spaces. But if $\Gamma(A)=C(\sigma(A))$ for some special $A$, it seems to tell a lot.