I'm not sure about how much of this goes through for general Banach algebras, but I can talk about the $C^*$ case.
For C$^*$-algebras (including $B(H)$), we can try to generalize the character space by using representation theory. If $A$ is an abelian C$^*$-algebra, then any character gives an irreducible representation of $A$. In fact, the character space is precisely the collection of all irreducible representations of $A$. So to generalize the character space (also sometimes called the spectrum of $A$), we gather up all of the irreducible representations of our algebra ($B(H)$ for example), and study that (after suitably topologizing it).
We can also study a related space, called the primitive spectrum of $A$, where we collect all of the kernels of irreducible representations instead of the representations themselves. Obviously there's a surjection from the spectrum of $A$ to the primitive spectrum, since we can just associate to each irreducible representation its kernel.
For commutative C$^*$-algebras, the character space, spectrum, and primitive spectrum all agree. So we really are generalizing the character space, at least in the C$^*$ setting.
One nice thing that we learn from Gelfand duality is that we can fully recover a commutative C$^*$-algebra $A$ from its spectrum, so the spectrum plays the role of a sort of dual object to $A$. Unfortunately, we can't recover C$^*$-algebras in general from their spectra: The spectrum of $K(H)$ is just a single point. In particular, we can't distinguish $M_n(\mathbb{C})$ from $M_m(\mathbb{C})$ by using their spectra.
For more, check out the wikipedia page here.