Follow up on subspectra: is the restriction a subalgebra This is a question that came up after thinking about one of my previous questions here.
My question is: If we consider the algebra $A$ of continuous linear operators $u: X \to X$ where $X$ is some Banach space can the algebra $A|_C = \{u|_C: u \in A\}$ where $C$ is some closed subset of $X$ and $u|_C$ denotes the restriction of $u$ to $C$ be viewed as a subalgebra of $A$ somehow? Basically I am asking if there exists some injective algebra homomorphism $\varphi : A |_C \hookrightarrow A$. I tried to produce such a homomorphism but I don't quite see how to make it injective.

I don't think you can expect anything like that if you just require $C$ to be a subset. Not even when it is a subspace, because you need $$ (ab)|_C=a|_C\,b|_C.$$ This requires $C$ to be an invariant subspace for all operators, which would never happen. 
This of course doesn't preclude the possible existence of some other homomorphism, but I wouldn't know how to address that. 
