$C(K)$ $C^*$-subalgebra of $C(K')$ if $K\subseteq K'$?

Question

Let $K, K'$ be compact Hausdorff spaces with $K\subseteq K'$. We obtain the $C^*$-algebras $C(K)$ and $C(K')$. I am wondering if we can think of $C(K)$ as a $C^*$-subalgebra of $C(K')$ in the sense that there is a $*$-embedding $C(K)\hookrightarrow C(K')$. Maybe it is obvious but I don't see how to extend a function $f\in C(K)$ suitably.

I appreciate any help.

Answer 1

Assuming the only embeddings we care about are extension operators then no, there need not be a multiplicative (linear) embedding. Note this:

If $T:C(K)\to C(K')$ is a multiplicative linear map with $(Tf)|_K=f$ for all $f$ then the Gelfand transform induces a continuous map $\hat T:K'\to K$ with $\hat T(k)=k$ for all $k\in K$.

If $K$ is the unit circle in the plane and $K'$ is the unit disk the no-retraction theorem says this is impossible.

Edit: A simpler and stronger counterexample, although less interesting: Say $K'$ is connected but $K$ is not. Then there is no embedding of $C(K)$ in $C(K')$, via an extension operator or otherwise, because $C(K')$ contains no non-trivial idempotent.