It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,

- For $k \ge 0$ and $K_n$ compact increasing with $\bigcup_n K_n =\mathbb{R}$,$$ \text{C}^k(K_1)\subseteq \text{C}^k(K_2) \subseteq \ldots \subseteq \text{C}^k(K_n) \subseteq \ldots \to \text{C}^k(\mathbb{R})$$ is the inductive limit of Banach spaces (an LB-space) where convergence is uniform in all $k$ derivatives on compact subsets. (It is also a Frechet space, but this is due to this explicitly construction, not for LB spaces in general)
- If $K\subseteq \mathbb{R}$ is compact then $$ \text{C}^\infty(K) \leftarrow \ldots \subseteq \text{C}^k(K) \subseteq \ldots \subseteq C^1(K)\subseteq C^0(K) $$ is the projective limit of Banach spaces (making it Frechet), where convergence is such that each derivative individually converges uniformly on $K$.
- Since we have $\text{C}^k(\mathbb{R})$ well defined for all $k \ge 0$, $$ \text{C}^\infty(\mathbb{R}) \leftarrow \ldots \subseteq \text{C}^k(\mathbb{R}) \subseteq \ldots \subseteq C^1(\mathbb{R})\subseteq C^0(\mathbb{R}) $$ is the projective limit of Frechet spaces (also a Frechet space), where convergence is such that each derivative individually converges uniformly on compact subsets of $\mathbb{R}$.
- For $K\subseteq \mathbb{R}$ compact, we take $\text{C}^k_c(K)$ to be the collection of $k$-continuously differentiable functions on $\mathbb{R}$ with support contained in $K$. Then $$ \text{C}^\infty_c(K) \leftarrow \ldots \subseteq \text{C}^k_c(K) \subseteq \ldots \subseteq C^1_c(K)\subseteq C^0_c(K) $$ is the projective limit of Banach spaces (making it Frechet), where convergence is such that each derivative individually converges uniformly on $K$. *We take $K_n$ compact increasing with $\bigcup_n K_n = \mathbb{R}$, $$ \text{C}^\infty_c(K_1)\subseteq \text{C}^\infty_c(K_2) \subseteq \ldots \subseteq \text{C}^\infty_c(K_n) \subseteq \ldots \to \text{C}^\infty_c(\mathbb{R}) $$ which is the inductive limit of Frechet spaces (an LF space), where convergence is such that each derivative individually converges uniformly on compact subsets.

**My question is how can the algebraic structures of these spaces be inherited from these constructions?** For instance, $\text{C}^k(K)$ has a nice structure as a $C^*$-algebra: can inductive or projective limits inherit this structure in one way or another to form some kind of Banach/Frechet/LB/LF-algebra?