Can you help me?
1: We know that $M_n(\mathbb{C}) $ is a Banach algebra.
a) Can we say the set of all upper triangular(down triangular) is a subalgebra of $ M_{n}(\mathbb{C}) $, but not an ideal?
b) suppose $ D=\{ (a_{ij} ) \in M_{n} (\mathbb{C}) :\ (a_{ij} ) = 0 , i \neq j \} $, so is $ D $ a subalgebra of $ M_{n} (\mathbb{C}) $, but not ideal?
2: Let $\mathbb D = \{ \lambda \in \mathbb{C} : | \lambda | < 1\} $, $ \mathbb T = \{ \lambda \in \mathbb{C} : | \lambda | = 1\} $, and $ K = \{ f \in C (\overline{\mathbb D}) : \quad f(\lambda) = 0 , \forall \lambda \in\mathbb T\}. $ Show that $ K $ is a closed ideal of $C (\overline {\mathbb D}) $?
(if $ \Omega $ is a compact Hausdorff space, the set $ C(\Omega) = \{ f : \Omega \longrightarrow \mathbb{C} :\ f\ \text{ is continuous on} \Omega\}$ is a algebra)
a) Sums and products of upper triangular matrices are again upper triangular, so the set of upper triangular matrices is indeed a subalgebra of $M_n(\mathbb C)$. It is not an ideal.
b) Indeed, $D$ is also a subalgebra, and also not an ideal. In fact, $M_n(\mathbb C)$ has no nontrivial two-sided ideals.
The argument for 2) is straightforward: if $f\in K$ and $g\in C(\overline{\mathbb D})$, then $(fg)(\lambda)=f(\lambda)g(\lambda)=0$ for any $\lambda\in\mathbb T$, since $f(\lambda)=0$.