Is the Banach space $M_{n\times n} (\mathbb{C})$ with normal structure?

Question

Is the Banach space $M_{n\times n} (\mathbb{C})$ with normal structure?

I know the Banach space $\oplus_{1}^{n} \mathbb{C}$ is with normal structure but I can't fine a subspace of $M_n(\mathbb{C})$ without nondiametral point.

Answer 1

Every finite-dimensional Banach space has a normal structure because every non-trivial compact convex set contains a non-diametral point. See also Lemma 4.1 in Goebel & Kirk's Topics in Metric Fixed Point Theory.

More generally, there is an easy criterion for a reflexive space to not have a normal structure.

Suppose that $X$ is a reflexive Banach space. If $X$ fails to have a normal structure then you will find in $X$ a sequence $(x_n)_{n=1}^\infty$ of unit vectors that converges to 0 weakly and such that ${\rm diam}\{x_1, x_2, \ldots\} \leqslant 1$.