Hi, I am interested in the following question:

Fix $n$. Let $M_n$ be matrix algebra over the field of complex numbers with normalized trace $tr_n$. Let $M_n^{\omega}$ be an ultrapover of $M_n$, namely we consider the algebra of all bounded (in norm) sequences in $M_n$,say $l_{\infty}(M_n)$, and take a quotient of this space by sequences $(a_i)_{i\in\mathbb{N}}$ with $lim_{\omega} tr_n(a_i^*a_i)=0$. Is $M_n^{\omega}$ is finite-dimensional? What is the structure of $M_n$?

Also, if $F_i$ is $n$-dimensional Banach space, what is the dimension of the Banach space ultraproduct of $\{F_i\}$.