strictly positive element vs positive definite matrix

Question

If $A=\prod_{n=1}^{\infty}M_{k(n)}(\mathbb{C})$,$x=(x_1,\cdots,x_n,\cdots)$ is strictly positive in $A$,does this mean that each $x_n\in M_{k(n)}\mathbb{C}$ is a positive definite matrix?

Answer 1

In a unital C$^*$-algebar, "strictly positive" is the same as "positive and invertible", which is exactly "positive definite". So yes, if $x$ is strictly positive then each $x_n$ is positive and invertible, so positive definite.