Is each finite dimensional $C^*$-algebra an $AW^*$-algebra?

Question

I know that a finite dimensional $C^*$-algebra is a $W^*$- algebra(von neumann algebra). But is it true for $AW^*$-algebras?

Answer 1

If $A$ is a finite-dimensional C$^*$-algebra, then it is a von Neumann algebra. And von Neumann algebras are AW$^*$.

If $A$ is a finite-dimensional AW$^*$-algebra, it is in particular a finite-dimensional C$^*$-algebra, so it is von Neumann.