What Exactly is a Finite-Dimensional C$^{*}$-Algebra?

Question

I am aware that if $A$ is a finite-dimensional C$^{*}$-algebra, then $A$ is isomorphic to

$$ M_{n_{1}}(\mathbb{C})\oplus\cdots\oplus M_{n_{k}}(\mathbb{C}) $$

for some positive integers $k$ and $n_{1},\ldots,n_{r}$. But what exactly is the precise definition of finite-dimensional here? Does it simply mean that there are finitely many $a_{1},\ldots,a_{m}$ with $A\cong C^{*}(a_{1},\ldots,a_{m})$?

Silly question, I know, but thank you.

Answer 1

Recall that a $C^*$-algebra is a vector space. Of course, it has the algebraic properties of multiplication and the $*$-involution and the topological properties to do with the norm, but beneath all that we are able to add two elements and multiply by a scalar.

A $C^*$ algebra is finite dimensional if, as a vector space, it has a finite basis.

Answer 2

"Finite-dimensional" just means finite-dimensional as a vector space over $\mathbb{C}$ (using the vector space structure over $\mathbb{C}$ that is part of the structure of a $C^*$-algebra). That is, there is a finite subset $S\subset A$ such that every element of $A$ is a $\mathbb{C}$-linear combination of elements of $S$.