TITLE: Characterisation of finite dimensional C*-algebras?
QUESTION [7 upvotes]: $\DeclareMathOperator\Spec{Spec}$Let $A$ be a finite dimensional $*$-algebra over $\mathbb C$.
(Namely, an associate algebra equipped with an involution $*:A\to A$ satisfying  $(ab)^*=b^*a^*$ and $(\lambda a)^*=\bar\lambda a^*$.)

Assume that for $\forall a\in A$ we have $\Spec(a^*a)\subset\mathbb R_+$. Does it follow that $A$ is a C*-algebra?

Here, the spectrum $\Spec(x)$ of an element $x$ is the set of scalars $\lambda\in \mathbb C$ such that $x-\lambda$ is not invertible.

REPLY [7 votes]: Let $V$ be a complex vector space equipped with an involutive anti-linear star operation (e.g. a C*-algebra whose multiplication has been forgotten).  Equip $V$ with the identically zero multiplication, namely
$xy=0$
for all $x$ and $y$ in $V$.  Then the unitization of $V$ is a counter-example.  In fact, every element $a$ of $V$ is nilpotent so $\text{spec}(a) = \{0\}$.  Consequently the spectrum of any element of the form $a-\lambda$ is $\lambda$ from where one easily checks the required condition.
However $a^*a=0$ for every $a$ in $V$, so $\tilde V$ cannot possibly be a C*-algebra.