$\DeclareMathOperator\Spec{Spec}$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?
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> 
> 


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.



Characterisation of finite dimensional C*-algebras?
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.


