Let $A$ be a unital involutive algebra over $\mathbb{C}$. A $C^*$-seminorm is a seminorm $p$ such that $p(x^*x) = p(x)^2, \forall x \in A$.
I understand that if the spectral radius of $x \in A$ given by $\sup \{ |\lambda| \in \mathbb{C} : x - \lambda \mathbb{1} \text{ is not invertible} \}$ is a seminorm (i.e. real-valued) then this is a $C^*$-seminorm on $A$.
I also understand that if $\pi$ is a unital $^*$-representation of $A$, then $x \rightarrow || \pi(x)||$, where $|| \cdot||$ is the operator norm associated with $\pi$, is a $C^*$-seminorm on $A$.
But what about the case where the spectral radius of $A$ is not real valued? Does there still exist a non-trivial $C^*$-seminorm on $A$? Or does there always exist a non-trivial $^*$-representation of $A$?
It appears the answer is negative. In Fragoulopoulou's Topological Algebras with Involution Section 14.(2), for example, a unital $^*$-algebra with no non-trivial positive linear forms is given. Since every $^*$-representation is associated with a positive linear form, this $^*$-algebra has no non-trivial $^*$-representations. And since every non-trivial $C^*$-seminorm gives a non-trivial $^*$-representation (see here), then this algebra must not admit any non-trivial $C^*$-seminorms.