The terminology can be confusing because in the phrases "state vector", "pure state", and "mixed state", the word "state" is being used in different ways.
The terms pure state and mixed state do not directly refer to a state vector in the Hilbert space, but to our state of knowledge about the physical system. If we say a system is in a pure state, that means that we can represent the system as a single state vector in Hilbert space.$^\star$ If we say the system is in a mixed state, it means that we do not know specifically what state vector represents the system; we can only assign probabilities for the system to be represented by any given state vector.
Pure states can be represented by a state vector in the Hilbert space. Mixed states cannot. On the other hand, while only a pure state can be represented by a state vector, both pure and mixed states can be represented by the density matrix.
The density matrix is an operator, not a state vector, by definition. A ket is a map that takes a bra and produces a complex number. A linear operator is a map that takes one bra and one ket and produces a complex number, and is a bilinear function of the bra and ket. The density matrix is the latter kind of mathematical object. This is clear from the usual way to represent the density matrix as an expansion over state vectors$^\dagger$
\begin{equation}
\hat{\rho} = \sum_a p_a |\Psi_a\rangle \langle \Psi_a|
\end{equation}
since given a bra $\langle b |$ and a ket $| k \rangle$, we see that $\langle b | \hat{\rho} | k \rangle$ is a complex number, and $\hat{\rho}$ is also bilinear function of the bra and ket.
The definitions of pure and mixed states have nothing in particular to do with $C^\star$-algebras, which are a way of formalizing the idea of an algebra of observables. You can understand the density matrix without that language.
$^\star$If the system is in a pure state, sometimes people will also refer to the state vector itself as the pure state. That's fine so long as the meaning is understood. In this answer, I want to try to clearly distinguish the concepts of a pure state, mixed state, and state vector, so I will intentionally avoid saying this.
$^\dagger$ Note this is not a definition of the density matrix, for example see this answer from Emilio Pisanty
