# On some definitions involving C* algebra representations

I am trying to understand different definitions of equivalence of representations. Given an algebra $$\textbf{A}$$, given two different representations $$\pi_1$$ and $$\pi_2$$ on two spaces $$\textbf{B}(\textit{H}_1)$$ and $$\textbf{B}(\textit{H}_2)$$, respectively. There are the following definitions:

• $$\pi_1$$ and $$\pi_2$$ are said to be unitarily equivalent if there is a unitary opearator U such that U $$\textbf{B}(\textit{H}_1)$$ $$U^{-1}$$ = $$\textbf{B}(\textit{H}_2)$$. Otherwise, they are said to be not (unitarily) equivalent
• $$\pi_1$$ and $$\pi_2$$ are said to be almost equivalent if every normal state for $$\pi_1$$ is also normal for $$\pi_2$$ and viceversa
• $$\pi_1$$ and $$\pi_2$$ are said to be disjoint if there is no normal state for both representations

My doubts are the following:

1. If both $$\pi_1$$ and $$\pi_2$$ are irreducible, $$\pi_1$$ and $$\pi_2$$ are unitarily equivalent $$\iff$$ $$\pi_1$$ and $$\pi_2$$ are almost equivalent?
2. $$\pi_1$$ and $$\pi_2$$ are unitarily inequivalent $$\iff$$ $$\pi_1$$ and $$\pi_2$$ are disjoint?

Update: Theorem 2.4.26 from Bratteli-Robinson vol. I proves that almost equivalence $$\iff$$ existence of a isomorphism relating $$\pi_1$$ and $$\pi_2$$ as pointed out by @QuantumFieldMedalist. Moreover, it also proves that $$\pi_1$$ and $$\pi_2$$ are irreducible and almost equivalent $$\iff$$ unitarily equivalent
Unitary equivalence of representations is a stronger statement than almost equivalence. To see this, for two unitarily equivalent representations $$\pi_{1}, \pi_{2}$$, any normal state on one is unitarily related to a normal state on the other. Almost equivalence is weaker as the relation between the normal states is not necessarily implementable by a unitary.
Disjoint representations $$\pi_{1}, \pi_{2}$$ occur when the representations are not even quasi-equivalent. This condition means that there is no *-isomorphism between $$\pi_{1}(A)'', \pi_{2}(A)''$$. A unitary is only a special type of *-isomorphism, so two representations need not be disjoint if they are not unitarily equivalent.