Let M be a finite dimensional von Neumann algebras with a normal faithful trace. Let e and f be two projections with rank 1. I want to know if e and f have identical traces. (This is obviously true if M is a factor.)
I guess it is false, while i have no counterexample.
By the Artin–Wedderburn theorem, $$M=M_{n_1}(\mathbb{C}) \oplus \ldots \oplus M_{n_k}(\mathbb{C})$$ for some natural numbers $n_j$. Denote by ${\rm tr}_{n_j}$ the normalised trace on $M_{n_j}(\mathbb{C})$. Now take any convex linear combination $\psi=\sum_j \lambda_j {\rm tr}_{n_j}$ with non-zero coefficients. Then $\psi$ is a n.f. trace yet the value of $\psi$ on a rank-one projection $p$ is nothing but $\frac{1}{n_j}\lambda_j$ where $j$ depends on the full matrix algebra that $p$ belongs to.
Your guess is correct. You can only say $e$ and $f$ have the same trace if your $M$ is a factor. To see this, any finite dimensional $M$ is isomorphic to a direct sum of full matrix algebras, say $$ M = \bigoplus_{j=1}^k M_{m_j}(\mathbb{C}). $$ By uniqueness of the normalized trace on $M_n(\mathbb{C})$, we see that normalized faithful traces on $M$ are in bijective correspondence with vectors $(t_1,\dots, t_k)$ with strictly positive entries such that $$ (m_1,\dots, m_k) \cdot (t_1,\dots, t_k) = \operatorname{tr}(1) = 1. $$ Note that $t_j$ is the trace of a minimal projection in the summand $M_{m_j}(\mathbb{C})$ of $M$. (Here, I've used the notation from Jones' Index for subfactors [MR696688], Section 3.2.)
So now we can build a concrete counterexample. Let $M = M_2(\mathbb{C}) \oplus \mathbb{C}$, with trace vector $\vec{t} = (3/8,1/4)$. Let $e$ be a minimal projection in the $M_2(\mathbb{C})$ summand, which has trace $3/8$, and let $f$ be the minimal projection in the $\mathbb{C}$ summand, which has trace $1/4$.
