Let $A$ be the C$^{*}$-algebra $C(X,M_{n})$, where $X$ is a compact, connected metric space. Let $$ E= \begin{pmatrix} 1 & 0 & \cdots & 0\\ 0 & 0 &\cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\cdots & 0 \end{pmatrix}. $$ Suppose $P\in A$ is a projection, which is Murray-von-Neumann equivalent to $E$. I.e., there is a $W\in A$ such that $P=WW^{*}$ and $E=W^{*}W$.
Can we conclude that $P$ and $E$ are unitarily equivalent?
This question is a a more general version of a question I posted a few days ago: Are These Projections Unitarily Equivalent?
The answer is no. Let $W : S^2 \to M_3(\mathbb{C})$ be the inclusion $$(x,y,z) \mapsto \begin{pmatrix}x & 0 & 0 \\ y & 0 & 0\\ z & 0 & 0\end{pmatrix},$$ so that $W \in M_3(C(S^2))$ is a partial isometry with $W^\ast W = E$ and $P := WW^\ast$ the projection corresponding to the complexified normal bundle of $S^2$ as the unit sphere in $\mathbb{R}^3$. Suppose, by contradiction, that $E$ and $P$ are unitarily equivalent, so that $P = UEU^\ast$ for some unitary $U \in C(S^2,U(3))$. Then $$ I - P = I-UEU^\ast = U(I-E)U^\ast, $$ so that $I-P$ and $I-E$ are unitarily equivalent, and hence that the corresponding vector bundles over $S^2$ are isomorphic. On the one hand, under Serre–Swan, the projection $I-E$ corresponds to the trivial subbundle $(\{0\} \times \mathbb{C}^2) \times S^2 \subset \mathbb{C}^3 \times S^2$, which has trivial top Chern class $c_2(\mathbb{C}^2 \times S^2) = 0$. On the other hand, under Serre–Swan, the projection $I-P$ corresponds to the complexified tangent bundle $T_{\mathbb{C}}S^2$ of $S^2$, but $$ \int_{S^2} c_2(T_{\mathbb{C}}S^2) = \int_{S^2}e(TS^2) = \chi(S^2) = 2 \neq 0, $$ so that $T_{\mathbb{C}}S^2$ has non-vanishing top Chern class, and hence can't possibly be trivial. It is possible to rephrase all this in operator-algebraic terms, but you'd need $K$-theory and $K$-homology to do so. To cut a long story short, the projections $I-P$ and $I-E$ define classes $[I-P]$ and $[I-E]$ in the $K$-theory $K_0(C(S^2))$ of $C(S^2)$, which you can show are distinct by computing their pairings with the appropriate class in the $K$-homology $K^0(C(S^2))$.