# Properties of image of a projection under *-homomorphism into Tensor Product

Let $$A$$ be a finite dimensional $$\mathrm{C}^*$$-algebra.

Suppose that $$q$$ is a projection and $$T:A\rightarrow A\otimes A$$ is a *-homomorphism.

Suppose we write:

$$T(q)=\sum_{j=1}^n q_j\otimes p_j.$$

What properties do the $$q_j$$ and $$p_j$$ have? Is there some presentation of $$T(q)$$ such that

1. The $$p_j$$ and $$q_j$$ are projections?
2. The $$(p_j)$$ are linearly independent projections.
3. The $$(p_j)$$ and $$(q_j)$$ are linearly independent projections.
4. The $$(p_j)$$ are (mutually) orthogonal projections $$p_ip_j=\delta_{i,j}p_j$$.
5. The $$(p_j)$$ and $$(q_j)$$ are (mutually) orthogonal projections.

This is a question additional to this question.

Already the answer to 1 is "no". Let $$A=M_2(\mathbb C)$$. You can see $$A\otimes A$$ canonically as $$M_4(\mathbb C)$$ where $$a\otimes b=\begin{bmatrix} ab_{11}&ab_{12}\\ ab_{21}&ab_{22}\end{bmatrix}$$. Let $$T:A\to A\otimes A$$ be given by $$T\left(\begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix} \right)=\begin{bmatrix} a_{11}&0&0&a_{12}\\ 0&0&0&0\\ 0&0&0&0\\ a_{21}&0&0&a_{22}\end{bmatrix}.$$ Now let $$q=\tfrac12\,\begin{bmatrix} 1&1\\1&1\end{bmatrix}$$. You can check in this answer that $$T(q)$$ cannot be written as a sum of tensors of positive elements, let alone projections.