Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{n \times n})$ denote the subalgebra of all $\phi \in \mathcal{L}(\mathbb{C}^{n \times n})$ which satisfy $$ \phi(U^*AU) = U^*\phi(A)U $$ for all $A \in \mathbb{C}^{n \times n}$ and all unitary $U \in \mathbb{C}^{n \times n}$ (in other words, $\mathcal{C}$ is the commutant of the set of all conjugations by unitary matrices).

**Question.** Is there an explicit description of $\mathcal{C}$?

Of course, there is some freedom for interpretation of the word "explicit"; I would be most happy with a set of mappings in $\mathcal{L}(\mathbb{C}^{n \times n})$ which spans $\mathcal{C}$.

**Remarks:**

Clearly, the identity $\operatorname{id}_{\mathbb{C}^{n \times n}}$ is an element of $\mathcal{C}$.

The operator $\tau: \mathbb{C}^{n \times n}\to \mathbb{C}^{n \times n}$ given by $$ \tau(A) = \operatorname{tr}(A) \cdot \operatorname{id}_{\mathbb{C}^{n \times n}} $$ is an element of $\mathcal{C}$ (where $\operatorname{tr}(A)$ denotes the trace of the matrix $A$).

The span of $\operatorname{id}_{\mathbb{C}^{n \times n}} $ and $\tau$ is a subalgebra of $\mathcal{C}$ (since $\tau^2 = n\tau$), but I don't know whether $\mathcal{C}$ is larger than this span.