Uniqueness of conditional expectations defined on $C(\mathbb{T})$ Let $C(\mathbb{T})$ be the $C^*$-algebra of continuous functions on the one-dimensional torus $\mathbb{T}$. For a fixed natural number $n_0$, consider the $C^*$-subalgebra $\mathcal{B}\subset C(\mathbb{T})$ which is generated by the function
$f_{n_0}(z):=z^{n_0}$. It is easy to verify that $E[z^k]:=0$ if $k$ is not a multiple of $n_0$ and $E[z^k]:=z^k$ if $k$ is a multiple of $n_0$ can be extended to a faithful conditional expectation from
$C(\mathbb{T})$ onto $\mathcal{B}$. The question is whether this is the only conditional expectation
from $C(\mathbb{T})$ onto $\mathcal{B}$ or not.
 A: The conditional expectation it is not unique. Basically the different positive definite functions on the quotient group $\mathbb{Z}/k\mathbb{Z}$ will produce other conditional expectations.  Here is an elementary description of that idea.
Fix $k>1$ and consider the element in $M_k(C(\mathbb{T}))$ defined by
\begin{equation*}
U=\left[  \begin{array}{cccc}  0 & 0 & \cdots & z\\
                                    1 & 0 & \cdots & 0\\
                                    \vdots& \cdots &\ddots&\vdots\\
                                    0 & \cdots & 1 & 0  \end{array} \right]
\end{equation*}
Then $U$ is clearly unitary and $U^k=\textrm{diag}(z,z,\cdots,z).$  We identify $C^*(U)$ with $C(\mathbb{T}).$ Then $C^*(U^k)$ are precisely those elements of $C^*(U)$ that are constant on the diagonal. Now let $A\in M_k(\mathbb{C})$ be any positive matrix with $\textrm{Trace}(A)=1.$ Then for $x\in C^*(U)$ define
\begin{equation*}
\mathbb{E}_A(x)=\textrm{diag}(\textrm{Trace}(Ax),...,\textrm{Trace}(Ax))
\end{equation*}
Then $\mathbb{E}_A$ is a conditional expectation onto $C^*(U^k).$  Moreover $\mathbb{E}_A$ is the conditional expectation you defined if and only if $A$ is a diagonal matrix.
