conditional expectations on $C^*$-algebras

Question

I am trying to get a feel for conditional expectations on arbitrary $C^*$-algebras but I am not able to find many examples. Obviously I can find conditional expectations from an $C^*$-algebra into $\mathbb{C}$ by taking linear functionals but i'd like to have examples to think about in general. What are some examples of conditional expectations to keep in mind? I am also particularly interested in the simple case so would be happy if you can give me some examples when we have an inclusion $A\subset B$ of $C^*$-algebras and a conditional expectation from $B$ into $A$ for $B$ simple.

Answer 1

(Caution: very loose answer, no effort in being technical here)

When $A$ is a II$_1$-factor, you have a conditional expectation, that preservers the trace, onto any von Neumann subalgebra $B$. Without details, this is basically considering $L^2(A,\tau)$, the GNS construction for the trace, and then you have the Hilbert space projection $e$ onto $L^2(B,\tau)$. With a bit of effort one can show that $e$ maps $A$ onto $B$, and that's the conditional expectation. Here, if $A$ and $B$ are II$_1$-factors, they are both simple (if non-separable as C$^*$-algebras).

In finite-dimension, you basically can only get conditional expectation by compressing ("pinching", if you are doing matrix analysis). If $A=M_n(\mathbb C)$ and $B\subset A$ is given by $\bigoplus_j p_jAp_j$, where $p_1,\ldots,p_n$ are pairwise orthogonal projections with sum 1, you get $E:A\to B$ by $E(a)=\sum_jp_jap_j$.

Let $G$ be a locally compact group, $\gamma$ the left-regular representation, and $A=C^*(\gamma(G))$ (this is the reduced group C$^*$-algebra). If $H\subset G$ is a normal subgroup, you can take $B=C^*(\gamma(H))\subset A$. You get a conditional expectation $E:A\to B$ the following way (one needs to check that this works fine for limits): $$ E(\sum_{g\in g}\alpha_g\,u_g)=\sum_{h\in H}\alpha_h u_h. $$

On tensor products, the slice maps (also, partial traces) are conditional expectations. That is, if $A=B\otimes C$, you get a conditional expectation $E:A\to B$ induced by $E(b\otimes c)=\psi(c)\,b$ for a state $\psi$ of $C$.