(Context: my lecturer's notes on Von Neumann's mean ergodic theorem)
Given a measure-preserving map $T$ on $(X,\mathcal{A},\mu)$ and an $L^2(X,\mathcal{A},\mu)$ function $f$, we define the linear isometry $U:L^2(X,\mathcal{A},\mu)\to L^2X,\mathcal{A},\mu$ by letting $f\mapsto f\circ T$. We then define $W=\{\phi-U\phi:\phi\in L^2(X,\mathcal{A},\mu)\}$, called the subspace of co-boundaries.
What is this subspace of co-boundaries? Is it a common construction in analysis? What is the motivation behind the name? Googling gave me nothing, and the only place where I have heard the term co-boundaries before is in algebraic topology, however I don't immediately see the connection with this.
This language just comes from group cohomology. If you have a linear isometry $ U $ acting on your vector space, this naturally gives a group action of $ \mathbb Z $ on your vector space in the usual manner (by taking powers of your operator), and since your vector space then becomes a $ \mathbb Z $-module you can form the chain complex of group cohomology and look at the coboundary groups. The group $ B^1(\mathbb Z, V) $ of coboundaries where $ V $ is your function space is then equal to the subspace generated by the maps $ f_{\phi} : U \to \phi - U \phi $ for all $ \phi \in V $, which in this case you can canonically identify with the element $ \phi - U \phi $ in $ V $.