I'm working through Li's and Barlak's [Cartan Subalgebras and the UCT Problem](https://arxiv.org/pdf/1511.02697.pdf) but I'm stuck at one of the simpler proofs of the paper. On page 9 they deal with masas (maximal abelian subalgebras) of a C\*-algebra $A$ and look at the crossed products $A \rtimes\_\alpha \mathbb{Z}\_n$ where $\alpha\in Aut(A)$ with $\alpha^n=id$. I now have two questions:


1.) Before prop. 4.1 the paper says *"let $t$ be the unitary in (the multiplier algebra of) $A\rtimes\_\alpha \mathbb{Z}\_n$ implementing the $\mathbb{Z}\_n$-action."* I'm not quite familiar with theory of crossed products. What does this mean and why does such a unitary exist? (Do you have literature recommendations?)


2.) In the proof they compute $[t^k x\_k, b]$ and the result seems to be $t^k(x\_kb-\alpha^{-k}(b)x\_k)$. By construction of crossed products I would've guessed $[t^k x\_k, b]=t^k\left(x\_{k}\alpha^{k}\left(b\right)-bx\_{k}\right)$. Where's my mistake?



Crossed products and unitaries implementing $\mathbb{Z}_n$-actions
The unitary exists by the definition of the crossed product. In the case of a $\mathbb{Z}/n\mathbb{Z}$-action you can think of an element of the crossed product as a linear combination
$$
\sum\_{n = 0}^{n-1} a\_n t^n
$$
with $a\_n \in A$ and where $t$ is a unitary (i.e. $t^\* = t^{-1}$), such that the following relations hold 
$$
tat^\* = tat^{-1} = \alpha(a)
$$
In particular $t^kat^{-k} = \alpha^k(a)$ and $t^{-k}at^{k} = \alpha^{-k}(a)$. In the case $A = \mathbb{C}$ and $\alpha$ trivial, you obtain the group $C^\*$-algebra of $\mathbb{Z}/n\mathbb{Z}$ this way, which you might be more familiar with. Concerning the second question we have
$$
[t^kx\_k, b] = t^kx\_kb - bt^kx\_k = t^kx\_kb - t^k(t^{-k}bt^k)x\_k = t^k(x\_kb - \alpha^{-k}(b)x\_k)\ .
$$
Therefore their calculation seems to be correct. A good source of information about crossed products is [this book](https://books.google.co.uk/books?id=Fgr0BwAAQBAJ&printsec=frontcover&dq=crossed+product&hl=en&sa=X&ved=0ahUKEwjKibnI7eXVAhVBBMAKHQtRDfUQ6AEILjAB#v=onepage&q=crossed%20product&f=false) by Dana P. Williams.


Here are some more details about crossed products. Formally the crossed product is defined as follows: Let $G = \mathbb{Z}/n\mathbb{Z}$. Consider the algebra $C(G,A)$ of functions on $G$ with values in $A$ equipped with the following twisted multiplication
$$
(f \cdot h)(x) = \sum\_{g \in G} f(g)\alpha\_g(h(g^{-1}x))
$$
and the $\*$-operation
$$
f^\*(x) = \alpha\_x(f(x^{-1})^\*).
$$
This $\*$-algebra has several interesting faithful representations, for example a regular and a maximal one. These correspond to (possibly different) completions to $C^\*$-algebras, e.g. the reduced or maximal crossed product. 


To obtain the picture described above, we can then identify the unitary $t$ with the function 
$$
t(x) = \begin{cases} 1\_A & if x = 1 \\ 0 & else \end{cases}
$$
where $1 \in \mathbb{Z}/n\mathbb{Z}$ refers to the generator on the right hand side and $1\_A$ is the unit of $A$. Strictly speaking this only works if $A$ is unital. Otherwise $t$ will not be an element in the crossed product, but in its multiplier algebra. Then we have 
$$
t\cdot a \cdot t^\*(x) = \sum\_{g,h} t(g)\alpha\_g(a(h)\alpha\_h(t^{-1}((gh)^{-1}x))) = \alpha(a(x))
$$
which corresponds to the relation mentioned above. 


The crossed product therefore is a $C^\*$-algebra that contains the original algebra $A$ (in the unital case) and in which the action becomes inner.


