What papers or references  have been devoted for  a  noncommutativization of "Fixed point theory". Here the terminology Noncommutativiztion, as usual, indicates to that famous table with 2  columns: first column is the classic(space) and the second is "operator algebra". See for example table in page 26 [of Connes book](http://www.alainconnes.org/docs/book94bigpdf.pdf)  or  page 6 of [this  note](http://arxiv.org/pdf/math/0408416v1.pdf)

For $A=C(X)$ the  algebraic translation is that "For every  unital morphism $\phi$ on $A$, there is  a maximal ideal $I$ which is invariant under $\phi$.  Now what is  an appropriate NC analogy?

 In this line, is  it reasonable to search  for  a particular  type of $\textit{index}$ as  a  NC  analogue for  Lefschetz index?