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 or page 6 of this note

For $A=C(X)$ the algebraic translation is that "For every 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?