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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 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?

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  • $\begingroup$ Is $\phi$ an automorphism or any endomorphism? Under $\phi$ or under $\phi^{-1}$? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 10, 2014 at 18:47
  • $\begingroup$ @მამუკაჯიბლაძე a unital endomorphism. by invariant I mean $\phi(I)\subset I$.(an algebraic translation of the classical fix point property, according to Gelfand duality. $\endgroup$
    – Ali Taghavi
    Commented Nov 10, 2014 at 19:00
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    $\begingroup$ I see. Though in fact for a maximal $I$ it is equivalent to $\phi^{-1}(I)=I$... $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Nov 10, 2014 at 19:31

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