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$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in R$, where we consider $r$ as the linear endomorphism of the corresponding module, then we know that $M\cong N$ by the Brauer–Nesbitt theorem.

My question is, assuming that we have specific values for traces (some homomorphism $g: R \rightarrow \mathbb{C}$ with $g(rr')=g(r'r)$ for every $r, r'\in R$), do we know that there exists some module $M$ with $Tr_M(r)=g(r)$ for every $r$? If not, can we somehow efficiently describe the functions that can appear as traces?

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    $\begingroup$ Your statement of the Brauer--Nesbitt theorem is incorrect: you need $M$ and $N$ to be semisimple. $\endgroup$
    – Aurel
    Jun 10, 2018 at 9:55

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Not always — e.g. $g(1)$ should be an integer. The desired description is given in

Helling, H., Eine Kennzeichnung von Charakteren auf Gruppen und assoziativen Algebren, Commun. Algebra 1, 491-501 (1974). ZBL0288.16019.

Edit as requested: To a linear $g:R\to\mathbf C$ with $g(rs)=g(sr)$ Helling attaches $g_n:R^n\to\mathbf C$ by $$ g_n(r_1,\dots,r_n)= \sum_{\sigma\in\mathfrak S_n}(-1)^\sigma\prod_{(i_1,\dots,i_l):\text{ cycle of }\sigma}g(r_{i_1}\!\cdots r_{i_l}) $$ and proves (Satz p. 496): $g$ is a character iff $g_n=0$ for some integer $n$.

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  • $\begingroup$ Is there an English version? I do not know German... Or any reference that contains the results of this paper in English? Sorry to be picky and thanks for the answer in any case! $\endgroup$
    – Ioannis Zolas
    Jun 9, 2018 at 2:54
  • $\begingroup$ The Zentralblatt link gives more recent papers quoting this one. $\endgroup$
    – Francois Ziegler
    Jun 9, 2018 at 2:56
  • $\begingroup$ I checked them but still everybody just references the paper, nobody explains the actual result (they are all of the type "this problem has been studied there") Could you possibly elaborate a little on your answer on what the results of the paper are? $\endgroup$
    – Ioannis Zolas
    Jun 9, 2018 at 14:13
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    $\begingroup$ In arithmetic geometry, this question has been studied for a while under the name of "pseudo-representation" (apparently introduced by Wiles in that context in "On ordinary λ-adic representations associated to modular forms", Invent. Math 94 (1988), 529-573). Searching for this gives many results, e.g, math.harvard.edu/~kisin/notes/notes2.pdf has some references to proofs by Taylor and by Rouquier of this result. (Interestingly, the paper of Helling is not mentioned in either Wiles's or Taylor's, so they must have re-discovered the result). $\endgroup$
    – Denis Chaperon de Lauzières
    Jun 9, 2018 at 16:34
  • $\begingroup$ Thank you both a lot, these results are exactly what I needed! $\endgroup$
    – Ioannis Zolas
    Jun 10, 2018 at 15:22

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