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It is well known that the equation $$(*)\;\;\;\;ab-ba=1$$ is unsolvable in a Banach algebra.

I search for some reasonable generalization of this equation in higher variable for investigation of solvability of such generalized equations. The above equation can be read as $$(**)\;\;\;\;\;\sum_{\sigma \in S_2} s(\sigma)\prod a_{\sigma_i}=1$$ provided we put $a_1=a,\;a_2=b.$

So our first question is the following:

Is there a Banach algebra with three elements $a,b,c$ which satisfy the following equation?$$(***)\;\;\;\;\;abc+bca+cab-bac-cba-acb=1$$

Our next question:

Apart from $(***)$, What would be some other generalization of $(*)$ whose solvability in a Banach algebra would be an interesting and non trivial question?

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In the algebra of real $2\times 2$ matrices, take $$ a=\left[ \begin{array}{cc} 0 & 1\\ 0 & 0\end{array} \right],\ b=\left[ \begin{array}{cc} 0 & 0\\ 1 & 0\end{array}\right],\ c=\left[ \begin{array}{rr} 1/3 & 0\\ 0 & -1/3\end{array}\right]. $$ In the other direction, is it true that if for all $n$ an equation has no solutions in $n\times n$ real matrices, then it has no solutions in any Banach algebra? What are the equations (or finite sets of equations) that for all $n$ have no solutions in $n\times n$ real matrix algebras?

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    $\begingroup$ If you start out with a non-trivial finitely presented group $G$ that has no finite-dimensional representations, then you get relations that are no satisfiable in matrices but are satisfied in $\ell^1(G)$. On the other side, in a suitable form there is no such example known where one would be able to show that relations cannot be "almost" satisfied in matrices (given a suitable norm like the operator norm). $\endgroup$
    – Andreas Thom
    Commented May 8, 2018 at 15:27
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    $\begingroup$ Can this be put in equational form: $xy = 1 \neq yx$? No matrix algebra has such elements, but there are plenty of Banach algebras that do. More generally, there are traces on any subalgebra of a product of matrix algebras, but not for Banach algebras in general. $\endgroup$
    – David Handelman
    Commented May 8, 2018 at 19:33
  • $\begingroup$ @RichardStanley Dear Professor Stanley thank you very much for this interesting example. $\endgroup$
    – Ali Taghavi
    Commented May 9, 2018 at 7:49
  • $\begingroup$ @DavidHandelman Dear Prof. Handelman, according to your comment it would be interesting to find an equation which does not have a solution in finite dimensional $C^*$ algebras but it has a solution in a $C^*$ algebra which admit a faithful trace.Is there an equation with this property? $\endgroup$
    – Ali Taghavi
    Commented May 9, 2018 at 8:19
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    $\begingroup$ @DavidHandelman: You can express stronger conditions like the Cuntz algebra relations: $xy+wz=1$ and $yx=zw=1$. $\endgroup$
    – Andreas Thom
    Commented May 9, 2018 at 10:40

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