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:



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



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> 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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A generalization of unsolvable equation $ab-ba=1$ in a Banach algebra
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?


