Is there universal $C^*$ algebra generated by self adjoint elements $x,y$ subject to relation $x^2+y^2=1+{(xy-yx)}^2$? What is a precise description of such algebra?
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4$\begingroup$ I think it's nicer to write this as $x^2+y^2+z^2=1$ with $x,y,z$ all self-adjoint and $[x,y]=iz$. One can check that in any representation by $2\times 2$ matrices we have $xz+zx=yz+zy=0$, and if we impose that relation then the monomials $x^iy^jz^k$ with $k\in\{0,1\}$ span everything, but I doubt whether that is too useful. $\endgroup$– Neil StricklandCommented Jul 4, 2017 at 15:12
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2$\begingroup$ @Neil, your observation is very usefull because it shows that in any concrete representation of the above, one has that $\Vert x\Vert, \Vert y\Vert, \Vert z\Vert \leq 1$, whence the above relations are "admissible" in the sense of Blackadar [Shape theory for C*-algebras, Math. Scand. 56(1985), 249-275], and hence the existence of this algebra is guaranteed. $\endgroup$– RuyCommented Jul 4, 2017 at 21:30
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$\begingroup$ @NeilStrickland Thank you very much for your comment. May I ask you to elaborate your comment(As an answer)? $\endgroup$– Ali TaghaviCommented Jul 5, 2017 at 7:34
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$\begingroup$ @Ruy Thank you very much for your comment. Could you please more explain on your comments or add add an answer to my question? To be honest I do not have background on "Universal C* algebra associated to some relation". $\endgroup$– Ali TaghaviCommented Jul 5, 2017 at 7:36
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