Universal $C^*$ algebra subject to $ x^*x+y^*y=xx^*+yy^*$

Question

What is the precise description for the universal unital $C^*$ algebra generated by two elements $x,y$ subject to relation $$ x^*x+y^*y=xx^*+yy^*$$

Answer 1

It doesn't exist. A requirement to have a universal C$^*$-algebra in terms of generators is that the relations give a bound for the norm of the generators. That's not the case here, as given any $x,y$ as you want you can replace them with $nx$ and $ny$ for any $n\in\mathbb N$ and still keep the relation.