Modular theory, non-commutative geometry and quantum gravity.
Multiplicative Cauchy functional equation and the equation of ratios on the Lorentz cone
It is proved that the solution of the multiplicative Cauchy functional equation on the Lorentz cone of dimension greater than two is a power function of the determinant. The equation is solved in full generality, i.e. no smoothness assumptions on the unknown function are imposed. Also the functional equation of ratios, of a similar nature, is solved in full generality.