Algebraic structure of \(k\)-mode fluctuations (Q1381954)

Consider a family of operators \(A(x)\), \(x\in Z^v\), where \(A(x)\) is an element of a \(C^*\)-algebra \(B\) representing an observable at site \(x\). A state \(\rho\) on \(B\) defines a corresponding product state on the family. The paper presents a ``central limit'' approach to the asymptotic behaviour of \(\sum_{x\in\Lambda_n} A(x)\cos kx\) under such a product state (with \(k\neq 0\) and \(\Lambda_n= [-n, n]^v\), \(n= 1,2,\dots\)), demonstrating the Gaussian behaviour of the scaled fluctuation operators in the limit and describing the algebraic structure involved.