Unitary representations of the groups of measurable and continuous functions with values in the circle (Q457636)

Strongly continuous unitary representations of the (Abel, Polish, typically not locally compact) multiplicative group of measurable functions of a standard Borel probability space \((X,\mu)\) into the circle group \(\mathbb T\subset\mathbb C\) endowed with the topology of convergence in measure are described. They are shown to be direct sums of ``building blocks'' each being, up to unitary equivalence, some simply defined homomorphism \(\sigma(\kappa,\lambda)\), depending on finitely many parameters \(\kappa\in\mathbb Z^n\) and a Borel measure \(\lambda\) on \(X^n\), of the group into functions on a finite power \(X^n\). These functions define, by multiplication, the corresponding ``building block operators'' on \(L^2(X^n,\lambda)\). This description of the representation is found such that it fulfils appropriate additional properties which ensure that it is in a sense (the same building blocks) unique.NEWLINENEWLINEThe basic step of the proof of the existence result uses a version of the spectral theorem for a commuting family of unitary operators, a transformation of particular continuous homomorphisms into linear operators over \(\mathbb R\) and Kwapień's representation of linear operators between spaces of measurable real functions. The requested description of the representation and its uniqueness is then deduced by further analysis.NEWLINENEWLINEA similar result for the group \(C(M,\mathbb T)\) of continuous functions on a null-dimensional compact metrizable space is deduced from the above-mentioned one using the spectral theorem as before and a combination of some known factorization theorems for linear operators.