Wick order, spreadability and exchangeability for monotone commutation relations

Abstract

We exhibit a Hamel basis for the concrete $\ast$-algebra ${\mathfrak{M}}_o$ associated to monotone commutation relations realised on the monotone Fock space, mainly composed by Wick ordered words of annihilators and creators. We apply such a result to investigate spreadability and exchangeability of the stochastic processes arising from such commutation relations. In particular, we show that spreadability comes from a monoidal action implementing a dissipative dynamics on the norm closure $C^{\ast }$-algebra $\mathfrak{M}=\overline{{\mathfrak{M}}_o}$. Moreover, we determine the structure of spreadable and exchangeable monotone stochastic processes using their correspondence with sp\-reading invariant and symmetric monotone states, respectively.

Ann. Henri Poincarè, to appear