Mathematics

In my talk I will consider the following topics:

• (1) Generalized q-Fock spaces F_{q}(H) for real q and complex q, |q|<1; on a real Hilbert space H.
• (2) Anyonic Fock spaces, i.e. q-Fock spaces for |q|=1.
• (3) Functor of second quantization Γ_{q}(S) from the von Neumann algebra VN (q,H) generated by q-Gaussian operators G(f) = a(f) + a^{+}(f), where f is in a real Hilbert space H and S is a real contraction.
• (4) Wick order and Bozejko-Haagerup inequalities for q-creation/annihilation operators.
• (5) Proof that the generalized Ornstein-Uhlenbeck semigroup U_{t} = Γ_{q} (exp(-t) Id) = exp(-tN), where N is the number operator, is completely positive on VN(q,H)=L(\infty) and is also ultra-contractive, i.e., U_{t} maps L^2 into VN(q,H)= L^∞.

The talk is based on join papers with W.Bozejko, W.Ejsmont, S.Gal, T.Hasebe, E.Lytvynov, W.Mlotkowski, I.Rodionova, Q.Xu and J.Wysoczański.