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Main purpose of this paper is to formulate canonical quantum continuity equations for white noise operators and to find their explicit unique solutions. By applying the continuity equations for white noise functionals and canonical topological isomorphisms between the spaces of white noise operators and the spaces of two-variable white noise functionals, we formulate canonical quantum continuity equations for white noise operators, and then we induce time dependent quantum evolution systems which are equivalent to the quantum continuity equations. Then we find explicit forms, in terms of the quantum analogues of generalized Fourier-Mehler transforms, of the unique solutions of the quantum continuity equations by solving the quantum evolution systems for white noise operators. [ABSTRACT FROM AUTHOR]

We study the q-creation and q-annihilation derivatives of operators on q-Fock space and then the derivatives are [lowercase_phi_synonym]-derivations with respect to the q-Wick product, where [lowercase_phi_synonym] is a *-homomorphism with respect to the q-Wick product, which provide interesting non-trivial examples of abstract [lowercase_phi_synonym]-derivation. [ABSTRACT FROM AUTHOR]

We discuss pairs ([lowercase_phi_synonym], [uppercase_phi_synonym]) of maps, where [lowercase_phi_synonym] is a map between C*-algebras and [uppercase_phi_synonym] is a [lowercase_phi_synonym]-module map between Hilbert C*-modules, which are generalization of representations of Hilbert C*-modules. A covariant version of Stinespring's theorem for such a pair ([lowercase_phi_synonym], [uppercase_phi_synonym]) is established, and quantum stochastic processes constructed from pairs ({[lowercase_phi_synonym]t}, {[uppercase_phi_synonym]t}) of families of such maps are studied. We prove that the quantum stochastic process J = {Jt} constructed from a [lowercase_phi_synonym]-quantum dynamical semigroup [uppercase_phi_synonym] = {[uppercase_phi_synonym]t} is a j-map for the quantum stochastic process j = {jt} constructed from the given quantum dynamical semigroup [lowercase_phi_synonym] = {[lowercase_phi_synonym]t}, and that J is covariant if the [lowercase_phi_synonym]-quantum dynamical semigroup [uppercase_phi_synonym] is covariant. [ABSTRACT FROM AUTHOR]