Monotone Closed Operators and Quantum Semigroup Related to Free Products

Guided by the construction of the unital *-algebra of closed operators which are ‘affiliated’ with a given unital *-algebra, we introduce the notion of the ‘monotone closure’ for certain increasing sequences of unital *-algebras. Then we study an example of a unital *-algebra F0(A) constructed from a countable number of copies of a unital *-algebra A. We endow F0(A) with a quantum semigroup structure, which can be lifted to its ‘monotone closure’ F(A), using the ‘monotone tensor product’ F(A)⊗F(A). We show that the quantum semigroup (F(A),∆, ǫ) is related to the additive free convolution. Namely, if μ, ν are states on A and μ̂, ν̂ are the associated states on F(A), then the additive free convolution μ⊞ ν is obtained from the quantum semigroup convolution μ̂ ⋆ ν̂ = (μ̂ ⊗ ν̂) ◦ ∆ by restriction. A similar relation is established between the free product of states and the tensor product of their extensions. Finally, monotone closed operators lead to ‘quantum orthogonal series’ representations of free random variables. Mathematics Subject Classification (2000): 46L54, 81R50