In quantum mechanics, creation operators, which add a particle to a system, and annihilation operators, which remove one, describe the basic rules governing the universe at the microscopic level. They are operators on either boson or fermion Fock spaces, and the $C^*$-algebras generated by these creation and annihilation operators---namely, the CCR (Weyl) algebra and the CAR (Clifford) algebra, respectively---attracted a lot of attention.

With the introduction of different non-commutative independences (the free independence at the forefront), more Fock-type spaces emerged, each one with its own creation and annihilation operators as well as $C^*$-algebras generated by them. In some cases, they turned out to be tightly linked to graph algebras: First, D. Evans observed the relation between the algebras of creators on the free Fock spaces and the Cuntz algebras. More recently V. Crismale, S. Del Vecchio, S. Rossi, and J. Wysoczański showed the algebras of infinitely many creators on the (weakly) monotone Fock spaces to be $*$-isomorphic to Exel-Laca algebras or their quotients.

During the talk, I will discuss a generalization of the latter results. Namely, we study the algebra of creators on the discrete boolean-anti-monotone Fock space and, for the index set having a tree-like structure, we link it to Exel-Laca algebras and further to graph algebras. This enables us to deduce many of the properties of the algebras in question.

This is joint work with Jack Spielberg and Janusz Wysoczański.