Operator algebras and representations from commuting semigroup actions

Boundary Representations for Operator Algebras

All operator algebras have (not necessarily irreducible) boundary representations. A unital operator algebra has enough such boundary representations to generate its C-envelope.

Irreducible Representations of Operator Algebras

Representations of vertex operator algebras

This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras An(V ) and their bimodules. A new result on the rationality is given. That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is a semisimple associative algebra. 2000MSC:17B69

Representations of Hecke Algebras and Dilations of Semigroup Crossed Products

We consider a family of Hecke C∗-algebras which can be realised as crossed products by semigroups of endomorphisms. We show by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group. Suppose that M is a subgroup of a gro...

Twisted representations of vertex operator algebras and associative algebras

Let V be a vertex operator algebra and g an automorphism of order T . We construct a sequence of associative algebras Ag,n(V ) with n ∈ 1 T Z nonnegative such that Ag,n(V ) is a quotient of Ag,n+1/T (V ) and a pair of functors between the category of Ag,n(V )-modules which are not Ag,n−1/T (V )-modules and the category of admissible V -modules. These functors exhibit a bijection between the sim...