Martijn Caspers, Utrecht University
“Absence of Cartan subalgebras for right angled Hecke von Neumann algebras.”
Hecke algebras are ∗-algebras generated by self-adjoint operators T(s) with s in some generating set that satisfy the Hecke relation (T (s) + q)(T (s) − 1/q) = 0 as well as suitable types of commutation relations. They generate a von Neumann algebra called the Hecke von Neumann algebra studied by Dymara et al. It was proved by Garncarek that in the right- angled case these von Neumann algebras are actually factors if the parameter q is in a certain interval around 1 and for q outside of this interval it is a direct sum of a factor and the complex numbers (so the center is always very small). In particular the isomorphism type of these algebras depends on q.
In this talk we first find approximation properties of right-angled Hecke von Neumann algebras: we prove that they are non-injective, have the completely contractive approximation property and the Haagerup property. We then turn to the existence of Cartan subalgebras and show that in the hyperbolic case these algebras are strongly solid and hence cannot have a Cartan subalgebra. In the general case, these algebras need not be strongly solid but still we are able to prove the non-existence of Cartan subalgebras.
MC 5417
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