Some remarks on actions of compact matrix quantum groups onC^∗-algebras

Abstract

In this paper we construct an action of a compact matrix quantum group on a Cuntz algebra or a UHF-algebra, and investigate the fixed point subalgebra of the algebra under the action. Especially we consider the action of μ U(2) on the Cuntz algebra <&i and the action of S μ U(2) on the UHF-algebra of type 2°° . We show that these fixed point subalgebras are generated by a sequence of Jones 9 projections.

Actions of compact quantum groups on 𝐶*-algebras

“…. , ψ d by a matrix v = [v ij ] i,j=1,...,d we can determine an action α of G on O d by defining it on generators (see [12], [8]): …”

“…5.3.3 and 5.3.4 show that the family {σ l (SS * ) : l ∈ N 0 } can be considered as the system of Jones' projections (see [12] Proof. Let C * (S, ρ) denote the smallest C * -algebra containing S and invariant under ρ.…”

“…Then is spanned by elements of the form (5.7) (see (4.1)), so E α (X) = 0 for every X ∈ P We conclude this section with a discussion of the structure of the fixed point algebra O α 2 . To this end let us recall that the authors of [12] have considered the twisted unitary group q U (2) and its action α on O 2 is. They have proved that the fixed point algebra O Proof.…”

Coactions of Hopf algebras on Cuntz algebras and their fixed point algebras

“…We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of the unitary matrix group U (d) as automorphisms of the Cuntz algebra O d .In particular we study the fixed point subalgebra under the coaction of the quantum compact groups Uq(d) on the Cuntz algebra O d by extending to any dimension d < ∞ a result of Konishi (1992).Furthermore we give a description of the fixed point subalgebra under the coaction of SUq(d) on O d in terms of generators. …”

Braided Endomorphisms of Cuntz Algebras

“…Moreover there is some evidence for them to be related to the dual object of a deformation of some group (possibly U 2 ) at a primitive fourth root of unity acting on o 2 , cf. [23]. Such fusion rules might describe the physical spectrum of some non-rational conformal quantum theory in low-dimensional space-time.…”