[Paper Review] The Martin boundary of a discrete quantum group

Sergey Neshveyev, Lars Tuset|ArXiv.org|Sep 20, 2002

Advanced Operator Algebra Research15 references3 citations

TL;DR

This paper develops the Martin boundary theory for discrete quantum groups by defining a C*-algebra $ A_φ $ of harmonic elements via convolution with a $ q $-tracial state $ \phi $. It establishes a representation theorem linking positive harmonic elements to positive linear functionals on $ A_\phi $, and proves that the Martin boundary of the dual quantum group $ \widehat{SU_q(2)} $ is isomorphic to the quantum homogeneous sphere of Podleř, unifying results from Biane and Izumi.

ABSTRACT

We consider the Markov operator P_phi on a discrete quantum group given by convolution with a q-tracial state phi. In the study of harmonic elements x, P_phi(x)=x, we define the Martin boundary A_phi. It is a separable C*-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on A_phi. The C*-algebra A_phi has a natural time evolution, and the unit can always be represented by a KMS state. Any such state gives rise to a u.c.p. map from the von Neumann closure of A_phi in its GNS representation to the von Neumann algebra of bounded harmonic elements, which is an analogue of the Poisson integral. Under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual. Next we apply these results to identify the Martin boundary of the dual of SU_q(2) with the quantum homogeneous sphere of Podles. This result extends and unifies previous results by Ph. Biane and M. Izumi.

Motivation & Objective

• To generalize the classical Martin boundary theory to the non-commutative setting of discrete quantum groups.
• To define a C*-algebraic Martin boundary $ A_\phi $ that carries canonical actions of the quantum group and its dual.
• To establish a representation theorem linking positive harmonic elements to positive linear functionals on $ A_\phi $.
• To identify the Martin boundary of $ \widehat{SU_q(2)} $ with the quantum homogeneous sphere of Podleř, unifying prior results.
• To construct a Poisson integral map from the GNS representation of $ A_\phi $ to the von Neumann algebra of bounded harmonic elements, and show it is an isomorphism under suitable conditions.

Proposed method

• Define the Markov operator $ P_\phi = (\phi \otimes \iota)\Delta $ via convolution with a $ q $-tracial state $ \phi $ on a discrete quantum group.
• Construct the Martin boundary $ A_\phi $ as a separable C*-algebra of $ P_\phi $-harmonic elements.
• Use the balayage theorem to approximate harmonic elements by potentials $ \sum_{n=0}^\infty P_\phi^n(x) $, enabling boundary construction.
• Introduce a natural time evolution on $ A_\phi $, and show the unit can be represented by a KMS state.
• Construct a u.c.p. map from the GNS von Neumann closure of $ A_\phi $ to the von Neumann algebra of bounded harmonic elements, analogous to the Poisson integral.
• Prove that under additional assumptions, this map is an isomorphism respecting the actions of the quantum group and its dual.

Experimental results

Research questions

• RQ1 How can the Martin boundary be defined in the context of discrete quantum groups, generalizing the classical theory?
• RQ2 What is the structure of the Martin boundary $ A_\phi $, and how do the quantum group and its dual act on it?
• RQ3 Can positive harmonic elements be represented via positive linear functionals on $ A_\phi $, and what is the role of the KMS state?
• RQ4 Is the Poisson integral map from $ A_\phi $ to the bounded harmonic elements an isomorphism, and under what conditions?
• RQ5 What is the Martin boundary of the dual quantum group $ \widehat{SU_q(2)} $, and how does it relate to the quantum homogeneous sphere of Podleř?

Key findings

• The Martin boundary $ A_\phi $ is a separable C*-algebra carrying canonical actions of the discrete quantum group and its dual.
• Positive harmonic elements correspond bijectively to positive linear functionals on $ A_\phi $, establishing a representation theorem.
• The C*-algebra $ A_\phi $ admits a natural time evolution, and the unit is representable by a KMS state.
• The u.c.p. map from the GNS von Neumann closure of $ A_\phi $ to the von Neumann algebra of bounded harmonic elements is an isomorphism under additional assumptions.
• The Martin boundary of $ \widehat{SU_q(2)} $ is isomorphic to the quantum homogeneous sphere $ C(S^2_{q,0}) $ of Podleř.
• The isomorphism $ \sigma: C(S^2_{q,0}) \to \Psi / \hat{A} $ intertwines the right coactions of the dual quantum group $ (\hat{A}, \hat{\Delta}) $, and the state $ \nu $ on $ \Psi / \hat{A} $ is quasi-invariant with Radon-Nikodym cocycle $ (K \otimes \iota)\hat{\Delta}(I_0) $.