离散量子群的Martin边界
The Martin boundary of a discrete quantum group
AI总结：
本文研究离散量子群上由q-迹态卷积定义的马尔可夫算子的Martin边界，建立正调和元的表示定理，构造类似泊松积分的u.c.p.映射，并将SU_q(2)对偶的Martin边界等同于Podleś量子齐性球面，推广统一了前人结果。
摘要：
我们研究离散量子群上由与q-迹态φ卷积给出的马尔可夫算子P_φ。在对调和元x（满足P_φ(x)=x）的研究中，我们定义了Martin边界A_φ，它是一个可分C*-代数，承载着该量子群及其对偶的典范作用。我们建立了一个表示定理，表明正调和元对应于A_φ上的正线性泛函。C*-代数A_φ具有自然的时间演化，且单位元总能由一个KMS态表示。任何此类态都会生成一个从A_φ在其GNS表示中的冯·诺依曼闭包到有界调和元冯·诺依曼代数的u.c.p.映射，这是泊松积分的类似物。在附加假设下，该映射是一个同构，且与量子群及其对偶的作用相容。随后我们应用这些结果，将SU_q(2)的对偶的Martin边界与Podleś的量子齐性球面等同起来。该结果推广并统一了Ph. Biane和M. Izumi的先前结果。
英文摘要：
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.
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