${\rm VN}(G)$上的双模、调和算子和非交换Poisson边界
Bimodules over ${\rm VN}(G)$, harmonic operators and the non-commutative Poisson boundary
摘要：
从$L^1(G)$的一个左理想$J$出发，我们考虑$L^{\infty}(G)$中的零化子$J^{\perp}$和$\mathcal{B}(L^2(G))$中的生成${\rm VN}(G)$ -双模，${\rm Bim}(J^{\perp})$ .我们证明了当$G$是弱顺从的离散的、紧的或阿贝尔的时，${\rm Bim}(J^{\perp})=({\rm Ran} J)^{\perp}$是等价的，其中${\rm Ran} J$是$J$在迹类中的一个适当的饱和.我们定义了联合调和函数和联合调和算子，并证明了对于这几类群，联合调和算子空间是由联合调和函数空间生成的${\rm VN}(G)$ -双模.利用这一点，我们证明了Izumi和Jaworski-Neufang的结果：非交换Poisson边界同构于调和函数空间的$G$交叉积.
英文摘要：
Starting with a left ideal $J$ of $L^1(G)$ we consider its annihilator $J^{\perp}$ in $L^{\infty}(G)$ and the generated ${\rm VN}(G)$-bimodule in $\mathcal{B}(L^2(G))$, ${\rm Bim}(J^{\perp})$. We prove that ${\rm Bim}(J^{\perp})=({\rm Ran} J)^{\perp}$ when $G$ is weakly amenable discrete, compact or abelian, where ${\rm Ran} J$ is a suitable saturation of $J$ in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the ${\rm VN}(G)$-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski - Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by $G$.
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