与Berezin量子化和自守形式有关的自由群因子中的非交换马氏过程
Non-Commutative Markov Processes in Free Group Factors, Related to Berezin's Quantization and Automorphic Forms
摘要：
本文用自由群因子的描述作为上半平面Berezin变形模PSL $(2,{\Bbb Z})$的von Neumann代数。在相应的代数的产品的变形参数的衍生物，是一个积极的Hochschild 2-上循环，定义在一个稠密的子代数。通过分析上循环的结构，证明了在强稠密子代数上实现上循环的量子动力学半群存在生成元$\cal L$。对于稠密子代数中的$x$，${\cal L}(x)$是（扩散）算子$$ {\cal L}(x)=Λ(x)-(1/2)\{T,x\}, $$，其中$Λ$是逐点（Schur）乘法算子，其符号函数与自守形式$Δ$的对数相关。算子$T$是正的，并且隶属于代数${\cal A}_t$，并且$T$对应于${\cal L}(1)$，在某种意义上在本文中是精确的。经过适当的规范化，对应于主值型方法，适用于II $_1$因素，$Λ$成为（完全）积极的工会弱稠密子代数。此外，与变形相关的2-循环上同调上循环可以用$Λ$表示。
英文摘要：
In this paper we use the description of free group factors as the von Neumann algebras of Berezin's deformation of the upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $$ {\cal L}(x)=Λ(x)-(1/2)\{T,x\}, $$ where $Λ$ is the pointwise (Schur) multiplication operator with a symbol function related to the logarithm of the automorphic form $Δ$. The operator $T$ is positive and affiliated with the algebra ${\cal A}_t$ and $T$ corresponds to ${\cal L}(1)$, in a sense to be made precise in the paper. After a suitable normalization, corresponding to a principal-value type method, adapted for II$_1$ factors, $Λ$ becomes (completely) positive on a union of weakly dense subalgebras. Moreover the 2-cyclic cohomology cocycle associated to the deformation may be expressed in terms of $Λ$.
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