Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras
Oleg Aristov
备注：
English translation; Russian text see version 6 and earlier; in version 4: corrections in Th.4.4 and related results; in version 2: Sections 5 and 6 rewritten, Section 3.2 removed
发表信息：
Izv. RAN. Ser. Mat. 86:6 (2022), 5--46 (in Russian)
We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain~$C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. In addition, we discuss the theory of holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results.
$H^\infty$ Functional Calculus for a Commuting Pair of $\text{Ritt}_{\text{E}}$ Operators
Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究所特里凡得琅分校）；The Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）
A holomorphic functional calculus for finite families of commuting semigroups
设A是一个交换Banach代数，使得对于u $\in$ A \{0}，u A = {0}且u A具有稠密主理想.给出了定义F（-- $λ$ 1 $Δ$ T1，. ..，- $λ$ kPDMATHABMO T k）其中F属于定义在C k的适当开子集上的全纯函数的自然类，所述开子集包含（- $λ$ 1 $Δ$ T 1，. .. .，- $λ$ k $Δ$ T k），其中$Δ$ T 1，. ..，$Δ$ T k是A上的可交换单参数乘子半群的无穷小生成元，它们属于下列一类：（1）强连续半群T =（T（te ia）t>0，使得$\cup$ t> 0 T（te ia）A在A中稠密，其中一个$\in$ R. (2)半群类T =（T（$ζ$））$ζ$ $\in$ Sa，b在开扇区Sa，b上全纯，使得T（$ζ$）A在A中对某些或等价地对所有$ζ$ $\in$ Sa，b稠密.我们使用的概念quasimulator，介绍了作者在1981年长滩会议上的Banach代数：所考虑的半群的生成元将被定义为A上的拟乘子，并且对于Arveson预解集$σ$中的$ζ$，（$Δ$ T）解决方案（$Δ$ T -- $ζ$ I）将被定义为A上的规则准乘数，即A上的一个拟乘数S使得对于某些$λ$ > 0和某些生成A的稠密理想且属于Sn，n $\ge$ 1的域的交集的u，supn $\ge$ 1 $λ$ n S n u < +…
