Proper analytic free maps
Proper analytic free maps
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DOI：
10.1016/j.jfa.2010.11.007
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发表时间：
2010
影响因子：
1.7
通讯作者：
S. McCullough
中科院分区：
文献类型：
--
作者：
J. Helton;I. Klep;S. McCullough
本文研究解析自由映射。这些映射是多个复变量的经典解析函数的自由类似物，并且是根据非交换变量定义的，其中没有关系-它们是自由变量。解析自由映射包括自由（非交换）变量的向量值多项式，并形成从一个非交换域D到另一个非交换域D的映射的典范类。作为通常概念的自然推广，一个解析自由映射是恰当的，如果它把D的边界映射到D的边界上。假设两个整环都包含0，我们证明了：如果f：D→D → D是一个真解析自由映射，且f（0）=0，则f是一对一的.此外，若g=g，则f是可逆的，且f− 1也是解析自由映射。这些关于映射f的结论是在没有关于域D和D ∞的额外假设的情况下可能最强的。
This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations – they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain D˜ in g˜ variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of D˜. Assuming that both domains contain 0, we show that if f:D→D˜ is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also g=g˜, then f is invertible and f−1is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and D˜.