Proper analytic free maps

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• 作者：J. William Helton^a ; ¹ ; ^{helton@math.ucsd.edu} ; Igor Klep^b ; ^c ; ² ; ^{igor.klep@fmf.uni-lj.si} ; Scott McCullough^d ; ³ ; ^{sam@math.ufl.edu}
• 关键词：Non-commutative set and function ; Analytic map ; Proper map ; Rigidity ; Linear matrix inequality ; Several complex variables ; Free analysis ; Free real algebraic geometry
• 刊名：Journal of Functional Analysis
• 出版年：2011
• 出版时间：1 March 2011
• 年：2011
• 卷：260
• 期：5
• 页码：1476-1490
• 全文大小：162 K

文摘

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 in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f⁻¹ is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains and .