The implicit function theorem and free algebraic sets

Agler, Jim; M$^{\mathrm{c}}$Carthy, John

doi:10.1090/tran/6546

The implicit function theorem and free algebraic sets
HTML articles powered by AMS MathViewer

by Jim Agler and John E. M$^{\mathrm {c}}$Carthy PDF

Trans. Amer. Math. Soc. 368 (2016), 3157-3175 Request permission

Abstract:

We prove an implicit function theorem for non-commutative functions. We use this to show that if $p(X,Y)$ is a generic non-commuting polynomial in two variables and $X$ is a generic matrix, then all solutions $Y$ of $p(X,Y)=0$ will commute with $X$.

References

Jim Agler and John E. McCarthy, Global holomorphic functions in several noncommuting variables, Canad. J. Math. 67 (2015), no. 2, 241–285. MR 3314834, DOI 10.4153/CJM-2014-024-1
Jim Agler and John E. M\raise.45exc Carthy, Pick interpolation for free holomorphic functions, Amer. J. Math., to appear.
D. Alpay and D. S. Kalyuzhnyĭ-Verbovetzkiĭ, Matrix-$J$-unitary non-commutative rational formal power series, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2006, pp. 49–113. MR 2187742, DOI 10.1007/3-7643-7431-4_{2}
Robert F. V. Anderson, The Weyl functional calculus, J. Functional Analysis 4 (1969), 240–267. MR 0635128, DOI 10.1016/0022-1236(69)90013-5
Joseph A. Ball, Gilbert Groenewald, and Tanit Malakorn, Conservative structured noncommutative multidimensional linear systems, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkhäuser, Basel, 2006, pp. 179–223. MR 2187744, DOI 10.1007/3-7643-7431-4_{4}
Benson Farb and R. Keith Dennis, Noncommutative algebra, Graduate Texts in Mathematics, vol. 144, Springer-Verlag, New York, 1993. MR 1233388, DOI 10.1007/978-1-4612-0889-1
J. William Helton, “Positive” noncommutative polynomials are sums of squares, Ann. of Math. (2) 156 (2002), no. 2, 675–694. MR 1933721, DOI 10.2307/3597203
J. William Helton, Igor Klep, and Scott McCullough, Analytic mappings between noncommutative pencil balls, J. Math. Anal. Appl. 376 (2011), no. 2, 407–428. MR 2747767, DOI 10.1016/j.jmaa.2010.11.040
J. William Helton, Igor Klep, and Scott McCullough, Proper analytic free maps, J. Funct. Anal. 260 (2011), no. 5, 1476–1490. MR 2749435, DOI 10.1016/j.jfa.2010.11.007
J. William Helton and Scott McCullough, Every convex free basic semi-algebraic set has an LMI representation, Ann. of Math. (2) 176 (2012), no. 2, 979–1013. MR 2950768, DOI 10.4007/annals.2012.176.2.6
Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of free noncommutative function theory, Mathematical Surveys and Monographs, vol. 199, American Mathematical Society, Providence, RI, 2014. MR 3244229, DOI 10.1090/surv/199
J. E. Pascoe, The inverse function theorem and the Jacobian conjecture for free analysis, Math. Z. 278 (2014), no. 3-4, 987–994. MR 3278901, DOI 10.1007/s00209-014-1342-2
Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
Gelu Popescu, Free holomorphic functions on the unit ball of $B(\scr H)^n$, J. Funct. Anal. 241 (2006), no. 1, 268–333. MR 2264252, DOI 10.1016/j.jfa.2006.07.004
Gelu Popescu, Free holomorphic functions and interpolation, Math. Ann. 342 (2008), no. 1, 1–30. MR 2415313, DOI 10.1007/s00208-008-0219-2
Gelu Popescu, Free holomorphic automorphisms of the unit ball of $B(\scr H)^n$, J. Reine Angew. Math. 638 (2010), 119–168. MR 2595338, DOI 10.1515/CRELLE.2010.005
Gelu Popescu, Free biholomorphic classification of noncommutative domains, Int. Math. Res. Not. IMRN 4 (2011), 784–850. MR 2773331, DOI 10.1093/imrn/rnq093
R. R. Smith, Completely bounded maps between $C^{\ast }$-algebras, J. London Math. Soc. (2) 27 (1983), no. 1, 157–166. MR 686514, DOI 10.1112/jlms/s2-27.1.157
J. Sylvester, Sur l’équations en matrices $px = xq$, C.R. Acad. Sci. Paris 99:67–71, 1884.
Joseph L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1–34. MR 315446, DOI 10.1090/S0002-9904-1973-13077-0