Geodesic
The Relation “Commutator Equals Function” in Banach Algebras
Matematičeskie zametki, Tome 109 (2021) no. 3, pp. 323-337

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The relation $xy-yx=h(y)$, where $h$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of this equality, we assume that $x$ and $y$ are elements of a Banach algebra (or of an Arens–Michael algebra). We prove that the universal algebra generated by a commutation relation of this kind can be represented explicitly as an analytic Ore extension. An analysis of the structure of the algebra shows that the set of holomorphic functions of $y$ degenerates, but at each zero of $h$, some local algebra of power series remains. Moreover, this local algebra depends only on the order of the zero. As an application, we prove a result about closed subalgebras of holomorphically finitely generated algebras.
Mots-clés : commutation relation
Keywords: quantum group, Banach algebra, Arens–Michael algebra, analytic Ore extension, holomorphically finitely generated algebra. 
O. Yu. Aristov. The Relation “Commutator Equals Function” in Banach Algebras. Matematičeskie zametki, Tome 109 (2021) no. 3, pp. 323-337. http://geodesic.mathdoc.fr/item/MZM_2021_109_3_a0/
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[1] K. Kassel, Kvantovye gruppy, Fazis, M., 1999 | MR

[2] N. Aizawa, H. Sato, “Quantum affine transformation group and covariant differential calculus”, Progr. Theoret. Phys., 91:6 (1994), 1065–1079 | DOI | MR

[3] V. L. Ostrovskii, Yu. S. Samoilenko, “O parakh operatorov, svyazannykh kvadratichnym sootnosheniem”, Funkts. analiz i ego pril., 47:1 (2013), 82–87 | DOI | MR | Zbl

[4] Yu. V. Turovskii, “O parakh operatorov, svyazannykh kvadratichnym sootnosheniem”, Spektralnaya teoriya operatorov i ee prilozheniya, 6, Elm, Baku, 1985, 144–181 | MR

[5] V. S. Shulman, “Invariant subspaces and spectral mapping theorems”, Functional Analysis and Operator Theory, Banach Center Publ., 30, Polish Acad. Sci. Inst. Math., Warszawa, 1994, 313–325 | MR

[6] A. Yu. Pirkovskii, “Obolochki Arensa–Maikla, gomologicheskie epimorfizmy i otnositelno kvazisvobodnye algebry”, Tr. MMO, 69, no. 1, 2008, 34–125 | MR

[7] N. Lao, R. Whitley, “Norms of powers of the Volterra operator”, Integr. Equ. Oper. Theory, 27:4 (1997), 419–425 | DOI | MR

[8] D. Kershaw, “Operator Norms of Powers of the Volterra Operator”, J. Integral Equations Appl., 11:3 (1999), 351–36 | DOI | MR

[9] B. Thorpe, “The norm of powers of the indefinite integral operator on $(0,1)$”, Bull. London. Math. Soc., 30:5 (1998), 543–548 | DOI | MR

[10] G. Little, J. B. Reade, “Estimates for the norm of the $n$th indefinite integral”, Bull. London. Math. Soc., 30:5 (1998), 539–542 | DOI | MR

[11] A. Yu. Pirkovskii, “Noncommutative analogues of Stein spaces of finite embedding dimension”, Algebraic Methods in Functional Analysis, Oper. Theory Adv. Appl., 233, Birkhäuser Verlag, Bassel, 2014, 135–153 | DOI | MR

[12] A. Yu. Pirkovskii, “Holomorphically finitely generated algebras”, J. Noncommut. Geom., 9:1 (2015), 215–264 | DOI | MR

[13] A. Mallios, Topological Algebras. Selected Topics, North-Holland Math. Stud., 124, North-Holland, Amsterdam, 1986 | MR

[14] W. Rudin, Real and Complex Analysis, McGraw-Hill Book, New York, 1987 | MR

[15] A. Ya. Khelemskii, Banakhovy i polinormirovannye algebry: obschaya teoriya, predstavleniya, gomologii, Nauka, M., 1989 | MR | Zbl

[16] J. L. Taylor, “A general framework for a multi-operator functional calculus”, Adv. Math., 9 (1972), 183–252 | DOI | MR

[17] J. L. Taylor, “Functions of several noncommuting variables”, Bull. Amer. Math. Soc., 79 (1973), 1–34 | DOI | MR

[18] A. Grotendik, “O prostranstvakh $(\mathscr F)$ i $(\mathscr{DF})$”, Matematika, 2:3 (1958), 81–128 | MR | Zbl

[19] P. Pérez Carreras, J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud., 131, North-Holland, Amsterdam, 1987 | MR

[20] Kh. Shefer, Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR | Zbl

[21] R. Meise, D. Vogt, Introduction to Functional Analysis, Oxf. Grad. Texts Math., 2, The Clarendon Press, New York, 1997 | MR

[22] J. Eschmeier, M. Putinar, Spectral Decompositions and Analytic Sheaves, London Math. Soc. Monogr. (N.S.), 10, The Clarendon Press, New York, 1996 | MR

[23] A. Yu. Pirkovskii, “Biprojective topological algebras of homological bidimension $1$”, J. Math. Sci. (New York), 111:2 (2002), 3476–3495 | DOI | MR

[24] A. Pietsch, “Zur Theorie der topologischen Tensorprodukte”, Math. Nachr., 25 (1963), 19–30 | DOI | MR

[25] O. Yu. Aristov, Arens–Michael Envelopes of Nilpotent Lie Algebras, Functions of Exponential Type, and Homological Epimorphisms, 2018, arXiv: 1810.13213