Polynomial extension property in the classical Cartan domain $${\mathcal {R}_{II}}$$ R II
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Archiv der Mathematik
Polynomial extension property in the classical Cartan domain RII Krzysztof Maciaszek
Abstract. In this work, it is shown that for the classical Cartan domain RII consisting of symmetric 2 × 2 matrices, every algebraic subset of RII , which admits the polynomial extension property, is a holomorphic retract. Mathematics Subject Classification. 32D15. Keywords. Polynomial extension property, Holomorphic retracts, Classical Cartan domain.
1. Introduction. Denote by RII the classical Cartan domain of type II, i.e. RII = A ∈ M2×2 (C) : A = AT , I − A∗ A > 0 . Here M2×2 (C) stands for the space of 2 × 2 matrices with complex elements and I is the unit matrix. In this paper, we examine certain varieties V of RII for which the restrictions of polynomials to V can be extended to holomorphic functions on RII without increasing their supremum norm. The origin of that sort of studies goes back to Rudin’s book [13] and one of the goals is to determine whether such a set V is a holomorphic retract. The important results were obtained by Agler and McCarthy [2] for the subsets of the bidisc. The authors also provided motivations concerning connections of the extension property to the Nevanlinna-Pick interpolation and to the von Neumann inequality. Generalizations of the problem on the bidisc have split in particular into studies of sets in the higher dimensional polydisc on the one hand, and on the other, in the symmetrized bidisc, a domain which is an image of the bidisc under the map (z, w) → (z + w, zw). In case of the polydisc, there are known partial results concerning relatively polynomially convex (or even algebraic) The author was supported by the NCN Grant SONATA BIS No. 2017/26/E/ST1/00723.
K. Maciaszek
Arch. Math.
subsets V, see [9,10], and [11]. The problem in D3 is completely solved only with the additional assumption that the extension operator may be chosen to be linear (cf. [6]). The description of varieties admitting the extension property in the symmetrized bidisc was obtained in [1] and independently in [4]. The symmetrized bidisc, as well as the other domain, the tetrablock, are related to the μ-synthesis problem (cf. [15]). These domains are also of interest for the theory of invariant distances. Recall that the latter can be described as z11 z12 . an image π(RII ) of a mapping π(z) = (z11 , z22 , det z), where z = z21 z22 Therefore the Cartan domain RII plays a similar role for the tetrablock as the bidisc plays for the symmetrized bidisc. Cartan domains are the open unit balls of Cartan factors, which appear as important examples of J ∗ -algebras, the class that from the operator theoretic point of view may be considered as a generalization of C ∗ -algebras (see [7]). We remark that the classification of holomorphic retracts in the open unit ball of any commutative C ∗ -algebra with identity was obtained in [12]. 2. Main theorem. Let V be an arbitrary subset of Cn . We will say that a function f : V → C is holomorphic on V if, for every point ζ ∈ V , it can be extended to
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