Multipliers of embedded discs (Q2258312)

Let \(1 \leq d \leq \infty\). Let \(\mathbb{B}_d\) be the open unit ball in \(\mathbb{C}^d\) (in \(l^2\) if \(d = \infty\)). Let \(V\) be a variety in \(\mathbb{B}_d\) and \[ \mathcal{H}_V = \overline{\text{span}}\{k_y : y \in V\}, \] where \(k\) is the reproducing kernel \(k(x, y) = (1 - \langle x, y \rangle)^{-1}\), \(x, y \in \mathbb{B}_d\). Let \(\mathcal{M}_V\) denote the multiplier algebra corresponding to the reproducing kernel Hilbert space \(\mathcal{H}_V\). In [Trans. Am. Math. Soc. 367, No. 2, 1121--1150 (2015; Zbl 1312.47092)], \textit{C. Ramsey} with the first and third author obtained a complete classification of isomorphic multiplier algebras of the above type: Let \(d < \infty\). Let \(V\) and \(W\) be varieties in \(\mathbb{B}_d\). Suppose that the multiplier algebras \(\mathcal{M}_V\) and \(\mathcal{M}_W\) are isomorphic. Then \(V\) and \(W\) are biholomorphic. The summary of the paper under review reads: ``We consider a number of examples of multiplier algebras on Hilbert spaces associated to discs embedded into a complex ball in order to examine the isomorphism problem for multiplier algebras on complete Nevanlinna-Pick reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many discs in the ball of \(l^2\) which are multiplier biholomorphic but have non-isomorphic multiplier algebras. We also show that there are closed discs in the ball of \(l^2\) which are varieties, and examine their multiplier algebras. In finite balls, we provide a counterpoint to a result of Alpay, Putinar and Vinnikov [\textit{D. Alpay} et al., Commun. Pure Appl. Anal. 2, No.~2, 139--145 (2003; Zbl 1045.32015)] by providing a proper rational biholomorphism of the disc onto a variety \(V\) in \(\mathbb{B}_2\) such that the multiplier algebra is not all of \(H^\infty(V)\). We also show that the transversality property, which is one of their hypotheses, is a consequence of the smoothness that they require.'' Note that the proof of Theorem 2.4, and therefore Corollaries 2.6 and 2.7, are incomplete in the case where \(d = \infty\); see the erratum [ibid. 9, No. 2, 323--327 (2015; Zbl 1345.47064)].