Duality, convexity and peak interpolation in the Drury-Arveson space (Q271730)

Let \(d\geq 1\) be an integer; in this paper, the authors deal with the closed algebra \(\mathcal{A}_d\) generated by the polynomial multipliers on the Drury-Arveson space. After giving a partial analogue of the Valskii decomposition, they establish one of the main results of this paper: an isometric identification of the first dual and the second dual of \(\mathcal{A}_d\). Namely, \(\mathcal{A}_{d}^{\ast\ast}\) is weak-* homeomorphic and completely isometrically isomorphic to the direct sum of the full multiplier algebra and a commutative von Neumann algebra \(\mathfrak{W}\). Consequently, \(\mathcal{A}_{d}^{\ast}\) is identified as a direct sum of the preduals of the full multiplier algebra and of the previous commutative von Neumann algebra : NEWLINE\[CARRIAGE_RETURNNEWLINE\mathcal{A}_{d}^{\ast\ast}\simeq\mathcal{M}_{d}\oplus_{\infty}\mathfrak{W}\,, \qquad\mathcal{A}_{d}^{\ast}\simeq\mathcal{M}_{d\ast}\oplus_{1}\mathfrak{W}_{\ast}.CARRIAGE_RETURNNEWLINE\]NEWLINENEWLINENEWLINEThe notion of \textit{\(\mathcal{A}_d\)-totally null sets} is introduced and many concrete examples of such spaces are provided, for instance \textit{the countable subsets of \(\mathbb{S}_d\)}, where \(\mathbb{S}_d\) is the unit sphere of \(\mathbb{C}^d\). Moreover, a characterization of \(\mathcal{A}_d\)-totally null sets in terms of \(\mathcal{A}_d\)-Henkin measures is given.NEWLINENEWLINEThe authors further prove a Choquet type theorem for linear functionals on subspaces of \(C(X)\) which do not necessarily contain the constant functions. Finally, they establish a version of peak interpolation adapted to multipliers in \(\mathcal{A}_d\) on closed \(\mathcal{A}_d\)-totally null sets.