Classification of noncommutative domain algebras (Q447889)

In this short article, the authors complete their classification of noncommutative domain algebras [\textit{A. Arias} and \textit{F. Latrémolière}, J. Oper. Theory 66, No.~2, 425--450 (2011; Zbl 1265.46096)], a special class of operator algebras which are noncommutative analogues of algebras of holomorphic functions on domains in Hermitian spaces. These algebras have been introduced by \textit{G. Popescu} [Mem. Am. Math. Soc. 964 (2010; Zbl 1194.47001)] and generalize Hardy algebras.NEWLINENEWLINEEach domain algebra \(A_f\) is associated with a polynomial \(f\) (with complex coefficients) on \(n\) variables \(X_1,\dots, X_n\), called the symbol of \(A_f\). The main result of the present paper states that, if \(f,g\) are two such polynomials with \(n\) and \(m\) variables, respectively, then \(A_f\) is completely isometrically isomorphic to \(A_g\) if and only if \(n=m\) and \(f\) and \(g\) are scale-permutation equivalent, meaning that there is a permutation \(\sigma\) of \(\{1,\dots, n\}\) and scalars \(\lambda_1,\dots,\lambda_n\in \mathbb{R}\) with \(f(X_1,\dots, X_n)=g(\lambda_1X_{\sigma(1)},\dots,\lambda_nX_{\sigma(n)})\).