Truncation and Duality Results for Hopf Image Algebras

Teodor Banica ¹

¹ Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise, France

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) no. 2, pp. 161-179

Résumé

Associated to an Hadamard matrix $H\in M_N(\mathbb C)$ is the spectral measure $\mu\in\mathcal P[0,N]$ of the corresponding Hopf image algebra, $A=C(G)$ with $G\subset S_N^+$. We study a certain family of discrete measures $\mu^r\in\mathcal P[0,N]$, coming from the idempotent state theory of $G$, which converge in Cesàro limit to $\mu$. Our main result is a duality formula of type $\int_0^N(x/N)^p\,d\mu^r(x)=\int_0^N(x/N)^r\,d\nu^p(x)$, where $\mu^r,\nu^r$ are the truncations of the spectral measures $\mu,\nu$ associated to $H,H^t$. We also prove, using these truncations $\mu^r,\nu^r$, that for any deformed Fourier matrix $H=F_M\otimes_QF_N$ we have $\mu=\nu$.

Détail

DOI : 10.4064/ba62-2-5

Keywords: associated hadamard matrix mathbb spectral measure mathcal corresponding hopf image algebra subset study certain family discrete measures mathcal coming idempotent state theory which converge ces limit main result duality formula type int int where truncations spectral measures associated prove using these truncations deformed fourier matrix otimes have