Truncation and duality results for Hopf image algebras

Abstract

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