On the connection between Bochner's theorem on positive definite maps and Choi theorem on complete positivity (2509.02529v1)

Abstract: In this work, we establish a connection between Bochner's theorem on positive definite maps and Choi theorem on complete positivity. We begin by defining a convolution product between maps from the contracted semigroup algebra $\mathbb{C}0[S]$ of a semigroup $S$ to an arbitrary associative algebra $\mathcal{A}$. The convolution product makes the space $L(\mathbb{C}_0[S],\mathcal{A})$ of linear maps from $\mathbb{C}_0[S]$ to $\mathcal{A}$ an associative algebra. We prove that the convolution algebra $L(\mathbb{C}_0[S],\mathcal{A})$ and the tensor product algebra $\mathbb{C}_0[S] \otimes \mathcal{A}$ are isomorphic. As a consequence, in the specific case of the inverse semigroup of matrix units, we identify the product in the space of maps on the matrix algebras which is preserved by the Choi-Jamio{\l}kowski isomorphism as convolution. Then, by defining the Fourier transform of a map from $\mathbb{C}_0[S]$ to $M_n(\mathbb{C})$, we derive the Fourier inversion formula when $S$ is a finite inverse semigroup. As a corollary of this formula, we show that in the case of the inverse semigroup of matrix units, the Fourier transformation of a map with respect to the identity representation becomes the Choi matrix of the map and the Fourier inversion formula becomes the Choi inversion formula. Then, by defining the notion of matrix valued positive definite maps, we prove Bochner's theorem in the context of finite inverse semigroup. It is demonstrated that Bochner's theorem reduces to Choi theorem on completely positive maps when the inverse semigroup of matrix units is considered. Additionally, the necessary and sufficient condition on a representation $\rho:M_m \to M{d_{\rho}}(\mathbb{C})$ such that the Complete positivity vs. positivity correspondence holds between a linear map $\Phi: M_m(\mathbb{C}) \to M_n(\mathbb{C})$ and its Fourier Transform $\widehat{\Phi}(\rho)$ is obtained.