A BANACH ALGEBRA AND ITS EQUIVALENT SPACES OVER PATHS WITH A POSITIVE MEASURE

Commun. Korean Math. Soc. Published online March 10, 2020

Dong Hyun Cho
Kyonggi University

Abstract : Let $C[0,T]$ denote the space of continuous, real-valued functions on the interval $[0,T]$ and let $C_0[0,T]$ be the space of functions $x$ in $C[0,T]$ with $x(0)=0$. In this paper, we introduce a Banach algebra $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ on $C[0,T]$ and its equivalent space $\bar{\mathcal F}(\mathcal H)$, a space of transforms of equivalence classes of measures, which generalizes the Fresnel class ${\mathcal F}(\mathcal H)$. We also investigate their properties and derive an isomorphism between $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ and $\bar{\mathcal F}(\mathcal H)$. As applications, we evaluate generalized analytic Feynman integrals of the functions in $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ and the Fresnel integrals of functions in $\bar{\mathcal F}(\mathcal H)$, which play significant roles in Feynman integration theories and quantum mechanics. When $C[0,T]$ is replaced by $C_0[0,T]$, $\bar{\mathcal F}(\mathcal H)$ and $\bar{\mathcal S}_{\alpha,\beta;\varphi}$ reduce to ${\mathcal F}(\mathcal H)$ and the Cameron-Storvick's Banach algebra $\mathcal S$, respectively, which is the space of generalized Fourier-Stieltjes transforms of the complex-valued, finite Borel measures on $L^2[0,T]$.

Keywords : analytic Feynman integral, Banach algebra, Fresnel class, It\^o integral, Paley-Wiener-Zygmund integral, Wiener space