On Hardy-Like Spaces Induced by Tracial Unital Banach \(*\)-Probability Spaces

Starting from a unital Banach \(*\) $\ast$ -probability space \(\left( A,\tau \right) \) $\left(A,,,\tau \right)$ , where \(\tau \) $\tau$ is a trace on a Banach \(*\) $\ast$ -algebra A equipped with its unity \(1_{A}\) $1_A$ , we construct the corresponding Banach space \(\left( A_{0},\left[ ,\right] _{\tau }\right) \) $\left(A_0,,,{\left[,\right]}_{\tau }\right)$ equipped with its definite, or indefinite inner product \(\left[ ,\right] _{\tau }\) ${\left[,\right]}_{\tau }$ (depending on both A and \(\tau \) $\tau$ ), where \(A_{0}=A/ker\left( \tau \right) \) $A_0=A/ker\left(\tau \right)$ is the quotient Banach space. And then, we define a pure-algebraic vector space \(A_{0}\left[ \left[ X\right] \right] \) $A_0\left[\left[X\right]\right]$ , and topological vector spaces \(\mathscr {A}_{0}\left[ \left[ \Omega \right] \right] \) ${\mathcal{A}}_0\left[\left[\mathrm{\Omega }\right]\right]$ called the \(A_{0}\) $A_0$ -analytic algebras for suitable open domains \(\Omega \) $\mathrm{\Omega }$ of \(\left( A_{0},[,]_{\tau }\right) \) $\left(A_0,,,{[,]}_{\tau }\right)$ , and \(\textbf{H}_{A_{0}:2}\left( D_{1}\right) \) ${\mathbf{H}}_{A_0:2}\left(D_1\right)$ , called the \(A_{0}\) $A_0$ -Hardy space, where \(D_{1}\) $D_1$ is the unit open ball of \(A_{0}\) $A_0$ . The algebra and analysis on such vector spaces are considered. As applications, (i) we study certain operators on \(\textbf{H}_{A_{0}2}\left( D_{1}\right) \) ${\mathbf{H}}_{A_{0}2}\left(D_1\right)$ , especially, we define and consider block-Toeplitz operators as in classical Toeplitz theory, (ii) the operator theory and operator algebra of the Banach \(*\) $\ast$ -algebra \(\mathscr {T}\) $\mathcal{T}$ generated by the block-Toeplitz operators of (i), (iii) we investigate how semicircular elements of \(\left( A,\tau \right) \) $\left(A,,,\tau \right)$ act on \(\textbf{H}_{A_{0}:2}\left( D_{1}\right) \) ${\mathbf{H}}_{A_0:2}\left(D_1\right)$ , as elements of \(\mathscr {T}\) $\mathcal{T}$ of (ii), and (iv) a canonical free-probabilistic structure on \(\mathscr {T}\) $\mathcal{T}$ is considered.