For a fixed unital Banach \(*\) -probability space \(\left( A,\tau \right) \) , we construct a definite, or indefinite inner product space \(\left( A_{0},\left[ ,\right] _{\tau }\right) \) , where \(A_{0}=A/ker\left( \tau \right) \) is the quotient Banach space and \(\left[ ,\right] _{\tau }\) is a definite, or indefinite inner product on \(A_{0}\) determined by the trace \(\tau \) on the unital Banach \(*\) -algebra A. From this Banach space \(\left( A_{0},\left[ ,\right] _{\tau }\right) \) , a functional vector space, called the \(A_{0}\) -Hardy space \(\textbf{H}_{A_{0}:2}\left( D_{1}\right) \) , is constructed, where \(D_{1}\) is the open unit ball of \(A_{0}\) . Similar to the classical Toeplitz-operator theory, one can define Toeplitz-like adjointable Banach-space operators acting on \(\textbf{H}_{A_{0}:2}\left( D_{1}\right) \) . Our main results characterizes operator-theoretic properties of those Toeplitz-like operators over \(\left( A,\tau \right) \) . In particular, self-adjointness, projection-property, normality, isometry-property, and unitarity are characterized as in the usual operator theory on Hilbert spaces. As application, we study free distributions of certain types of our operator-valued Toeplitz-like operators.
