For a fixed tracial unital Banach \(*\) -probability space \(\left( A,\tau \right) \) , we constructed the corresponding 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 \(A_{0}\) induced by the trace \(\tau \) on A. The \(A_{0}\) -Hardy space \(\textbf{H}_{A_{0}:2}\left( D_{1}\right) \) was constructed and adjointable Block Toeplitz Banach space operators over \(A_{0}\) acting on \(\textbf{H}_{A_{0}:2}\left( D_{1}\right) \) were studied in Cho (Block-Toeplitz operators on the hardy space induced by a Tracial Unital Banach \(*\) -probability space, 2024, to submitted). In this paper, we are interested in the cases where \(\left( A,\tau \right) \) is a free product Banach \(*\) -probability space, \(\underset{k\in \Lambda }{\star }\left( A_{k},\tau _{k}\right) \) of Banach \(*\) -probability spaces \(\left\{ \left( A_{k},\tau _{k}\right) \right\} _{k\in \Lambda }\) of A, where \(\Lambda \) is a countable (finite or infinite) index set. As applications, we consider a case where such A is a unital Banach \(*\) -algebra generated by mutually free multi-semicircular elements.
