Inner product on B∗B∗-algebras of operators on a free Banach space over the Levi-Civita field

Let CC be the complex Levi-Civita field and let c0(C)c0(C) or, simply, c0c0 denote the space of all null sequences z=(zn)n∈Nz=(zn)n∈N of elements of CC. The natural inner product on c0c0 induces the sup-norm of c0c0. In a previous paper Aguayo et al. (2013), we presented characterizations of normal projections, adjoint operators and compact operators on c0c0. In this paper, we work on some B∗B∗-algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators. We finish the paper with a characterization of closed subspaces of the B∗B∗-algebra of all adjoint and compact operators on c0c0 which admit normal complements.