Continuous Extension of Trace and Determinant

This is one of the main chapters in this book. It contains the extension theory of trace and determinant of finite rank operators to wider algebras of linear operators. Since the functionals trace and determinant are not continuous with respect to the operator norm on the finite rank operators acting in infinite dimensional spaces, we study extensions of these functions to algebras of operators with other norms. In this chapter we also investigate the main properties of the extended determinant. In particular the connection between zeros of the determinant and eigenvalues of the corresponding operators is given.