On field algebras in quantum theory with indefinite metric

The paper consists of two parts. The first discusses the problem of formulating the algebraic approach directly in a space with indefinite metric. One of the possible ways of constructing a net of local field algebras $\mathscr F(O)$ in the $J$ space (Krein space) is considered; it is based on the Bisognano–Wichmann formalism. The second part establishes a number of properties of "$WJ^*$ algebras" (weakly closed algebras with $J$ involution and unit in the $J$ space); the algebras $\mathscr F(O)$ belong to this class. The methods of the proof use generalizations of the fundamental concepts of Tomita–Takesaki theory for algebras with $J$ involution.