Motivated by formal similarities between the continuum limit of the Ising model and the Unruh effect, this paper connects the notion of an Ishibashi state in boundary conformal field theory with the Tomita–Takesaki theory for operator algebras. A geometrical approach to the definition of Ishibashi states is presented, and it is shown that, when normalizable, the Ishibashi states are cyclic separating states, justifying the operator state corespondence. When the states are not normalizable Tomita–Takesaki theory offers an alternative approach based on left Hilbert algebras, making possible extensions of our construction and the state-operator correspondence.