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Motivated by formal similarities between the continuum limit of the Isingmodel and the Unruh effect, this paper connects the notion of an Ishibashistate in boundary conformal field theory with the Tomita--Takesaki theory foroperator algebras. A geometrical approach to the definition of Ishibashi statesis presented, and it is shownthat, when normalisable the Ishibashi states arecyclic separating states, justifying the operator state correspondence. Whenthe states are not normalisable Tomita--Takesaki theory offers an alternativeapproach based on left Hilbert algebras, opening the way to extensions of ourconstruction and the state-operator correspondence

Boundary conformal fields and Tomita–Takesaki theory