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A Morita class of symmetric special Frobenius algebras A in the modular tensor category of a chiral CFT determines a full CFT on oriented world sheets. For unoriented world sheets, A must in addition possess a reversion, i.e. an isomorphism from A^opp to A squaring to the twist. Any two reversions of an algebra A differ by an element of the group Aut(A) of algebra automorphisms of A. We establish a group homomorphism from Aut(A) to the Picard group of the bimodule category C_AA, with kernel consisting of the inner automorphisms, and we refine Morita equivalence to an equivalence relation between algebras with reversion.

Ribbon categories and (unoriented) CFT: Frobenius algebras, automorphisms, reversions