Abstract. We prove that the Hochschild homology (and cohomology) of a symmetric open Frobenius algebra A has a natural coBV and BV structure. The underlying coalgebra and algebra structure may not be resp. counital and unital. Moreover we prove that the product and coproduct satisfy the Frobenius compatibility condition i.e. the coproduct on HH∗(A) is a map of left and right HH∗(A)-modules. If A is commutative, we also introduced a natural BV structure on the relative Hochschild homology ˜HH∗(A) after a shift in degree. We anticipate that the product of this BV structure to be related to the Goresky-Hingston product on the cohomology of free loop spaces
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