Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^* = Hom_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^*$ is an *invertible* $A$-$A$ bimodule. That is, there is another $A$-$A$ bimodule $L$ and $A$-$A$ bimodule isomorphisms $L \otimes_A A^* \cong A \cong A^* \otimes_A L$.

One class of algebras which has this property are the Frobenious algebras. One of the classical definitions of a Frobenius algebra is that it is an algebra with an isomorphism of right $A$-modules ${A^*}_A \cong A_A$. If this is an isomorphism of *bimodules* then this is a symmetric Frobenius algebra. More generally we have ${}_A{A^*}_A \cong {}_A{}^\sigma A_A$, where the right-hand side is simply $A$ as a bimodule but where the left action is twisted by the Nakayama isomorphism $\sigma$. In particular since the Nakayama isomorphism is an isomorphism, $A^*$ is an invertible bimodule.

**Question**: If $A$ is an algebra such that $A^*$ is an invertible bimodule, does $A$ admit the structure of a Frobenius algebra?

Upon reviewing some old notes to myself, apparently at one time I believed that the answer to the above question is yes. However I don't remember the reasoning and didn't record a reference. Further, I am suspicious of my old self because in general there are certainly invertible bimodules which do not come from twisting the left action of the trivial bimodule. I would be happy to understand a counterexample or to find out that my old self was right.

One motivation for studying these algebras is that they arise naturally in extended topological field theory. There is a certain variant of 2D framed tqfts (the "non-compact" variant) and these algebras are in bijection with those tqfts with values in the Morita 2-category. So I would also be interested in anything further that could be said about these algebras, even with further assumptions like $k$ being characteristic zero.