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Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is multiplication. The *stable center* is $\underline{Z}(A)=Z(A)/I$. Equivalently, $\underline{Z}(A)=\underline{Hom}_{A^{op}\otimes A}(A,A)$ in the stable category of $A$-bimodules while ${Z}(A)={Hom}_{A^{op}\otimes A}(A,A)$.

I wanted to show that the natural projection $Z(A)\twoheadrightarrow \underline{Z}(A)$ induces a surjection between groups of units $Z(A)^\times\twoheadrightarrow \underline{Z}(A)^\times$. I can show it when $A$ is indecomposable as a bimodule over itself since in that case, if $A$ is not separable then $I$ is nilpotent, and if $A$ is separable then the stable center is trivial.

Any clue about the general case?