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I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra.

Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$ the set of $A-A$-bimodules which are invertible for the tensor products over $A$ (this means that $X \in \mathrm{Pic}(A)$ if there exists a $A-A$-bimodule $Y$ such that $X \otimes_A Y \simeq A$ and $Y \otimes_A X \simeq A$ as $A-A$ bimodules and these isomorphisms are compatibles with some natural maps).

The set $\mathrm{Pic}(A)$ is an algebraic group and I denote by $\mathrm{Pic}^0(A)$ the connected component of $\mathrm{Pic}(A)$ containing $A$.

I am looking for a description of the center of $\mathrm{Pic}^0(A)$. In some cases (for instance if $A$ is Azumaya), $\mathrm{Pic}(A)$ is commutative, so that the center is all $\mathrm{Pic}^0(A)$. But I don't know what is the situation in general.

Is it possible to relate the center of $\mathrm{Pic}^0(A)$ to $\mathrm{Pic}(Z(A))$? (where $Z(A)$ is the center of $A$). For the precise situation I am interested in, I may assume that the map $Z(A) \rightarrow A$ is split (if that's of any help).

Thanks a lot!

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