Differential graded Brauer groups

We consider central simple $K$-algebras which happen to bedifferential graded $K$-algebras. Two such algebras $A$ and $B$are considered equivalent if there are bounded complexes of finite dimensional$K$-vector spaces $C_A$ and $C_B$ such that the differential graded algebras $A\otimes_K {\rm End}_K^\bullet(C_A)$ and $B\otimes_K {\rm End}_K^\bullet(C_B)$ are isomorphic.Equivalence classes form an abelian group, which we call thedg Brauer group.We prove that this group is isomorphic to the ordinary Brauer group of the field $K$.