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We consider moduli spaces of Azumaya algebras on K3 surfaces and construct anexample. In some cases we show a derived equivalence which corresponds to aderived equivalence between twisted sheaves. We prove if $A$ and $A'$ areMorita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) -c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stackof Azumaya algebras with the same underlying gerbe, if non empty, is a properalgebraic space.

An Explicit Derived Equivalence of Azumaya Algebras on K3 Surfaces via Koszul Duality