Regular abelian Banach algebras of linear maps of operator algebras

With H a complex Hilbert space we study regular abelian Banach subalgebras of the Banach algebra of bounded linear maps of B(H) into itself. If a ? b denotes the map x → axb, a, b, x ? B(H), it is shown that normalized positive maps in algebras of the form A ? A with A an abelian C¹-algebra, can be described by a generalized Bochner theorem.