A new C*-enlargement of a C*-algebra A nested between the local multiplier
algebra Mloc​(A) of A and its injective envelope I(A) is
introduced. Various aspects of this maximal C*-algebra of quotients,
Qmax​(A), are studied, notably in the setting of AW*-algebras. As a
by-product we obtain a new example of a type I C*-algebra A such that
Mloc​(Mloc​(A))î€ =Mloc​(A).Comment: 37 page