The Symmetric Algebra of Quotients and its Bounded Analogue

Abstract

We now set up the scenario for local multipliers of C*-algebras. In the first section, the symmetric ring of quotients Q _s (R) of a semiprime ring R is introduced and some of the basic properties of Q _s (R), as well as of its centre C(R), called the extended centroid of R, will be studied. If R carries an involution, then this can be extended to Q _s (R); this is the main advantage of the symmetric ring over other, one-sided rings of quotients. In the case of a C*-algebra A, Q _s (A) becomes a *-algebra with positive-definite involution. Hence, it is possible to define order-bounded elements in the sense of Han- delman and Vidav and to distinguish the *-subalgebra Q _b (A) of all bounded elements within Q _s (A) (Section 2.2). Using the identity element as an order-unit, Q _b (A)sa is an order-unit space and so Q_b(A) is a pre- C*-algebra; its completion is the algebra of local multipliers of A, denoted by M _loc (A) and introduced in Section 2.3.