Local Multipliers of *C**-Algebras
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# The Symmetric Algebra of Quotients and its Bounded Analogue

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## 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.

## Keywords

Prime Ring Functional Calculus Double Centraliser Dense Open Subset Semiprime Ring## Preview

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