The Symmetric Algebra of Quotients and its Bounded Analogue
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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 Qb(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.
KeywordsPrime Ring Functional Calculus Double Centraliser Dense Open Subset Semiprime Ring
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