Multipliers of Pedersen’s ideal

Double multipliers on topological algebras

“…Introduction. The theory of double multipliers (i.e., of double centralizers) was developed for topological algebras by Johnson [7] and further investigated in the case of Banach algebras and C * -algebras by Busby [2], Fontenot [5], Taylor [17], Tomiuk [18], Argün and Rowlands [1], and others; see the monographs [9,12] for additional references. If A is a commutative C * -algebra, that is A = C 0 (X)− the algebra of all complex-valued continuous functions which vanish at infinity on a locally compact Hausdorff space X−, then M d (A), the algebra of all double multipliers of A, is C b (X)− the algebra of all complex-valued bounded continuous functions on X [21].…”

“…If A is a commutative C * -algebra, that is A = C 0 (X)− the algebra of all complex-valued continuous functions which vanish at infinity on a locally compact Hausdorff space X−, then M d (A), the algebra of all double multipliers of A, is C b (X)− the algebra of all complex-valued bounded continuous functions on X [21]. The noncommutative generalization of the relationship between C 0 (X) and C(X) was found to be useful in the work of Busby [2], Taylor [17], Davenport [4], and Lazar and Taylor [12].…”

The strong bidual of Γ(K)

“…Let x e Γ(K Λ ) + and {e λ } a positive increasing approximate identity for A contained in K A . By virtue of [12,Chapter 5], {x 1/2 e λ x 1/2 } is a /c-bounded subset of Γ(K) + ; moreover, by [12, 3.4, p. 8…”

“…It is known that J is a self adjoint two-sided ideal of Γ(K A ) [12,Chapter 8]. Now define J J -= {/ef(^y:/|J r ={0}} and P-= {/eA':/|I= {0}} .…”

“…In fact the main result is obtained by regarding the C*-algebra A with σ-compact spectrum as a sort of inductive limit of PCS-algebras. The reader is referred to [10], [17], [18] for basic definitions and concepts of topological vector spaces and to [5], [12], [14], [15], [16] for basic concepts and definitions of multipliers and Pedersen's ideal.…”