Vol. 103, No. 1, 1982

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The splitting of operator algebras. II

Steve Wright

Vol. 103 (1982), No. 1, 243–249

Let {Aα : α A} be a family of C-algebras (resp., W-algebras). For α0 A, we let Pα0 : αAα Aα0 denote the canonical coordinate projection of αAα onto Aα0. If B is a C-(resp., W-) subalgebra of αAα, we say that B splits if B = αPα(B). In this note, we give conditions both necessary and sufficient for B to split. In the C-category, these conditions are given in terms of separation properties of the spectrum and primitive ideal space of B, and in the W-category, the conditions are expressed in terms of disjointness of certain subsets of the center of B. We also give examples to show that these conditions cannot be weakened, and are hence the best possible of their kind.

Mathematical Subject Classification 2000
Primary: 46L05
Received: 12 June 1980
Revised: 24 June 1981
Published: 1 November 1982
Steve Wright