The alternative Dunford-Pettis property in <IMG >-algebras and von Neumann preduals期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

The alternative Dunford-Pettis property in -algebras and von Neumann preduals

Authors:	Leslie J. Bunce Antonio M. Peralta

Affiliation:	Department of Mathematics, University of Reading, Reading RG6 2AX, Great Britain ; Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Abstract:	A Banach space is said to have the alternative Dunford-Pettis property if, whenever a sequence $x_{n} rightarrow x$ weakly in with $Vert x_{n}Vert rightarrow Vert xVert$ , we have $rho_{n} (x_{n}) rightarrow 0$ for each weakly null sequence $(rho_{n})$ in X. We show that a -algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for -algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.

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