On order structure and operators in L ∞(μ)

Irina Krasikova 1 , Miguel Martín 2 , Javier Merí 2 , Vladimir Mykhaylyuk 3 ,  and Mikhail Popov 3
  • 1 Department of Mathematics, Zaporizhzhya National University, Zaporizhzhya, Ukraine
  • 2 Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Granada, Spain
  • 3 Department of Mathematics, Chernivtsi National University, Chernivtsi, Ukraine

Abstract

It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

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