Completely Continuous Banach Algebras

Publish Year: 1398
نوع سند: مقاله ژورنالی
زبان: English
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شناسه ملی سند علمی:

JR_IJNAA-10-1_006

تاریخ نمایه سازی: 11 آذر 1401

Abstract:

For a Banach algebra \mathfrak{A}, we introduce ~c.c(\mathfrak{A}), the set of all \phi\in \mathfrak{A}^* such that \theta_\phi:\mathfrak{A}\to \mathfrak{A}^* is a completely continuous operator, where \theta_\phi is defined by \theta_\phi(a)=a\cdot\phi for all a\in \mathfrak{A}. We call \mathfrak{A}, a completely continuous Banach algebra if c.c(\mathfrak{A})=\mathfrak{A}^*. We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J.E Gale, T.J. Ransford and M. C. White: Is there exist an infinite-dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.

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Faculty of Mathematical Sciences, Malayer University, P. O. Box ۱۶۸۴۶-۱۳۱۱۴, Malayer, Iran