International Journal For Multidisciplinary Research

E-ISSN: 2582-2160   •   Impact Factor: 9.24

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Volume 8 Issue 5
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Defect Methods for Amenability and Ultrapower Stability in Banach Algebras

Author(s) Mohit Kumar Jha, Dr. Kamla Nand Jha
Country India
Abstract This paper develops a defect-based approach to several themes appearing in the proposed doctoral study of non-standard Banach-algebra structures. The central idea is to replace an exact algebraic requirement by a family of quantitative defects measured on finite subsets. For amenability, the defect of a tensor records both its failure to commute with the algebra and its failure to act as an approximate identity after multiplication. This formulation is equivalent to the bounded approximate-diagonal criterion and gives a practical route for distinguishing amenable from non-amenable behavior. The same viewpoint is applied to approximately inner derivations, Arens regularity, Gelfand theory, group algebras, and ultrapowers. Particular attention is paid to the fact that tensor products do not automatically commute with ultrapower constructions, so uniform estimates must be separated from unjustified lifting arguments. Finite-dimensional and finite-group examples are included, together with a research programme for quotient algebras, semigroup algebras, operator algebras, and spectral applications. The paper therefore supplies a new title and a focused mathematical direction while remaining entirely within the scope of the original synopsis.
Keywords Banach algebra; amenability defect; approximate diagonal; derivation; Arens regularity; ultrapower; Gelfand transform; group algebra
Field Mathematics
Published In Volume 8, Issue 1, January-February 2026
Published On 2026-01-03
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E-ISSN 2582-2160
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CrossRef DOI prefix of IJFMR is 10.36948/ijfmr

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