Sajjad Khan, Choonkil Park

Hyers-Ulam stability of fuzzy Hilbert $C^{*}$-module homomorphisms and fuzzy Hilbert $C^{*}$-module derivations

Introduction

Hyers-ulam stability of fuzzy hilbert $c^{*}$-module homomorphisms and fuzzy hilbert $c^{*}$-module derivations. Study Hyers-Ulam stability of fuzzy Hilbert C*-module homomorphisms & derivations. Introduces fuzzy Hilbert C*-modules and uses the fixed point method for analysis.

Abstract

In the present paper, we introduce the notion of a fuzzy Hilbert $C^*$-module and study the Hyers-Ulam stability of fuzzy Hilbert $C^{*}$-module homomorphisms and fuzzy Hilbert $C^{*}$-module derivations in fuzzy Hilbert $C^*$-modules using the fixed point method.

Review

The paper's title clearly articulates its focus on the Hyers-Ulam stability of specific mappings within fuzzy algebraic structures. The abstract indicates the introduction of the "fuzzy Hilbert $C^*$-module" concept, which is a novel and potentially significant extension of classical Hilbert $C^*$-modules into the realm of fuzzy set theory. This work addresses a relevant area at the intersection of functional analysis, fuzzy mathematics, and stability theory, offering a valuable contribution by expanding the theoretical framework for analyzing operator-related structures under uncertainty. The core contribution appears to be the establishment of Hyers-Ulam stability for fuzzy Hilbert $C^*$-module homomorphisms and derivations. The chosen methodology, the fixed point method, is a well-established and powerful technique often employed in proving various types of functional equations' stability. By applying this method, the authors aim to demonstrate that approximate homomorphisms and derivations are close to true ones within the newly defined fuzzy Hilbert $C^*$-module setting. This application of a robust technique to a novel fuzzy structure is a central strength of the research. The paper presents an interesting and timely generalization of stability results to fuzzy settings, which has implications for modeling systems with inherent imprecision. The introduction of fuzzy Hilbert $C^*$-modules is a foundational step that could pave the way for further research in fuzzy functional analysis. While the abstract is concise, the work appears to be theoretically sound and rigorously executed given the use of the fixed point method. Future work could explore applications of these stability results in areas such as quantum information or signal processing where fuzzy $C^*$-algebras and their modules play a role. Overall, this paper offers a solid theoretical advancement and is recommended for publication.

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