Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hyers-Ulam stability of homomorphisms and *-homomorphisms on Hilbert C*-modules

Sajjad Khan, Choonkil Park

TL;DR

This paper addresses the Hyers-Ulam stability of homomorphisms and $\ast$-homomorphisms on Hilbert $C^{*}$-modules by employing a fixed-point approach. The authors introduce $\ast$-homomorphisms on Hilbert $C^{*}$-modules and prove the existence and uniqueness of stabilizing maps $H$ under contractive control functions, with explicit error bounds $\|f(x)-H(x)\|\le \frac{1}{2-2L}\varphi(x,x,0)$. They also establish Hyers-Ulam-Rassias stability for the $p$-type controls and provide analogous results for $\ast$-homomorphisms, ensuring that $H$ preserves the module operations and star-structure. The results extend classical stability theory to the setting of Hilbert $C^{*}$-modules and offer quantitative stability criteria via generalized metric spaces and fixed-point theory.

Abstract

In this paper, we introduce the idea of $\ast$-homomorphism on a Hilbert $C^{*}$-module. Furthermore, we prove the Hyers-Ulam stability of homomorphisms and $\ast$-homomorphisms on Hilbert $C^{*}$-modules using the fixed point method.

Hyers-Ulam stability of homomorphisms and *-homomorphisms on Hilbert C*-modules

TL;DR

This paper addresses the Hyers-Ulam stability of homomorphisms and -homomorphisms on Hilbert -modules by employing a fixed-point approach. The authors introduce -homomorphisms on Hilbert -modules and prove the existence and uniqueness of stabilizing maps under contractive control functions, with explicit error bounds . They also establish Hyers-Ulam-Rassias stability for the -type controls and provide analogous results for -homomorphisms, ensuring that preserves the module operations and star-structure. The results extend classical stability theory to the setting of Hilbert -modules and offer quantitative stability criteria via generalized metric spaces and fixed-point theory.

Abstract

In this paper, we introduce the idea of -homomorphism on a Hilbert -module. Furthermore, we prove the Hyers-Ulam stability of homomorphisms and -homomorphisms on Hilbert -modules using the fixed point method.

Paper Structure

This paper contains 4 sections, 10 theorems, 50 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Stability of Hilbert $C^{*}$-module homomorphisms
  3. Stability of Hilbert $C^{*}$-module $\ast$-homomorphisms
  4. Conclusion

Key Result

Theorem 1.3

Cadariu1 Let (X,d) be a complete generalized metric space and $J: X\rightarrow X$ be a strictly contractive mapping with Lipschitz constant $L<1$. Then, for all $x\in X$, either for all nonnegative integers $n$ or there exists a positive integer $n_0$ such that $(1)$$d(J^n x,J^{n+1}x)<\infty$ for all $n\geq n_0$; $(2)$ the sequence $\{J^n x\}$ converges to a fixed point $y^*$ of $J$; $(3)$$y^*$ i

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 10 more