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Exact and Approximate Operator Parallelism

  • Mohammad Sal Moslehian,
    Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad 91775, Iran
  • Ali Zamani,
    Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad 91775, Iran
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Abstract

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module.
Keywords: $C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-module $C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-module
MSC Classifications: 47A30, 46L05, 46L08, 47B47, 15A60 show english descriptions Norms (inequalities, more than one norm, etc.)
General theory of $C^*$-algebras
$C^*$-modules
Commutators, derivations, elementary operators, etc.
Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
47A30 - Norms (inequalities, more than one norm, etc.)
46L05 - General theory of $C^*$-algebras
46L08 - $C^*$-modules
47B47 - Commutators, derivations, elementary operators, etc.
15A60 - Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
 

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