A note on topological divisors of zero and division algebras

Marcos, Juan Carlos López; Rodríguez-Palacios, A.; Velasco, María V.

doi:10.1007/s13398-014-0168-4

Cited by 1 publication

(2 citation statements)

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“…Trivial examples of absolutely semi-normed rings are the absolutely semi-valued rings and the algebras of continuous functions in view of Example 3. On the other hand, almost absolutely semi-normed rings contain absolutely invertibles in view of Lemma 1 (6), so in particular they must be unital ( 1 = 1) in virtue of Lemma 1 (5).…”

Section: Example 3 In Everymentioning

confidence: 99%

“…In this manuscript we generalize the point, continuous and residual spectra of an operator to algebras. We will also analyse the topological properties of the group of invertibles [1][2][3] and the topological divisors of zero [4][5][6] in general topological rings. We also refer the reader to [7][8][9] for further information about extending the classical Operator Spectral Theory to the scope of Banach algebras through the Gelfand Theory and the Continuous Functional Calculus.…”

Section: Introductionmentioning

confidence: 99%

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Invertibles in topological rings: a new approach

García-Pacheco

Miralles

Murillo‐Arcila

2021

RACSAM

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Every element in the boundary of the group of invertibles of a Banach algebra is a topological zero divisor. We extend this result to the scope of topological rings. In particular, we define a new class of semi-normed rings, called almost absolutely semi-normed rings, which strictly includes the class of absolutely semi-valued rings, and prove that every element in the boundary of the group of invertibles of a complete almost absolutely semi-normed ring is a topological zero divisor. To achieve all these, we have to previously entail an exhaustive study of topological divisors of zero in topological rings. In addition, it is also well known that the group of invertibles is open and the inversion map is continuous and C-differentiable in a Banach algebra. We also extend these results to the setting of complete normed rings. Finally, this study allows us to generalize the point, continuous and residual spectra to the scope of Banach algebras.

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Section: Example 3 In Everymentioning

confidence: 99%