Banach Algebras in Which Every Element is a Topological Zero Divisor

Bhatt, S. J.; Dedania, H. V.

doi:10.2307/2160793

Cited by 4 publications

(5 citation statements)

References 2 publications

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“…In [3] S. J. Bhatt and H. V. Dedania established the following result concerning complex Banach algebras in which every element is a TDZ: Theorem 1.1. [3, Theorem 1] Every element of a complex Banach algebra A is a TDZ, if at least one of the following holds:…”

Section: Identities Approximate Identities and Tdzmentioning

confidence: 99%

“…However, it is significantly more difficult to decide about the containment for the latter two classes of Banach algebras in Theorem 1.1. To emphasize this, we revisit some further examples which appear in [3]:…”

Section: Identities Approximate Identities and Tdzmentioning

confidence: 99%

“…Consequently, it follows from [8,Theorem 2.11] that Fejér's kernel is an approximate identity in both algebras. The authors of [3] did not explicitly give an example of a Banach algebra satisfying condition (ii) in Theorem 1.1. However, by the Gelfand-Naimark Theorem it can straightforwardly be established that every non-unital commutative C * -algebra A satisfies ∂A = M(A).…”

Section: Identities Approximate Identities and Tdzmentioning

confidence: 99%

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Identities, approximate identities and topological divisors of zero in Banach algebras

Schulz

Brits

Melanie

2017

Journal of Mathematical Analysis and Applications

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In [3] S. J. Bhatt and H. V. Dedania exposed certain classes of Banach algebras in which every element is a topological divisor of zero. We identify a new (large) class of Banach algebras with the aforementioned property, namely, the class of non-unital Banach algebras which admits either an approximate identity or approximate units. This also leads to improvements of results by R. J. Loy and J. Wichmann, respectively. If we observe that every single example that appears in [3] belongs to the class identified in the current paper, and, moreover, that many of them are classical examples of Banach algebras with this property, then it is tempting to conjecture that the classes exposed in [3] must be contained in the class that we have identified here. However, we show somewhat elusive counterexamples. Furthermore, we investigate the role completeness plays in the results and show, by giving a suitable example, that the assumptions are not superfluous. The ideas considered here also yields a pleasing characterization: The socle of a semisimple Banach algebra is infinite-dimensional if and only if every socle-element is a topological divisor of zero in the socle.2010 Mathematics Subject Classification. 43H15, 46H05, 46H10.

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Section: Identities Approximate Identities and Tdzmentioning

confidence: 99%

Section: Identities Approximate Identities and Tdzmentioning

confidence: 99%

Section: Identities Approximate Identities and Tdzmentioning

confidence: 99%

See 1 more Smart Citation

Identities, approximate identities and topological divisors of zero in Banach algebras

Schulz

Brits

Melanie

2017

Journal of Mathematical Analysis and Applications

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“…However every positive zero divisor is automatically a two sided zero divisor. It is well known that every element of a non unital C * algebra is a topological zero divisor, see [1]. But it is not true that every non unital C * algebra satisfies in the property that all its elements are zero divisor.…”

Section: Z * Algebrasmentioning

confidence: 99%

A Note on Z* algebras

Taghavi

2013

Preprint

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We study some properties of Z * algebras, those C * algebras which all positive elements are zero divisors. By means of an example we show that an extension of a Z * algebra by a Z * algebra is not necessarily a Z * algebra. However we prove that the extension of a non Z * algebra by a non Z * algebra is a non Z * algebra. We also prove that the tensor product of a Z * algebra by a C * algebra is a Z * algebra. As an indirect consequence of our methods we prove the following inequality type results: i)Let an be a sequence of positive elements of a C * algebra A which converges to zero. Then there are positive sequences bn of real numbers and cn of elements of A which converge to zero such that a n+k ≤ bnc k . ii)Every compact subset of the positive cones of a C * algebra has an upper bound in the algebra.

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“…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%

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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Banach Algebras in Which Every Element is a Topological Zero Divisor

Cited by 4 publications

References 2 publications

Identities, approximate identities and topological divisors of zero in Banach algebras

Identities, approximate identities and topological divisors of zero in Banach algebras

A Note on Z* algebras

Invertibles in topological rings: a new approach

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