Idempotents in an ultrametric Banach algebra
-
Alain Escassut
alain.escassut@uca.fr
Downloads
DOI:
https://doi.org/10.4067/S0719-06462021000100161Abstract
Let IK be a complete ultrametric field and let \(A\) be a unital commutative ultrametric Banach IK-algebra. Suppose that the multiplicative spectrum admits a partition in two open closed subsets.
Then there exist unique idempotents \(u,\ v\in A\) such that \(\phi(u)=1, \ \phi(v)=0 \ \forall \phi \in U, \ \phi(u)=0 \ \phi(v)=1 \ \forall \phi \in V\). Suppose that IK is algebraically closed. If an element \(x\in A\) has an empty annulus \(r<|\xi-a|<s\) in its spectrum \(sp(x)\), then there exist unique idempotents \(u,\ v\) such that \(\phi(u)=1, \ \phi(v)=0\) whenever \( \phi(x-a)\leq r\) and \(\phi(u)=0, \ \phi(v)=1\) whenever \(\phi(x-a)\geq s\).
Keywords
J. Araujo, Prime and maximal ideals in the spectrum of the ultrametric algebra, Contemporary of the AMS, vol. 704, 2018.
V. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, AMS Survey and Monographs, vol. 33, 1990.
M. Chicourrat, and A. Escassut, “Banach algebras of ultrametric Lipschitzian functions”, Sarajevo Journal of Mathematics, vol. 14, no. 2, pp. 1–12, 2018. (27)
M. Chicourrat, B. Diarra, and A. Escassut, “Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions”, Bulletin of the Belgium Mathematical Society, vol. 26, no. 3, pp. 413–420, 2019.
M. Chicourrat, and A. Escassut, “Ultrafilters and ultrametric Banach algebras of Lipschitzian functions”, Advances in Operator Theory, vol. 5, no. 1, pp. 115–142, 2020.
M. Chicourrat, and A. Escassut, “A survey and new results on Banach algebras of ultrametric functions”, p-adic Numbers, Ultrametic Analysis and Applications, vol. 12, no. 3, pp. 185–202, 2020.
A. Escassut, Analytic elements in p-adic analysis, World Scientific Publishing, 1995.
A. Escassut, “Algèbres de Banach ultramétriques et algèbres de Krasner-tate”, Astérisque, no. 10, pp. 1-107, 1973.
A. Escassut, Ultrametric Banach algebras, World Scientific Publishing, 2003.
A. Escassut, and N. Mainetti, “Spectrum of ultrametric Banach algebras of strictly differentiable functions”, t Contemporary Mathematics, vol. 704, pp. 139–160, 2018.
A. Escassut, “Survey on the Kakutani problem in p-adic analysis I”, Sarajevo Journal of Mathematics, vol. 15, no. 2, pp. 245–263, 2019.
A. Escassut, “Survey on the Kakutani problem in p-adic analysis II”, Sarajevo Journal of Mathematics, vol. 16, no. 1, pp. 55–70, 2020.
B. Guennebaud, “Alg`ebres localement convexes sur les corps valués”, Bulletin des Sciences Mathématiques, vol. 91, pp. 75–96, 1967.
B. Guennebaud, Sur une notion de spectre pour les algèbres normées ultramétriques, Thèse d‘Etat, Université de Poitiers, 1973.
Ch. E. Rickart, General Theory of Banach Algebras, Krieger Publishing Company, 2002.
P. Salmon, “Sur les séries formelles restrintes”, Bulletin de la Société Mathématique de France, vol. 92, pp. 385–410, 1964.
J. Tate, “Rigid analytic spaces”, Inventiones Mathematicae, vol. 12, pp. 257–289, 1971.
Similar Articles
- S. S. Dragomir, M. V. Boldea, M. Megan, Inequalities for Chebyshev functional in Banach algebras , CUBO, A Mathematical Journal: Vol. 19 No. 1 (2017): CUBO, A Mathematical Journal
- Mehdi Dehghanian, Choonkil Park, Yamin Sayyari, Stability of ternary antiderivation in ternary Banach algebras via fixed point theorem , CUBO, A Mathematical Journal: Vol. 25 No. 2 (2023)
- D. Ebrahimi Bagha, M. Amini, Module amenability for Banach modules , CUBO, A Mathematical Journal: Vol. 13 No. 2 (2011): CUBO, A Mathematical Journal
- Amar Kumar Banerjee, Pratap Kumar Saha, Semi Open sets in bispaces , CUBO, A Mathematical Journal: Vol. 17 No. 1 (2015): CUBO, A Mathematical Journal
- Raúl Cordovil, David Forge, Gr¨obner and diagonal bases in Orlik-Solomon type algebras , CUBO, A Mathematical Journal: Vol. 7 No. 2 (2005): CUBO, A Mathematical Journal
- Irena Kosi-Ulbl, Joso Vukman, An identity related to derivations of standard operator algebras and semisimple H∗ -algebras , CUBO, A Mathematical Journal: Vol. 12 No. 1 (2010): CUBO, A Mathematical Journal
- Mohadeseh Rostamani, Shirin Hejazian, Maps preserving Fredholm or semi-Fredholm elements relative to some ideal , CUBO, A Mathematical Journal: Vol. 17 No. 1 (2015): CUBO, A Mathematical Journal
- N. Neuburg, César Burgueño, A. Suazo, Pivotal idempotents in the Bernstein algebras , CUBO, A Mathematical Journal: No. 6 (1990): CUBO, Revista de Matemática
- Violeta Petkova, Spectral results for operators commuting with translations on Banach spaces of sequences on Zᴷ and Z⺠, CUBO, A Mathematical Journal: Vol. 14 No. 3 (2012): CUBO, A Mathematical Journal
- Paolo D‘alessandro, An immediate derivation of maximum principle in Banach spaces, assuming reflexive input and state spaces , CUBO, A Mathematical Journal: Vol. 14 No. 2 (2012): CUBO, A Mathematical Journal
You may also start an advanced similarity search for this article.