LEADER 04190nam 2200589Ia 450 001 9910782281003321 005 20230617002024.0 010 $a1-281-92826-7 010 $a9786611928261 010 $a981-277-560-9 035 $a(CKB)1000000000537924 035 $a(DLC)2005297859 035 $a(StDuBDS)AH24684598 035 $a(SSID)ssj0000263646 035 $a(PQKBManifestationID)11207838 035 $a(PQKBTitleCode)TC0000263646 035 $a(PQKBWorkID)10273541 035 $a(PQKB)11428018 035 $a(MiAaPQ)EBC1681773 035 $a(WSP)00005128 035 $a(Au-PeEL)EBL1681773 035 $a(CaPaEBR)ebr10255411 035 $a(CaONFJC)MIL192826 035 $a(OCoLC)879025657 035 $a(EXLCZ)991000000000537924 100 $a20030613d2003 uy 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt 182 $cc 183 $acr 200 10$aUltrametric Banach algebras$b[electronic resource] /$fAlain Escassut 210 $aSingapore ;$aRiver Edge, NJ $cWorld Scientific$dc2003 215 $a1 online resource (xiii, 275 p.) 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a981-238-194-5 320 $aIncludes bibliographical references (p. 265-267) and indexes. 327 $a1. Basic properties in commutative algebra -- 2. Tree structure -- 3. Ultrametric absolute values -- 4. L-productal vector spaces -- 5. Multiplicative semi-norms and Shilov boundary -- 6. Spectral semi-norm -- 7. Hensel Lemma -- 8. Infraconnected sets -- 9. Monotonous filters -- 10. Circular filters -- 11. Tree structure and metric on circular filters -- 12. Rational functions and algebras R(D) -- 13. Simple convergence on Mult(K[x]) -- 14. Topologies on Mult(K[x]) -- 15. Spectral properties and Gelfand transforms -- 16. Analytic elements -- 17. Holomorphic properties on infraconnected sets -- 18. T-filters and T-sequences -- 19. Applications of T-filters and T-sequences -- 20. Analytic elements on classic partitions -- 21. Holomorphic properties on partitions -- 22. Shilov boundary for algebras H(D, O) -- 23. Holomorphic functional calculus -- 24. Uniform K-Banach algebras and properties (s) and (q) -- 25. Properties (o) and (q) in uniform Banach K-algebras -- 26. Properties (o) and (q) and strongly valued fields -- 27. Multbijective Banach K-algebras -- 28. Pseudo-density of Mult[symbol] -- 29. Polnorm on algebras and algebraic extensions -- 30. Definition of affinoid algebras -- 31. Algebraic properties of affinoid algebras -- 32. Jacobson radical of affinoid algebras -- 33. Salmon's theorems -- 34. Separable fields -- 35. Spectral norm of affinoid algebras -- 36. Spectrum of an element of an affinoid algebra -- 37. Krasner-Tate algebras -- 38. Universal generators in Tate algebras -- 39. Mappings from H(D) to the tree Mult(K[x]) -- 40. Continuous mappings on Mult(K[x]) -- 41. Examples and counterexamples -- 41. Associated idempotents -- 43. Krasner-Tate algebras among Banach K-algebras. 330 $aIn this book, ultrametric Banach algebras are studied with the help of topological considerations, properties from affinoid algebras, and circular filters which characterize absolute values on polynomials and make a nice tree structure. The Shilov boundary does exist for normed ultrametric algebras. In uniform Banach algebras, the spectral norm is equal to the supremum of all continuous multiplicative seminorms whose kernel is a maximal ideal. Two different such seminorms can have the same kernel. Krasner-Tate algebras are characterized among Krasner algebras, affinoid algebras, and ultrametric Banach algebras. Given a Krasner-Tate algbebra A=K{t}[x], the absolute values extending the Gauss norm from K{t} to A are defined by the elements of the Shilov boundary of A. 606 $aBanach algebras 606 $aTopological algebras 615 0$aBanach algebras. 615 0$aTopological algebras. 676 $a512.554 700 $aEscassut$b Alain$067801 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910782281003321 996 $aUltrametric Banach algebras$93831348 997 $aUNINA