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This is the Past Exam of Mathematical Tripos which includes Class Field Theory, Artin Map, Abelian Extension of Number Fields, Decomposition Group, Inertia Group, Factorisation of Prime Ideals, Version of Hensel’s Lemma, Hilbert Norm Residue Symbol etc. Key important points are: Banach Algebras, Open Subset, Invertible Elements, Unique Continuous, Unital Homomorphism, Runge Approximation Theorem, Unique Complex Number, Commutative Subalgebra, Banach Algebra with Identity
Typology: Exams
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Thursday 9 June, 2005 9 to 12
Attempt THREE questions.
There are FIVE questions in total. The questions carry equal weight.
All Banach algebras should be taken to be over the complex field, and to be non-zero.
Cover sheet None Treasury Tag Script paper
1 Let A be a Banach algebra with identity element 1 and let G be the set of all invertible elements of A. Prove that G is an open subset of A and that the mapping x 7 → x−^1 (x ∈ G) is a homeomorphism of G onto itself.
Let (xn) be a sequence in G and let xn → x as n → ∞. Prove that if x /∈ G then:
(i) ‖x− n 1 ‖ → ∞ as n → ∞; (ii) the element x has neither left nor right inverse.
Let a ∈ A and suppose that, for each λ ∈ C, 1 − λa has either a left inverse or a right inverse. Prove that Sp a = { 0 }.
2 Let A be a Banach algebra with identity, let x ∈ A and let U be an open neighbourhood of Sp x in C. Prove that there is a unique continuous, unital homomorphism Θx : O(U ) → A such that Θx(Z) = x (where Z is the function Z(λ) = λ (λ ∈ U )).
Prove also that, for every f ∈ O(U ), Sp Θx(f ) = f (Sp x). [Any form of the Runge approximation theorem may be quoted without proof .]
Let x ∈ A have the property that Sp x contains no real number t 6 0. Prove that there is a unique element y ∈ A such that both y^3 = x and | arg λ| < π/3 for every λ ∈ Sp y.
3 Let A be a complex Banach algebra with identity, let L be a maximal left ideal of A and let the element a of A be such that La ⊆ L. Prove that there is a unique complex number λ such that a − λ 1 ∈ L.
Let Z = {z ∈ A : zx = xz for all x ∈ A} (i.e. Z is the centre of A). Prove that Z is a closed, commutative subalgebra of A, containing 1. Prove also that if L is any maximal left ideal of A then L ∩ Z is a maximal ideal of Z.
Paper 7
