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== Let A be a unital Banach algebra, and let a Є A be a normal element (i.e., ||a|| = ||a* || if A is a C*- algebra). 1. Spectral Radius Formula: Prove that the spectral radius r(a) of a satisfies r(a) = lim ||a" ||1/n 818 2. Gelfand Transform and Functional Calculus: Suppose A is a C*-algebra. Show that the Gelfand transform provides a functional calculus for a, and use it to demonstrate that ||a|| = r(a). 3. Measure-Theoretic Integration in Functional Calculus: For a given continuous function f on the spectrum o(a), express f(a) using a suitable measure-theoretic integral with respect to the spectral measure associated with a. Prove that this integral is well-defined and preserves the algebraic operations. Requirements: • • Employ the properties of Banach algebras and C*-algebras in the context of spectral theory. Utilize the Gelfand transform and its relationship with the spectrum of an element. • Integrate measure-theoretic techniques within the functional calculus framework.

== Let A be a unital Banach algebra, and let a Є A be a normal element (i.e., ||a|| = ||a* || if A is a C*- algebra). 1. Spectral Radius Formula: Prove that the spectral radius r(a) of a satisfies r(a) = lim ||a" ||1/n 818 2. Gelfand Transform and Functional Calculus: Suppose A is a C*-algebra. Show that the Gelfand transform provides a functional calculus for a, and use it to demonstrate that ||a|| = r(a). 3. Measure-Theoretic Integration in Functional Calculus: For a given continuous function f on the spectrum o(a), express f(a) using a suitable measure-theoretic integral with respect to the spectral measure associated with a. Prove that this integral is well-defined and preserves the algebraic operations. Requirements: • • Employ the properties of Banach algebras and C*-algebras in the context of spectral theory. Utilize the Gelfand transform and its relationship with the spectrum of an element. • Integrate measure-theoretic techniques within the functional calculus framework.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.1: The Geometry And Algebra Of Vectors
Problem 24EQ
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== Let A be a unital Banach algebra, and let a Є A be a normal element (i.e., ||a|| = ||a* || if A is a C*- algebra). 1. Spectral Radius Formula: Prove that the spectral radius r(a) of a satisfies r(a) = lim ||a" ||1/n 818 2. Gelfand Transform and Functional Calculus: Suppose A is a C*-algebra. Show that the Gelfand transform provides a functional calculus for a, and use it to demonstrate that ||a|| = r(a). 3. Measure-Theoretic Integration in Functional Calculus: For a given continuous function f on the spectrum o(a), express f(a) using a suitable measure-theoretic integral with respect to the spectral measure associated with a. Prove that this integral is well-defined and preserves the algebraic operations. Requirements: • • Employ the properties of Banach algebras and C*-algebras in the context of spectral theory. Utilize the Gelfand transform and its relationship with the spectrum of an element. • Integrate measure-theoretic techniques within the functional calculus framework.
Transcribed Image Text:== Let A be a unital Banach algebra, and let a Є A be a normal element (i.e., ||a|| = ||a* || if A is a C*- algebra). 1. Spectral Radius Formula: Prove that the spectral radius r(a) of a satisfies r(a) = lim ||a" ||1/n 818 2. Gelfand Transform and Functional Calculus: Suppose A is a C*-algebra. Show that the Gelfand transform provides a functional calculus for a, and use it to demonstrate that ||a|| = r(a). 3. Measure-Theoretic Integration in Functional Calculus: For a given continuous function f on the spectrum o(a), express f(a) using a suitable measure-theoretic integral with respect to the spectral measure associated with a. Prove that this integral is well-defined and preserves the algebraic operations. Requirements: • • Employ the properties of Banach algebras and C*-algebras in the context of spectral theory. Utilize the Gelfand transform and its relationship with the spectrum of an element. • Integrate measure-theoretic techniques within the functional calculus framework.
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