An Introduction to Operator Algebras

Kehe Zhu

• Publisher: CRC Press
• Year: 2018
• eBook ISBN: 9781351467025
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

An Introduction to Operator Algebras is a concise text/reference that focuses on the fundamental results in operator algebras. Results discussed include Gelfand's representation of commutative C*-algebras, the GNS construction, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) functional calculus for normal operators, and type decomposition for von Neumann algebras. Exercises are provided after each chapter.

Part II

C*-ALGEBRAS

8

C*-Algebras

8.1 The DEFINITION

DEFINITION 8.1 A C* -algebra is a Banach algebra $\mathcal{A}$ together with a mapping x ↦ x* on $\mathcal{A}$ satisfying the following conditions:

(a) (x*)* = x for all x ∈ $\mathcal{A}$.

(b) (ax + by)* = āx* + b̄y* for all x, y ∈ $\mathcal{A}$ and a, b ∈ C.

(c) (xy)* = y*x* for all x, y ∈ $\mathcal{A}$.

(d) ‖x*x‖ = ‖x‖² for all x ∈ $\mathcal{A}$.

Any mapping x ↦ x* on an algebra satisfying (a), (b), and (c) is called an involution on the algebra. The element x* is usually called the adjoint of x.

We point out that condition (d) above is the most stringent condition in the definition of a C*-algebra.

The involution in a C* -algebra is an isometry. In fact, for every x in $\mathcal{A}$ we have

${\Vert x\Vert }^2=\Vert x^{\ast }x\Vert \le \Vert x^{\ast }\Vert \Vert x\Vert ,$

so that ‖x‖≤ ‖x*‖. Replacing x by x* we conclude that ‖x*‖≤ ‖x‖ and hence ‖x‖ = ‖x*‖.

It can also be shown that any involution x ↦ x* satisfying ‖x‖ ≤ ‖x*x‖ must be an isometry. In fact, this condition gives

${\Vert x\Vert }^2\le \Vert x^{\ast }x\Vert \le \Vert x^{\ast }\Vert \Vert x\Vert$

and hence ||‖x‖≤ ‖x*‖. This implies that ||‖x‖ = ‖x*‖.for all x in $\mathcal{A}$.

Let 1 be the (multiplicative) unit in a C*-algebra $\mathcal{A}$. For every x in $\mathcal{A}$ we have

$1^{\ast }x={\left(x^{\ast }1\right)}^{\ast }=x,x{1}^{\ast }={\left(1x^{\ast }\right)}^{\ast }=x.$

By the uniqueness of the unit in an algebra, 1* = 1.

It is easy to check that an element x in a C*-algebra is invertible if and only if x* is. In t...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• I Banach Algebras
• II C*-Algebras
• III Von Neumann Algebras
• Bibliography
• Index