---
title: C*-Algebras
description: C*-algebras are complex algebras of bounded linear operators on a Hilbert space, closed under the adjoint operation. Learners will understand their algebraic structure, representation theory, and applications to quantum mechanics and non-commutative geometry.
category: mathematics
subcategory: functional-analysis
difficulty: beginner, intermediate, advanced
url: /subject/c*-algebras
---

# C*-Algebras

C*-algebras are complex algebras of bounded linear operators on a Hilbert space, closed under the adjoint operation. Learners will understand their algebraic structure, representation theory, and applications to quantum mechanics and non-commutative geometry.

## Available Resources

1 Books • 1 Courses • 2 Websites

## Websites

### 1. nLab: C*-algebra

Reference wiki entry defining abstract and concrete C*-algebras and summarizing the central structural results: positive elements and the partial order, the Gelfand-Naimark theorem, the GNS construction, and Gelfand duality with compact Hausdorff spaces, followed by examples and an extensive reference list.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/C*-algebra

**Tags:** c*-algebras, operator-algebras, gelfand-duality, gns-construction, functional-analysis, nlab

### 2. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Operator Algebras

Learn C*-algebras with this comprehensive course by N. Christopher Phillips. Explore operator algebras and build a strong foundation in the topic!

**Difficulty:** Advanced | **Price:** Free

**Link:** https://www.youtube.com/playlist?list=PLuWnG60GozUmd4jcY9bKZ3K5t3pAdtwzM

**Tags:** operator-algebras, c*-algebras, von-neumann-algebras, functional-analysis, graduate-lectures

## Books

### 1. C*-Algebras and Their Automorphism Groups

**Author:** Gert K. Pedersen

Research monograph on the structure theory of C*-algebras: positive functionals, the enveloping von Neumann algebra, Borel structure, type I algebras, and crossed products, with a detailed treatment of automorphism groups and dynamical systems. Readers gain the standard reference toolkit for research-level work.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0125494505?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, functional-analysis

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