C*-algebra

In mathematics, a C^∗-algebra (pronounced "C-star") is a special kind of mathematical structure. It is used in an area called functional analysis. This structure mixes ideas from algebra and geometry. It helps mathematicians study complex functions and operators.

C*-algebras are important because they were first used to understand physical quantities in quantum mechanics. Scientists like Werner Heisenberg, Pascual Jordan, and later John von Neumann helped develop this idea. These algebras help describe how particles and energy behave in the tiny world of atoms and subatomic particles.

Today, C*-algebras are used in many areas of mathematics and physics. They help mathematicians understand groups of symmetries. They are also a key tool in modern theories about quantum mechanics. Researchers study these algebras to learn more about their types and properties.

Abstract characterization

A C*-algebra is a special kind of math object called a Banach algebra. It uses complex numbers and has a special rule called an involution. This rule acts like a mirror on the numbers.

The mirror operation has special properties. For example, using the mirror twice brings you back to the original number. Also, the "size" of a number times its mirror image matches the size of the number squared. These rules help mathematicians understand how these algebras work.

History: B*-algebras and C*-algebras

The term B*-algebra was introduced in 1946 by mathematician C. E. Rickart. It described special types of mathematical structures called Banach *-algebras that followed a specific rule.

Later, in 1947, mathematician I. E. Segal introduced the term C*-algebra. He used it for certain collections of operators) on a Hilbert space that were "closed" in a particular way. The letter "C" stands for "closed." Over time, the term C*-algebra became the more commonly used name for these mathematical structures.

Structure of C*-algebras

C*-algebras are special types of mathematical structures with useful properties.

One way to study them is by looking at special elements called "self-adjoint elements." These elements help us understand how the algebra behaves and can show relationships between elements.

C*-algebras also have something called "approximate identities." These are sets of elements that almost act like a single identity element. They help scientists study the algebra more easily and understand its structure better.

Examples

Finite-dimensional C*-algebras

We can make a C*-algebra using special number grids, called n × n matrices. When we use these matrices with certain rules, they form a C*-algebra. These are examples of C*-algebras that have a fixed, limited size.

C*-algebras of operators

Another example of a C*-algebra comes from special operations we can do on complex spaces. These operations follow specific rules and create a structure that is a C*-algebra.

C*-algebras of compact operators

We can also make C*-algebras from special operations on spaces that have infinitely many dimensions but are still manageable. These operations are called compact operators, and they form a C*-algebra.

Commutative C*-algebras

Commutative C*-algebras are connected to the idea of continuous functions on spaces. These functions can be added, multiplied, and combined in ways that follow the rules of C*-algebras.

C*-enveloping algebra

For some mathematical structures called Banach *-algebras, there is a special C*-algebra that “envelopes” or wraps around them. This helps us study these structures using the rules of C*-algebras.

Von Neumann algebras

Von Neumann algebras are a special type of C*-algebra that follow extra rules. They are important in advanced areas of mathematics.

Type for C*-algebras

A C*-algebra is called "type I" if special math rules apply when it connects to other math ideas. This helps mathematicians learn more about groups and algebras.

For locally compact groups, being "type I" depends on their related C*-algebras. Some C*-algebras can have representations that are not type I, and these can include types II and III, based on work by James Glimm.

C*-algebras and quantum field theory

In quantum mechanics, a C*-algebra helps us describe physical systems. Special parts of this algebra, called self-adjoint elements, stand for things we can measure in the system. A "state" of the system is a special way to measure that tells us what we expect to find.

This use of C*-algebras is important in local quantum field theory. In this theory, every open area of Minkowski spacetime connects to a C*-algebra, helping scientists learn about how particles and forces act in space and time.