C*-algebra

In mathematics, a C^∗-algebra (pronounced "C-star") is a special kind of mathematical structure used mainly in an area called functional analysis. It combines ideas from algebra and geometry, and 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 and Pascual Jordan started this work, and later John von Neumann developed it further. 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 and are a key tool in modern theories about quantum mechanics. Researchers also study these algebras to classify different types and understand their properties better.

Abstract characterization

A C*-algebra is a special kind of math object called a Banach algebra that works with complex numbers. It has a special rule called an involution, which acts like a mirror operation on the numbers inside the algebra.

This mirror operation has special properties that make the algebra behave in predictable ways. For example, applying the mirror operation twice brings you back to the original number. Also, the "size" of a number multiplied by its mirror image matches the size of the number squared. These rules help mathematicians understand how these algebras work and relate to each other.

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 that have many 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 be used to create orders and relationships between elements.

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

Examples

Finite-dimensional C*-algebras

A type of C*-algebra can be made using special kinds of number grids, called n × n matrices. When these matrices are used with rules for combining them, they create a C*-algebra. These are examples of C*-algebras that have a limited, fixed size.

C*-algebras of operators

Another example of a C*-algebra comes from special sets of operations that can be done 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 kinds of operations on spaces that have infinitely many dimensions but are still manageable in a certain way. These operations are called compact operators, and they too form a C*-algebra.

Commutative C*-algebras

Commutative C*-algebras are linked 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 certain kinds of mathematical structures called Banach *-algebras, there is a special C*-algebra that “envelopes” or wraps around them. This helps in studying 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 related to how they behave in weak topologies. They are important in advanced areas of mathematics.

Type for C*-algebras

A C*-algebra is called "type I" if certain special mathematical rules apply to it when it is connected to other mathematical structures. This idea helps mathematicians understand the properties of groups and algebras better.

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

C*-algebras and quantum field theory

In quantum mechanics, a C*-algebra is used to describe physical systems. The special elements of this algebra, called self-adjoint elements, represent measurable quantities of the system. A "state" of the system is a special kind of measurement that gives expected values for these quantities.

This way of using C*-algebras is important in local quantum field theory. In this theory, every open area of Minkowski spacetime is linked to a C*-algebra, helping scientists understand how particles and forces behave in space and time.