*-algebra

In mathematics, a *-algebra is a special kind of mathematical structure. It links two areas called involutive rings. One ring is commutative, and the other acts like an associative algebra.

This idea helps us understand many number systems with a process called conjugation.

For example, the complex numbers have an operation called complex conjugation. Also, matrices with complex numbers use something called conjugate transpose. And linear operators in a Hilbert space have Hermitian adjoints. These are all examples of *-algebras. Not all algebras have this special operation.

Definitions

A *-ring is a special mathematical structure with a map called an involution, written as *. This involution follows special rules, much like how complex numbers have a process called complex conjugation.

A *-algebra extends this idea by combining the *-ring with another structure called an associative algebra. This helps mathematicians study properties similar to those in complex numbers and matrices, but in a more general way. The *-operation acts like a mirror, flipping certain elements while keeping the structure balanced.

Examples

Some simple examples of *-algebras include the field of complex numbers, where the involution is complex conjugation. Another example is the matrix algebra of n × n matrices over the reals, with the involution given by the transposition.

More generally, quaternions, split-complex numbers, and dual numbers are *-rings with their built-in conjugation operation. The Hurwitz quaternions form a non-commutative *-ring with quaternion conjugation. The polynomial ring R[x] over a commutative trivially-*-ring R is also a *-algebra over R.

Non-Example

Not every algebra has an involution. For example, think about special kinds of 2×2 matrices made from complex numbers. Here, some operations do not work in a way that allows for an involution. This shows that some algebras do not have this special feature.

Additional structures

Many ideas from working with numbers and matrices also work in *-algebras. Special types of elements in a *-algebra can form new structures. When certain rules are followed, the algebra can be split into parts.

Skew structures

There is a way to study elements by changing their sign. In some cases, these changed elements have special names. For example, in complex numbers, real numbers are one type, while imaginary numbers are another type.