*-algebra

In mathematics, a *-algebra (also called an involutive algebra) is a special kind of mathematical structure. It connects two areas called involutive rings, where one ring is commutative and the other acts like an associative algebra over the first. This idea helps us understand many number systems that have a process called conjugation.

For example, the complex numbers have a operation known as complex conjugation. Similarly, 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 in action. However, not all algebras have such an involution, or "star" operation, that fits these rules.

Definitions

A *-ring is a special kind of mathematical structure that includes a map called an involution, written as *. This involution has special rules that make it behave in predictable ways, similar to how complex numbers have a process called complex conjugation.

A *-algebra builds on this idea, combining the *-ring with another mathematical structure called an associative algebra. This allows mathematicians to study properties that are similar to those found in complex numbers and matrices, but in a more general setting. The *-operation is like a mirror that flips certain elements while keeping the overall structure consistent.

Examples

Some simple examples of *-algebras include the field of complex numbers, where the involution is just 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, consider special types of 2×2 matrices made from complex numbers. In this case, certain operations do not behave in a way that allows for an involution. This shows that some algebras simply do not include this special feature.

Additional structures

Many ideas from working with numbers and matrices also work in *-algebras. For example, special types of elements in a *-algebra can form structures called Jordan algebras or Lie algebras. When certain conditions are met, the algebra can be split into parts of symmetric and anti-symmetric elements.

Skew structures

There is also a way to study elements using a map that changes their sign. In some cases, these changed elements are called skew Hermitian. For example, in complex numbers, real numbers are Hermitian, while imaginary numbers are skew Hermitian.