In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
However, it may happen that an algebra admits no involution.
Definitions
*-ring
In mathematics, a *-ring is a ring with a map that is an antiautomorphism and an involution.
More precisely, is required to satisfy the following properties:
for all in .
This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.
Elements such that are called self-adjoint.
Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring.
Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: and so on.
*-rings are unrelated to star semirings in the theory of computation.
*-algebra
A *-algebra is a *-ring, with involution * that is an associative algebra over a commutative *-ring with involution , such that .
The base *-ring is often the complex numbers (with acting as complex conjugation).
It follows from the axioms that * on is conjugate-linear in , meaning
:
for .
A *-homomorphism is an algebra homomorphism that is compatible with the involutions of and , i.e.,
- for all in .
<math display="block">\varphi_z\left[\begin{pmatrix}1&0\\0&0\end{pmatrix}\right] = \begin{pmatrix}1&z\\0&0\end{pmatrix} \quad \varphi_z\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}</math>
for any complex number <math>z\in\Complex</math>.
It follows that any nontrivial antiautomorphism fails to be involutive:
<math display="block">\varphi_z^2\left[\begin{pmatrix}0&1\\0&0\end{pmatrix}\right] = \begin{pmatrix}0&0\\0&0\end{pmatrix}\neq\begin{pmatrix}0&1\\0&0\end{pmatrix}</math>
Concluding that the subalgebra admits no involution.
Additional structures
Many properties of the transpose hold for general *-algebras:
- The Hermitian elements form a Jordan algebra;
- The skew Hermitian elements form a Lie algebra;
- If 2 is invertible in the *-ring, then the operators and are orthogonal idempotents,