*-algebra

Examples

• Any commutative ring becomes a *-ring with the trivial (identical) involution.
• The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers C where * is just complex conjugation.
• More generally, a field extension made by adjunction of a square root (such as the imaginary unit √−1) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.
• A quadratic integer ring (for some D) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings.
• Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *-rings (with their built-in conjugation operation) and *-algebras over reals (where * is trivial). None of the three is a complex algebra.
• Hurwitz quaternions form a non-commutative *-ring with the quaternion conjugation.
• The matrix algebra of n × n matrices over R with * given by the transposition.
• The matrix algebra of n × n matrices over C with * given by the conjugate transpose.
• Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra.
• The polynomial ring R[x] over a commutative trivially-*-ring R is a *-algebra over R with P *(x) = P (−x).
• If (A, +, ×, *) is simultaneously a *-ring, an algebra over a ring R (commutative), and (r x)* = r (x*) ∀r ∈ R, x ∈ A, then A is a *-algebra over R (where * is trivial).
  • As a partial case, any *-ring is a *-algebra over integers.
• Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring.
• For a commutative *-ring R, its quotient by any its *-ideal is a *-algebra over R.
  • For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ring.
  • The same about a commutative ring K and its polynomial ring K[x]: the quotient by x = 0 restores K.
• In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial.
• The endomorphism ring of an elliptic curve becomes a *-algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties).

Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:

• The group Hopf algebra: a group ring, with involution given by g ↦ g⁻¹.

Non-Example

Not every algebra admits an involution:

Regard the 2×2 matrices over the complex numbers. Consider the following subalgebra: $${\displaystyle \mathcal{A}:=\left\{\left(\begin{array}{cc}a& b\\ 0& 0\end{array}\right):a,b\in \mathbb{C}\right\}}$$

Any nontrivial antiautomorphism necessarily has the form: $${\displaystyle {\phi }_z\left[\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}1& z\\ 0& 0\end{array}\right){\phi }_z\left[\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)}$$ for any complex number ${\displaystyle z\in \mathbb{C}}$.

It follows that any nontrivial antiautomorphism fails to be involutive: $${\displaystyle {\phi }_z^2\left[\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\right]=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\ne \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)}$$

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 ⁠1/2⁠(1 + *) and ⁠1/2⁠(1 − *) are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the *-ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.

Skew structures

Given a *-ring, there is also the map −* : x ↦ −x*. It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as 1 ↦ −1, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where x ↦ x*.

Elements fixed by this map (i.e., such that a = −a*) are called skew Hermitian.

For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.