-algebra

Short description: Mathematical structure in abstract algebra

Algebraic structures

Group-like

* Group
* Semigroup / Monoid
* Rack and quandle
* Quasigroup and loop

* Abelian group
* Magma
* Lie group

Group theory

Ring-like

* Ring
* Semiring
* Near-ring
* Commutative ring
* Integral domain
* Field
* Division ring

Ring theory

Lattice-like

* Lattice
* Semilattice
* Complemented lattice
* Total order
* Heyting algebra
* Boolean algebra

* Map of lattices
* Lattice theory

Module-like

* Module
* Group with operators
* Vector space

* Linear algebra

Algebra-like

* Algebra

* Associative
* Non-associative
* Composition algebra

* Lie algebra
* Graded
* Bialgebra

* v
* t
* e

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. 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.[lower-alpha 1]

## Contents

* 1 Definitions
* 1.1 *-ring
* 1.2 *-algebra
* 1.3 Philosophy of the *-operation
* 1.4 Notation
* 2 Examples
* 3 Non-Example
* 4 Additional structures
* 4.1 Skew structures
* 5 See also
* 6 Notes
* 7 References

## Definitions

### *-ring

Algebraic structure → Ring theory
Ring theory

Basic concepts

Rings

• Subrings
• Ideal
• Quotient ring

• Fractional ideal
• Total quotient ring
• Product ring
• Free product of associative algebras
• Tensor product of rings

Ring homomorphisms

• Kernel
• Inner automorphism
• Frobenius endomorphism

Algebraic structures

• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings

• Initial ring [math]\displaystyle{ \mathbb{Z} }[/math]
• Terminal ring [math]\displaystyle{ 0 = \mathbb{Z}_1 }[/math]

Related structures

• Field

• Finite field
• Non-associative ring

• Lie ring
• Jordan ring
• Semiring

• Semifield

* * *

Commutative algebra

Commutative rings

• Integral domain

• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field

• Finite field
• Composition ring

• Polynomial ring
• Formal power series ring

Algebraic number theory

• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree

p-adic number theory and decimals

• Direct limit/Inverse limit
• Zero ring [math]\displaystyle{ \mathbb{Z}_1 }[/math]
• Integers modulo pn [math]\displaystyle{ \mathbb{Z}/p^n\mathbb{Z} }[/math]
• Prüfer p-ring [math]\displaystyle{ \mathbb{Z}(p^\infty) }[/math]
• Base-p circle ring [math]\displaystyle{ \mathbb{T} }[/math]
• Base-p integers [math]\displaystyle{ \mathbb{Z} }[/math]
• p-adic rationals [math]\displaystyle{ \mathbb{Z}[1/p] }[/math]
• Base-p real numbers [math]\displaystyle{ \mathbb{R} }[/math]
• p-adic integers [math]\displaystyle{ \mathbb{Z}_p }[/math]
• p-adic numbers [math]\displaystyle{ \mathbb{Q}_p }[/math]
• p-adic solenoid [math]\displaystyle{ \mathbb{T}_p }[/math]

Algebraic geometry

• Affine variety

* * *

Noncommutative algebra

Noncommutative rings

• Division ring
• Semiprimitive ring
• Simple ring
• Commutator

Noncommutative algebraic geometry

Free algebra

Clifford algebra

• Geometric algebra
Operator algebra

* v
* t
* e

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:[1]

* (x \+ y)* = x* + y*
* (x y)* = y* x*
* 1* = 1
* (x*)* = x

for all x, y in A.

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 x* = x are called self-adjoint.[2]

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: x ∈ I ⇒ x* ∈ I and so on.

*-rings are unrelated to star semirings in the theory of computation.

### *-algebra

A *-algebra A is a *-ring,[lower-alpha 2] with involution * that is an associative algebra over a commutative *-ring R with involution ′, such that (r x)* = r′ x* ∀r ∈ R, x ∈ A.[3]

The base *-ring R is often the complex numbers (with ′ acting as complex conjugation).

It follows from the axioms that * on A is conjugate-linear in R, meaning

(λ x \+ μ y)* = λ′ x* + μ′ y*

for λ, μ ∈ R, x, y ∈ A.

A *-homomorphism f : A → B is an algebra homomorphism that is compatible with the involutions of A and B, i.e.,

* f(a*) = f(a)* for all a in A.[2]

### Philosophy of the *-operation

The *-operation on a *-ring is analogous to complex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in complex matrix algebras.

### Notation

The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:

x ↦ x*, or
x ↦ x∗ (TeX: `x^*`),

but not as "x∗"; see the asterisk article for details.

## 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). Neither 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: [math]\displaystyle{ \mathcal{A} := \left\\{\begin{pmatrix}a&b\\\0&0\end{pmatrix} : a,b\in\Complex\right\\} }[/math]

Any nontrivial antiautomorphism necessarily has the form:[4] [math]\displaystyle{ \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]\displaystyle{ z\in\Complex }[/math].

It follows that any nontrivial antiautomorphism fails to be idempotent: [math]\displaystyle{ \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 1/2(1 + *) and 1/2(1 − *) are orthogonal idempotents,[2] 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.

## See also

* Semigroup with involution
* B*-algebra
* C*-algebra
* Dagger category
* von Neumann algebra
* Baer ring
* Operator algebra
* Conjugate (algebra)
* Cayley–Dickson construction
* Composition algebra

## Notes

1. ↑ In this context, involution is taken to mean an involutory antiautomorphism, also known as an anti-involution.
2. ↑ Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only.

## References

1. ↑ Weisstein, Eric W. (2015). "C-Star Algebra". http://mathworld.wolfram.com/C-Star-Algebra.html.
2. ↑ 2.0 2.1 2.2 Baez, John (2015). "Octonions". University of California, Riverside. Archived from the original on 26 March 2015. https://web.archive.org/web/20150326133405/http://math.ucr.edu/home/baez/octonions/node5.html. Retrieved 27 January 2015.
3. ↑ star-algebra in nLab
4. ↑ Winker, S. K.; Wos, L.; Lusk, E. L. (1981). "Semigroups, Antiautomorphisms, and Involutions: A Computer Solution to an Open Problem, I". Mathematics of Computation 37 (156): 533–545. doi:10.2307/2007445. ISSN 0025-5718. https://www.jstor.org/stable/2007445.

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