Involutive algebra

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 at all.

Look up ***star*** in
Wiktionary, the free dictionary.

## Terminology

### Involutive ring

In mathematics, an **involution**, or an **involutory function**, is a function f that is its own inverse,

*f*(*f*(*x*)) = *x*
In ring theory, the word *involution* is customarily taken to mean an antihomomorphism that is its own inverse function.
Examples of involutions in common rings:

* complex conjugation on the complex plane
* multiplication by j in the split-complex numbers
* taking the transpose in a matrix ring.

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:

* (*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**. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1\* is also a multiplicative identity, and identities are unique.

Elements such that *x*\* = *x* 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: *x* ∈ *I* ⇒ *x*\* ∈ *I* and so on.

### \*-algebra

A **\*-algebra** A is a \*-ring, with involution \* that is an associative algebra over a commutative \*-ring R with involution ′, such that (*r x*)\* = *r′* *x*\*  ∀*r* ∈ *R*, *x* ∈ *A*.

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.

### 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 GL*n*(**C**).

### 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). Note that 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 2x2 matrices over the complex numbers.

Consider the following subalgebra:

A

:=

{

(

a

b

0

0

)

:
a
,
b
∈

C

}

{\displaystyle \mathcal{A}:=\left\{\begin{pmatrix}a&b\\0&0\end{pmatrix}:a,b\in\mathbb{C}\right\}}

Any nontrivial antiautomorphism necessarily has the form:

φ
z

[

(

1

0

0

0

)

]

=

(

1

z

0

0

)

φ
z

[

(

0

1

0

0

)

]

=

(

0

0

0

0

)

{\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}}

for any complex number

z
∈

C

{\displaystyle z \in \C}

.

It follows that any nontrivial antiautomorphism fails to be idempotent:

φ
z
2

[

(

0

1

0

0

)

]

=

(

0

0

0

0

)

≠

(

0

1

0

0

)

{\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}}

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 12(1 + \*) and 12(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.

## Notes

1. ↑ Most definitions do not require a \*-algebra to have the unity, i.e. a \*-algebra is allowed to be a \*-rng only.