*-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.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map * : AA 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. 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.[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: xIx* ∈ I and so on.

*-algebra

A *-algebra A is a *-ring,[lower-alpha 1] with involution * that is an associative algebra over a commutative *-ring R with involution , such that (r x)* = r′x* rR, xA.[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 + μy)* = λx* + μy*

for λ, μR, x, yA.

A *-homomorphism f : AB 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 GLn(C).

Notation

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

xx*, or
xx (TeX: x^*),

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

Examples

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

Non-Example

Not every algebra admits an involution:

Regard the 2x2 matrices over the complex numbers.
Consider the following subalgebra:

Any nontrivial antiautomorphism necessarily has the form:

for any complex number .
It follows that any nontrivial antiautomorphism fails to be idempotent:

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 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 xx*.

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

Notes

  1. 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". Wolfram MathWorld.
  2. Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 25 March 2015. Retrieved 27 January 2015.
  3. star-algebra in nLab
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