*-algebra

Mathematical structure in abstract algebra 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 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.In this context, involution is taken to mean an involutory antiautomorphism, also known as an anti-involution. * star Definitions *-ring In mathematics, a *-ring is a ring A with a map * : A → A that is an antiautomorphism and an involution. More precisely, * is required to satisfy the following properties:http://mathworld.wolfram.com/C-Star-Algebra.html C-Star Algebra Wolfram MathWorld 2015 Eric W. Weisstein Eric W. Weisstein 120% (x + y)* = x* + y* 120% (x y)* = y* x* 120% 1* = 1 120% (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.http://math.ucr.edu/home/baez/octonions/node5.html Octonions 2015 27 January 2015 Department of Mathematics University of California, Riverside Baez John John Baez https://web.archive.org/web/20150326133405/http://math.ucr.edu/home/baez/octonions/node5.html 26 March 2015 live 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. *-objectsAlso, 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,Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only. with involution * that is an associative algebra over a commutative *-ring R with involution , such that (r x)* = r x*  ∀r ∈ R, x ∈ A.star-algebra 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 120% (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., 120% 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 complex matrix algebras. Notation The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line: 120% x ↦ x*, or 120% 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). 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: Any nontrivial antiautomorphism necessarily has the form:Winker S. K. Wos L. Lusk E. L. 1981 Semigroups, Antiautomorphisms, and Involutions: A Computer Solution to an Open Problem, I https://www.jstor.org/stable/2007445 Mathematics of Computation 37 156 533–545 10.2307/2007445 0025-5718 subscription for any complex number . It follows that any nontrivial antiautomorphism fails to be involutive: 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 120% 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 References Category:Algebras Category:Ring theory