Movatterモバイル変換

*-algebra

From Wikipedia, the free encyclopedia

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 Rng Semiring Near-ring Commutative ring Domain Integral domain Field Division ring Lie 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 Hopf algebra
v t e

Inmathematics, and more specifically inabstract algebra, a*-algebra (orinvolutive algebra; read as "star-algebra") is a mathematical structure consisting of twoinvolutive ringsR andA, whereR is commutative andA has the structure of anassociative algebra overR. Involutive algebras generalize the idea of a number system equipped with conjugation, for example thecomplex numbers andcomplex conjugation,matrices over the complex numbers andconjugate transpose, andlinear operators over aHilbert space andHermitian adjoints.However, it may happen that an algebra admits noinvolution.^[a]

Look up* orstar in Wiktionary, the free dictionary.

Definitions

[edit]

*-ring

[edit]

Algebraic structure → Ring theory Ring theory
Basic concepts Rings •Subrings •Ideal •Quotient ring •Fractional ideal •Total ring of fractions •Product of rings • Free product of associative algebras •Tensor product of algebras Ring homomorphisms •Kernel •Inner automorphism •Frobenius endomorphism Algebraic structures •Module •Associative algebra •Graded ring •Involutive ring •Category of rings •Initial ring $\mathbb {Z}$ •Terminal ring $0=\mathbb {Z} /1\mathbb {Z}$ 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 •Polynomial ring •Formal power series ring Algebraic number theory •Algebraic number field •Integers modulon •Ring of integers •p-adic integers $\mathbb {Z} _{p}$ •p-adic numbers $\mathbb {Q} _{p}$ •Prüferp-ring $\mathbb {Z} (p^{\infty })$
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

Inmathematics, a*-ring is aring with a map* :A →A that is anantiautomorphism and aninvolution.

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

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

for allx, y inA.

This is also called aninvolutive ring,involutory ring, andring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.

Elements such thatx* =x are calledself-adjoint.^[2]

Archetypical examples of a *-ring are fields ofcomplex numbers andalgebraic numbers withcomplex conjugation as the involution. One can define asesquilinear form over any *-ring.

Also, one can define *-versions of algebraic objects, such asideal andsubring, with the requirement to be *-invariant:x ∈I ⇒x* ∈I and so on.

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

*-algebra

[edit]

A*-algebraA is a *-ring,^[b] with involution * that is anassociative algebra over acommutative *-ringR with involution′, such that(r x)* =r′ x* ∀r ∈R,x ∈A.^[3]

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

It follows from the axioms that * onA isconjugate-linear inR, meaning

(λ x +μ y)* =λ′ x* +μ′ y*

forλ, μ ∈R,x, y ∈A.

A*-homomorphismf :A →B is analgebra homomorphism that is compatible with the involutions ofA andB, i.e.,

f(a*) =f(a)* for alla inA.^[2]

Philosophy of the *-operation

[edit]

The *-operation on a *-ring is analogous tocomplex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to takingadjoints in complexmatrix algebras.

Notation

[edit]

The * involution is aunary operation written with a postfixed star glyph centered above or near themean line:

x ↦x*, or

x ↦x^∗ (TeX:x^*),

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

Examples

[edit]

Anycommutative ring becomes a *-ring with the trivial (identical) involution.
The most familiar example of a *-ring and a *-algebra overreals is the field of complex numbersC where * is justcomplex conjugation.
More generally, afield extension made by adjunction of asquare root (such as theimaginary unit√−1) is a *-algebra over the original field, considered as a trivially-*-ring. The *flips the sign of that square root.
Aquadratic integer ring (for someD) 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 otherhypercomplex 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.
Thematrix algebra ofn × nmatrices overR with * given by thetransposition.
The matrix algebra ofn × nmatrices overC with * given by theconjugate transpose.
Its generalization, theHermitian adjoint in the algebra ofbounded linear operators on aHilbert space also defines a *-algebra.
Thepolynomial ringR[x] over a commutative trivially-*-ringR is a *-algebra overR withP *(x) =P (−x).
If(A, +, ×, *) is simultaneously a *-ring, analgebra over a ringR (commutative), and(r x)* =r (x*) ∀r ∈R,x ∈A, thenA is a *-algebra overR (where * is trivial).
- As a partial case, any *-ring is a *-algebra overintegers.
Any commutative *-ring is a *-algebra over itself and, more generally, over any its*-subring.
For a commutative *-ringR, itsquotient by any its*-ideal is a *-algebra overR.
- For example, any commutative trivially-*-ring is a *-algebra over itsdual numbers ring, a *-ring withnon-trivial *, because the quotient byε = 0 makes the original ring.
- The same about a commutative ringK and its polynomial ringK[x]: the quotient byx = 0 restoresK.
InHecke algebra, an involution is important to theKazhdan–Lusztig polynomial.
Theendomorphism ring of anelliptic curve becomes a *-algebra over the integers, where the involution is given by taking thedual isogeny. A similar construction works forabelian varieties with apolarization, in which case it is called theRosati involution (see Milne's lecture notes on abelian varieties).

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

Thegroup Hopf algebra: agroup ring, with involution given byg ↦g⁻¹.

Non-Example

[edit]

Not every algebra admits an involution:

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

Any nontrivial antiautomorphism necessarily has the form:^[4] $\varphi _{z}\left[{\begin{pmatrix}1&0\\0&0\end{pmatrix}}\right]={\begin{pmatrix}1&z\\0&0\end{pmatrix}}\quad \varphi _{z}\left[{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]={\begin{pmatrix}0&0\\0&0\end{pmatrix}}$ for any complex number $z\in \mathbb {C}$ .

It follows that any nontrivial antiautomorphism fails to be involutive: $\varphi _{z}^{2}\left[{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\right]={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\neq {\begin{pmatrix}0&1\\0&0\end{pmatrix}}$

Concluding that the subalgebra admits no involution.

Additional structures

[edit]

Many properties of thetranspose hold for general *-algebras:

TheHermitian elements form aJordan algebra;
The skew Hermitian elements form aLie algebra;
If 2 is invertible in the *-ring, then the operators⁠1/2⁠(1 + *) and⁠1/2⁠(1 − *) areorthogonal idempotents,^[2] calledsymmetrizing andanti-symmetrizing, so the algebra decomposes as a direct sum ofmodules (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 areoperators, not elements of the algebra.

Skew structures

[edit]

Given a *-ring, there is also the map−* :x ↦ −x*.It does not define a *-ring structure (unless thecharacteristic is 2, in which case −* is identical to the original *), as1 ↦ −1, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra wherex ↦x*.

Elements fixed by this map (i.e., such thata = −a*) are calledskew Hermitian.

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

Notes

[edit]

^In this context,involution is taken to mean an involutory antiautomorphism, also known as ananti-involution.
^Most definitions do not require a *-algebra to have theunity, i.e. a *-algebra is allowed to be a *-rng only.

References

[edit]

^Weisstein, Eric W. (2015)."C-Star Algebra".Wolfram MathWorld.
^^a ^b ^cBaez, John (2015)."Octonions".Department of Mathematics. University of California, Riverside.Archived from the original on 26 March 2015. Retrieved27 January 2015.
^star-algebra at thenLab
^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.

v t e Spectral theory and^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With anApproximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem