*-algebra

Last updated January 26, 2024

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.^{[lower-alpha 1]}

Definitions

*-ring

In mathematics, a *-ring is a ring with a map $* : A \to A$ 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: $x \in I \Rightarrow x * \in I$ 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 $(r x)* = r' x * \forall r \in R, x \in A$ .^[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 $λ, μ \in R, x, y \in A$ .

A *-homomorphism $f : A \to B$ 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 \mapsto x *

, or

x \mapsto 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 $n \times n$ matrices over R with * given by the transposition.
The matrix algebra of $n \times n$ 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 $g \mapsto g -1$ .

Non-Example

Not every algebra admits an involution:

Regard the 2×2 matrices 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 idempotent:

\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 the operators $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/2(1 + *)$ 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 $-* : x \mapsto - x *$ . It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as $1 \mapsto -1$ , neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where $x \mapsto x *$ .

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

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

Related Research Articles

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In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map $as additive identity and the identity map as multiplicative identity.$

<span class="mw-page-title-main">Transpose</span> Matrix operation which flips a matrix over its diagonal

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix $A$ by producing another matrix, often denoted by $A T$ .

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In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In mathematics, the special unitary group of degree $n$ , denoted $SU(n)$ , is the Lie group of $n \times n$ unitary matrices with determinant 1.

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In mathematics, an element $of a ring is called nilpotent if there exists some positive integer, called the index, such that .$

<span class="mw-page-title-main">Involution (mathematics)</span> Function that is its own inverse

In mathematics, an involution, involutory function, or self-inverse function is a function $f$ that is its own inverse,

In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices.

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In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector.

In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:

$.$

<span class="mw-page-title-main">Hermitian symmetric space</span> Manifold with inversion symmetry

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A² = I, where I is the n × n identity matrix. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.

In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called $-quadratic forms, particularly in the context of surgery theory.$

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1934). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.

In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.

In mathematics, a bioctonion, or complex octonion, is a pair (p,q) where p and q are biquaternions.

References

↑ Weisstein, Eric W. (2015). "C-Star Algebra". Wolfram MathWorld.
1 2 3 Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 26 March 2015. Retrieved 27 January 2015.
↑ star-algebra at the nLab
↑ 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.

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[1] In this context, involution is taken to mean an involutory antiautomorphism, also known as an anti-involution.

[4] Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only.

[2] Weisstein, Eric W. (2015). "C-Star Algebra". Wolfram MathWorld.

[:0-3] 1 2 3 Baez, John (2015). "Octonions". Department of Mathematics. University of California, Riverside. Archived from the original on 26 March 2015. Retrieved 27 January 2015.

[5] star-algebra at the nLab

[6] 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.

[lower-alpha 1]

[1]

[2]

[lower-alpha 2]

[3]

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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 an Approximate 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

*-algebra

Contents

Definitions

*-ring

*-algebra

Philosophy of the *-operation

Notation

Examples

Non-Example

Additional structures

Skew structures

See also

Notes

Related Research Articles

References