nLab star-algebra

Idea

A $*$-algebra is an algebra $A$ (associative or non-associative) equipped with an anti-involution, meaning a map $(-)^\ast:A \to A$ such that

• for all elements $a \in A$ and $b \in A$, $(a b)^\ast = b^\ast a^\ast$

• for all elements $a \in A$, $(a^\ast)^\ast = a$

• $1^\ast = 1$

Definition

In more detail, begin with a commutative ring (often a field, but possibly just a rig) $K$ equipped with an involution (a homomorphism whose square is the identity), written $x \mapsto \bar{x}$. (The usual example for $K$ is the field of complex numbers with involution given by complex conjugation, but the concept of $*$-algebra makes sense in more general contexts. (An example is given by any commutative ring $K$ with trivial involution $\bar{x} \coloneqq x$.)

Then a $K$-$*$-algebra (a $*$-algebra over $K$) is a $K$-module $A$ equipped with a $K$-bilinear map $A\times A \to A$, written as multiplication (and often assumed to be associative) and a $K$-antilinear map $A \to A$, written as $x \mapsto x^*$, such that

• $(x^\ast)^\ast = x$ for all $x$ in $A$ (so we have an involution on the underlying $K$-module), and

• $(x y)^* = y^* x^*$ for all $x,y$ in $A$ (so it is an anti-involution on $A$ itself)

• $1^\ast = 1$.

The hypothesis that the anti-involution is $K$-antilinear means that $(r x)^* = \overline{r} x^*$ for all $r$ in $K$ and all $x$ in $A$ (as well as $(x + y)^* = x^* + y^*$).

If a $K$-$*$-algebra $A$ is itself commutative, then it is in particular a commutative ring with involution, and one can consider $A$-$*$-algebras as well. On the other hand, a commutative ring with involution is simply a commutative $*$-algebra over the ring of integers (with trivial involution), and similarly for rigs and natural numbers.

$*$-Rings

A $*$-ring is simply a $*$-algebra over the ring of integers (with trivial involution). Similarly, a $*$-rig is a $*$-algebra over the rig of natural numbers.

Arguably, when we began this article with a commutative ring $K$ equipped with involution, we should have begun it with a ring with anti-involution instead. However, since the ring (or rig) is commutative, there is no difference.

Banach $*$-algebras

When $K$ is the field $\mathbb{C}$ of complex numbers (or the field $\mathbb{R}$ of real numbers, with trivial involution), we can additionally ask that the $*$-algebra be a Banach algebra; then it is a Banach $*$-algebra. Special cases of this are

• $C^*$-algebras (aka $B^*$-algebras)

• and von Neumann algebras (aka $W^*$-algebras)

Arguably, one should require that the map $*$ be an isometry (which follows already if it is required to be short); some authors require this and some don't. However, this is automatic in the case of $C^*$-algebras (and hence also von Neumann algebras).

Examples

General

Example

(trivial star-structure)
Any plain algebra becomes a star-algebra by equipping it with the identity anti-involution.

Example

(complex conjugation)
The complex numbers – regarded as an associative algebra over the real numbers – form a star-algebra with anti-involution (which here is an involution, since the product is commutative) given by complex conjugation.

Example

(Cayley-Dickson construction) The Cayley-Dickson construction takes any star-algebra over the real numbers to a new star algebra.

Applied to the real numbers trivially regarded as a star-algebra (via Example ), this yields, successively, the star-algebras of

• real numbers (Example )

• complex numbers (Example )

• quaternions

• octonions

(which are the four real normed division algebras) and then further the

• sedenions

• …

In each case the star-operation may be thought of as complex conjugation, given by changing the sign of the imaginary part and keeping the real part intact.

Example

(operators on a Hilbert space) The algebra of continuous operators on a Hilbert space is a star-algebra, with the Hermitian adjoint as the anti-involution.

Example

(C-star algebras) A C-star algebra is a star-algebra where the anti-involution is compatible with the norm of the underlying Banach algebra.

Involutive Hopf algebras

Example

(involutive Hopf algebras are star-algebras)
Any involutive Hopf algebra is a star-algebra, with star-involution given by the antipode (by this Prop.).

Example

(groupoid algebras are star-algebras) A group algebra and, more generally, a groupoid convolution algebra, is a star-algebra, with the star-involution given by pullback along the inversion operation of the groupoid.

Yet more generally, the category convolution algebra of a dagger-category is a $*$-algebra, with the involution being the pullback along the $\dagger$ operation.

All these are involutive Hopf algebras (since taking inverses and taking dagger-operations squares to the identity) and as such are special cases of Example

Example

(star-algebra of horizontal chord diagrams)
The algebra of horizontal chord diagrams is a star-algebra under reversal of orientation of strands (see here, CSS 21, Prop. 2.9).

Since horizontal chord diagrams are the homology of the loop space of configuration space and the homology of a loop space is an involutive Hopf algebra, this is a special case of Example .

Generalizations

This concept could be generalized from the category of $R$-modules to any monoidal category:

Let $(C, I, \otimes)$ be a monoidal category and let $(A, e:I \to A, \pi:A \otimes A \to A)$ be a unital algebra object in $C$. Then $A$ is a $*$-algebra object in $C$ if it comes with a morphism $\iota:A \to A$ such that

• $\iota \circ \iota = \mathrm{id}_A$

• $\iota \circ e = e$

• for all morphisms $a:I \to A$ and $b:I \to A$,

$$\iota \circ \pi \circ (a \otimes b) = \pi \circ ((\iota \circ b) \otimes (\iota \circ a))$$

In the category of sets, $*$-algebra objects are called anti-involutive unital magmas, and in the category of abelian groups, associative $*$-algebra objects are called *-rings.

Related concepts

• real part, imaginary part

• anti-linear map, anti-dual linear space

• Hermitian form

• state on a star-algebra

• star-representation

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

See also:

• Wikipedia, star-algebra

Discussion for group algebras:

• Antonio Giambruno, Cesar Polcino Milies and Sudarshan K. Sehgal, Star-group identities on units of group algebras: The non-torsion case, Forum Mathematicum Volume 30 Issue 1 (doi:10.1515/forum-2016-0266)

Discussion of the Example of horizontal chord diagrams:

• David Corfield, Hisham Sati, Urs Schreiber: Fundamental weight systems are quantum states (arXiv:2105.02871)