A **-algebra is an algebra AA (associative or non-associative) equipped with an anti-involution:

() *:AAs.t.a,bA(ab) *=b *a *,aA((a) *) *=a. (-)^\ast \;\colon\; A \longrightarrow A \;\;\;\; \text{s.t.} \;\;\;\; \underset{a,b \in A}{\forall} \; (a b)^\ast \;=\; b^\ast a^\ast \,, \;\;\;\;\;\; \underset{a \in A}{\forall} \; \left((a)^\ast\right)^\ast \;=\; a \,.


In more detail, begin with a commutative ring (often a field, or possibly just a rig) KK equipped with an involution (a homomorphism whose square is the identity), written xx¯x \mapsto \bar{x}. (The usual example for KK is the field of complex numbers with involution given by complex conjugation, but the concept of **-algebra makes sense in more general contexts. Note that we can take any commutative ring KK and simply define x¯x\bar{x} \coloneqq x.)

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

  • x **=xx^{**} = x for all xx in AA (so we have an involution on the underlying KK-module), and
  • (xy) *=y *x *(x y)^* = y^* x^* for all x,yx,y in AA (so it is an anti-involution on AA itself).

The claim that the anti-involution is KK-antilinear means that (rx) *=r¯x *(r x)^* = \overline{r} x^* for all rr in KK and all xx in AA (as well as (x+y) *=x *+y *(x + y)^* = x^* + y^*).

If a KK-**-algebra AA is itself commutative, then it is in particular a commutative ring with involution, and one can consider AA-**-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.


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 KK 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 KK 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

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 *C^*-algebras (and hence also von Neumann algebras).




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


(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.


(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

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

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.


(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


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


(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


(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 .

quantum probability theoryobservables and states


See also:

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:

Last revised on May 19, 2021 at 11:05:36. See the history of this page for a list of all contributions to it.