# nLab star-algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

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

$(-)^\ast \;\colon\; A \longrightarrow A \;\;\;\; \text{s.t.} \;\;\;\; \underset{a,b \in A}{\forall} \; (a b)^\ast \;=\; b^\ast a^\ast \,, \;\;\;\;\;\; \underset{a \in A}{\forall} \; \left((a)^\ast\right)^\ast \;=\; a \,.$

## Definition

In more detail, begin with a commutative ring (often a field, or 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. Note that we can take any commutative ring $K$ and simply define $\bar{x} \coloneqq x$.)

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

• $x^{**} = x$ 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).

The claim 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

(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.

###### 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 .