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star-algebra

Idea

A $*$-algebra is an associative algebra (or even a nonassociative algebra) $A$ equipped with an anti-involution.

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 $x\mapsto \overline{x}$. (The usual example for $K$ is the field of complex numbers, but the concept of $*$-algebra makes sense in more general contexts. Note that we can take any commutative ring $K$ and simply define $\overline{x}≔x$.)

A $K$-$*$-algebra (a $*$-algebra over $K$) is a $K$-module $A$ equipped with a $K$-bilinear map $A,A\to A$, 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
• $(xy{)}^{*}=y^{*}{x}^{*}$ 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 $(rx{)}^{*}=\overline{r}{x}^{*}$ 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 star-algebras

When $K$ is the field $ℂ$ of complex numbers (or the field $ℝ$ 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

• and $W^{*}$-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 $W^{*}$-algebras).

Examples

A groupoid convolution algebra is naturally a star-alegbra, with the involution given by pullback along the inversion operation of the groupoid.

More generally the category convolution algebra of a dagger-category is a star algebra, with the involution being the pullback along the dagger-operation.

Related concepts

• star-representation