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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 $x \mapsto \bar{x}$. (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

• A C-star algebra is in particular a star-algebra.

• A groupoid convolution algebra is naturally a $*$-algebra, 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 $*$-algebra, with the involution being the pullback along the $\dagger$ operation.

Related concepts

• real part, imaginary part

• state on a star-algebra

• star-representation

References

See also

• Wikipedia, star-algebra