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star-algebra

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Idea

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

Definition

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, 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.

**-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 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).

Examples

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

Revised on April 22, 2014 04:38:10 by Colin Tan (182.19.168.23)