symmetric monoidal (∞,1)-category of spectra
In more detail, begin with a commutative ring (often a field, or possibly just a rig) equipped with an involution (a homomorphism whose square is the identity), written . (The usual example for 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 and simply define .)
The claim that the anti-involution is -antilinear means that for all in and all in (as well as ).
If a --algebra is itself commutative, then it is in particular a commutative ring with involution, and one can consider --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.
Arguably, when we began this article with a commutative ring 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.
When is the field of complex numbers (or the field 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 -algebras (and hence also von Neumann algebras).