nLab anti-dual linear space

Idea

If the ground ring (ground field) is equipped with a star structure (an anti-involution, such as complex conjugation in the complex numbers), then the operation of forming dual linear spaces may be “twisted” by this involution to yield a notion of “anti-dual” spaces. These relate to Hermitian forms as ordinary dual spaces relate to ordinary inner products.

Definition

Let $\mathbb{K}$ be a star-ring, hence a unital ring (often a field, whence our notation, and here often the complex numbers) equipped with an anti-involution (for instance complex conjugation):

$$\overline{(-)} \,\colon\, \mathbb{K} \to \mathbb{K}$$

$$\overline{x \cdot y} \,=\, \overline{y} \cdot \overline{x} \,.$$

Consider a $\mathbb{K}$-module $\mathscr{V} \,\in\, \mathbb{K}Mod$ (say right modules for possibly non-commutative $\mathbb{K}$, but being just $\mathbb{K}$-vector spaces if $\mathbb{K}$ is a field).

Definition

The anti-dual space $\overline{\mathscr{V}^\ast}$ of $\mathscr{V}$ is the space of anti-linear maps to the ground ring (ground field):

$$\overline{\mathscr{V}^\ast} \;\coloneqq\; \Big\{ \phi \,\colon\, \mathscr{V} \to \mathbb{K} \,\Big\vert\, \underset{v, v' \in \mathscr{V}}{\forall} \, \phi(v + v') = \phi(v) + \phi(v') ,\; \underset{ { v \in \mathscr{V} } \atop { \lambda \in \mathbb{K} } }{\forall} \, \phi(v \cdot \lambda) \,=\, \overline{\lambda} \cdot v \Big\}$$

equipped itself with the structure of a (right) module by

$$\phi \in \overline{\mathscr{V}^\ast} ,\, \,\lambda \in \mathbb{K} \;\;\; \vdash \;\;\; (\phi \cdot \lambda)(c) \;=\; \phi(v) \cdot \lambda \,.$$

(eg. Karoubi & Villamayor 1973, Ex. 1; Karoubi 2010, §1)

As the notation already indicates, an equivalent definition is:

Definition

The anti-dual $\overline{\mathscr{V}^\ast}$ has as underlying abelian group the ordinary linear dual $\mathscr{V}^\ast$ but equipped with its anti-linear structure, namely with the (right) module action twisted by the involution as:

$$\phi \,\in\, Hom_{\mathbb{K}}(\mathscr{V}, \mathbb{K}) \;\;\; \vdash \;\;\; \underset{v \in \mathscr{V}}{\forall} \, (\phi \cdot \lambda)(v) \;=\; \overline{\lambda} \cdot \phi(v) \,.$$

(eg. Mishchenko 1976 §1.1).

An isomorphism between the two definitions

$$\array{ \mathscr{V}^\ast &\longrightarrow& \mathscr{V}^\ast \\ \phi &\mapsto& \overline{\phi} }$$

is established by postcomposing linear forms with the star-involution

$$\overline{\phi} \,\colon\, v \,\mapsto\, \overline{\phi(v)}$$

which respects the above (right) $\mathbb{K}$-actions due to

$$\array{ \big( v \,\mapsto\, \phi(v)\cdot \lambda \big) &\mapsto& \big( v \,\mapsto\, \overline{\phi(v) \cdot \lambda} \big) \\ & = & \big( v \,\mapsto\, \overline{\lambda} \cdot \overline{\phi(v)} \big) \mathrlap{\,.} }$$

Related concepts

• hyperbolic functor

References

Discussion in the context of Hermitian K-theory:

• Max Karoubi, Orlando Villamayor, Ex. 1 in: K-théorie algébrique et K-théorie topologique II, Math. Scand. 32 (1973) 57-86 [jstor:24490565]

• Alexandr S. Mishchenko, §1.1 (pp. 76) in: Hermitian K-Theory. The Theory of characteristic classes and methods of functional analysis, Uspeki Mat. Nauk 31 2 (1976) 69-134, Russ. Math. Surv. 31 71 (1976) [doi:10.1070/RM1976v031n02ABEH001478, pdf]

• Max Karoubi, §1 in: Le théorème de périodicité en K-théorie hermitienne, Quanta of Maths 1, AMS and Clay Math Institute Publications (2010) [arXiv:0810.4707]

See also:

• Wikipedia, Anti-dual space