Endomorphisms of C as a vector space and as an algebra

The same carrier $\mathbb C$ has different endomorphism and automorphism collections in the following categories. Here $k\text{-}\mathbf{Alg}_1$ means unit-preserving associative algebra maps.

Category	Endomorphisms of $\mathbb C$	Automorphisms
Complex vector spaces	$z\mapsto az$ , $a\in\mathbb C$	$a\ne0$ , hence $\mathbb C^\times$

| Real vector spaces | $z\mapsto az+b\bar z$ , $a,b\in\mathbb C$ | $|a|^2-|b|^2\ne0$ , hence $GL_2(\mathbb R)$ | | $\mathbb C\text{-}\mathbf{Alg}_1$ | identity only | identity only | | $\mathbb R\text{-}\mathbf{Alg}_1$ | identity and conjugation | identity and conjugation |

In the two arbitrary-map associative-algebra categories, add the zero map to the endomorphism sets in the last two rows. The automorphism groups do not change.

Verification of the linear rows

A complex-linear map is determined by its value at $1$ . A real-linear map is determined by $T(1)$ and $T(i)$ ; solving for $a,b$ gives

a=\frac{T(1)-iT(i)}2,\qquad b=\frac{T(1)+iT(i)}2.

The real determinant is $|a|^2-|b|^2$ . Thus all real-linear endomorphisms form $M_2(\mathbb R)$ after choosing the basis $(1,i)$ .

Verification of the algebra rows

A unital real algebra map fixes $\mathbb R$ , and its value at $i$ must square to $-1$ . The only choices in $\mathbb C$ are $i$ and $-i$ , producing identity and conjugation. A complex-linear unital map must fix $i$ , leaving only identity. For a nonzero multiplicative map, the image of $1$ is a nonzero idempotent in the field $\mathbb C$ , hence equals $1$ ; the only additional arbitrary-map possibility is zero.

Jordan views

Since $\mathbb C$ is commutative, its ordinary product is already its Jordan product. The same algebra classifications therefore apply in the corresponding real or complex Jord, UJord, and Jord1 categories, with their specified unit policies.