Set of complex numbers notation

Question

What is the meaning of the notation $\mathbb{C}^\times$ where $\mathbb{C}$ is the usual set of complex numbers? The context is that of continuous functions of the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ that are $\mathbb{C}^\times$-valued.

Answer 1

It means $\mathbb C\setminus\{0\}$, that is, the set of non-zero complex numbers.

Answer 2

The notation $\mathbb{C}^\times$ is the multiplicative group consisting of all of the nonzero elements of $\mathbb{C}$. That is, $\mathbb{C}^\times$ is the set $\mathbb{C}\setminus\{0\}$, along with the operation of complex multiplication. This suggests somewhat more structure than just the notation $\mathbb{C}\setminus\{0\}$ alone.

More generally, a field is a tuple $(k,+,\times)$ consisting of a set $k$ and two binary operations which have the following properties:

1. the addition operation $+$ is commutative and associative,

2. there is an additive identity $0\in k$ such that $x + 0 = x = 0 + x$ for all $x \in k$,

3. additive inverses exist: if $x \in k$, then there is some $-x \in k$ such that $x+(-x) = 0$

4. the multiplication operation $\times$ is commutative and associative,

5. there is a multiplicative identity $1\in k$ such that $x\times 1 = x = 1\times x$ for all $x \in k$;

6. multiplicative inverses exist: if $x \in k$ and $x \ne 0$, then there is some $x^{-1} \in k$ such that $x \times x^{-1} = 1$; and

7. multiplication distributes over addition; that is, if $x,y,z \in k$, then $x\times (y+z) = (x\times y) + (x\times z)$.

Properties 1–3 can be reduced to the statement "$(k,+)$ is an abelian group", while properties 4–6 can be restated as "$(k\setminus\{0\}, \times)$ is an abelian group". The last property is a compatibility condition—that is, the two operations need to be compatible in the manner indicated. As the multiplicative group appears with some frequency, it is useful to have a succinct notation for it. The usual convention it to write $$ k^\times = (k\setminus\{0\}, \times). $$ Again, it is worth noting that the notation indicates both a set (i.e. the set of nonzero elements of $k$), as well as an algebraic structure on that set (the multiplication operation).