Approximate unit and proper C star morphism

Let $A$ be a complex C*-algebra (C star algebra).

An approximate unit for $A$ is a net $(e_i{)}_{i\in I}$ — that is, a family indexed by a directed set $I$; directed sets are nonempty here, so there is no empty-net convention to add — such that each $e_i$ is a positive contraction $e_i=e_i^{\ast }$, $e_i=b^{\ast }b$ for some $b$, and $\mathrm{\parallel }{e}_i\mathrm{\parallel }\le 1$ (Self-adjoint positive unitary and normal elements), and

$\mathrm{\parallel }{e}_{i}a-a\mathrm{\parallel }\to 0\text{and}\mathrm{\parallel }a{e}_i-a\mathrm{\parallel }\to 0\text{for every }a\in A\mathrm{.}$

When $A$ is commutative the two conditions coincide, and one says simply $e_{i}a\to a$. The algebra $A$ may be the zero algebra, in which case the constant net $e_i=0$ is an approximate unit.

Let $A$ and $B$ be complex C*-algebras and let $\phi :A\to B$ be a bounded star-homomorphism in the sense of C star algebra, not required to be unital. Then $\phi$ is proper when it carries every approximate unit of $A$ to an approximate unit of $B$: for every approximate unit $(e_i)$ of $A$, the net $(\phi (e_i))$ is an approximate unit of $B$ in the above sense.

Finally, for topological spaces, a continuous map $g:Y\to X$ is proper when the inverse image of every compact subset of $X$ is a compact subset of $Y$.

• A nonzero target of a proper map from a unital algebra is unital. A unital C*-algebra has the constant net $e_i=1_A$ as an approximate unit. If $\phi :A\to B$ is proper, the constant net $\phi (1_A)$ is therefore an approximate unit of $B$, so $\phi (1_A)b=b=b\phi (1_A)$ for every $b\in B$. If $B\mathrm{\ne }\{0\}$, this identity is nonzero, so $B$ is unital in the convention $1_B\mathrm{\ne }0$ and $\phi (1_A)=1_B$. If $B=\{0\}$, every approximate unit of $A$ maps to the constant zero approximate unit, so the unique zero map is proper. The zero algebra is nevertheless nonunital under the stated convention.
• The two uses of "proper" are linked by the duality. Under Locally compact Gelfand duality proper maps of locally compact Hausdorff spaces correspond exactly to proper star-homomorphisms of commutative complex C*-algebras, and this is where the pullback of a compactly supported function uses the compact-preimage condition.
• No choice principle is used in the definition. The definition is a condition on nets and maps; existence for commutative C*-algebras is the theorem Every commutative C star algebra has an approximate unit.