nLab unitization of a C-star-algebra

Idea

The generalization of the process of one-point compactification of possibly non-compact topological spaces from topology to non-commutative topology is the process of adding units to possibly non-unital C*-algebras, thought of as formal duals of non-commutative spaces in noncommutative topology.

Definition

Definition

For $(A, {\Vert \cdot \Vert_{A}})$ a non-unital C*-algebra, its unitisation is the $C^{\ast}$-algebra whose underlying vector space is the direct sum

$$A^{+} \coloneqq A \oplus \mathbb{C}$$

of $A$ with the field of complex numbers, equipped with the multiplication law

$$(a_{1} \oplus z_{1}) \cdot (a_{2} \oplus z_{2}) \coloneqq (a_{1} a_{2} + z_{2} a_{1} + z_{1} a_{2}) \oplus z_{1} z_{2},$$

and the involution

$$(a \oplus z)^\ast \coloneqq a^{\ast} \oplus \overline{z}$$

(complex conjugation is taking place on the right). This really is a $C^\ast$-algebra. The norm can be characterised as

$${\Vert a \oplus z \Vert}_{A^{+}} = {\Vert L_{a} + z \cdot \operatorname{Id}_{A} \Vert}_{\mathcal{B}(A)} \coloneqq \sup_{b \in A, \Vert b \Vert_{A} \leq 1} \Vert a b + z b \Vert_{A}.$$

if $A$ does not have a multiplicative unit. (Note that the operatornorm vanishes on $1_A\oplus -1_{\mathbb{C}}$ if $A$ does have a multiplicative unit). For $C^\ast$-algebras with a multiplicative unit one can show that the norm equals

$$\Vert a\oplus z \Vert_{A^+} = \max\{\|a+z1_A\|,|z|\}$$

Examples

Example

For $X$ a locally compact Hausdorff topological space and $C(X)$ its possibly non-unital C*-algebra of functions, then

$$(C_0(X))^+ \simeq C_0(X) \oplus \mathbb{C} \simeq C(X^+)$$

this is equivalently the unital $C^\ast$-algebra of functions on the one-point compactification of $X$.

There is hence a canonical projection

$$i^\ast \colon C(X^+) \to \mathbb{C}$$

The topological K-theory of $X$ is the kernel of the induced map in operator K-theory

$$K(X) \simeq ker(i^\ast) \simeq K(C_0(X)) \,.$$

Properties

The point at infinity

Remark

The unitisation of $A$ comes with a canonical projection homomorphism of C*-algebras

$$i^\ast_A \colon A^+ \to \mathbb{C}$$

given by

$$(a + z) \mapsto z \,.$$

Dually this corresponds to the inclusion of the “point at infinity”.

K-theory with compact support on non-unital $C^\ast$-algebras

The map $i^\ast_A \colon A^+ \to \mathbb{C}$ of remark induces a homomorphism of operator K-theory groups of the form

$$K(i^\ast_A) \colon K(A^+) \to K(\mathbb{C}) \simeq \mathbb{Z} \,.$$

Definition

The kernel of this map is the operator K-theory of the original possibly non-unital $C^\ast$-algebra $A$:

$$K(A) \coloneqq ker(K(i^\ast_A)) \,.$$

Heuristically it is clear that this is the “compactly suppported” K-theory of the possibly non-compact non-commutative space given by the algebra $A$. This statement has been made precise for instance in (Emerson 07, theorem 3.8)):

Proposition

For $X$ a locally compact Hausdorff topological space, there is a natural isomorphism

$$K(X) \simeq [X,Fred]_{cs} \,,$$

where on the right we homotopy classes of maps of compact support into the classifying space for K-theory (space of Fredholm operators).

References

• Heath Emerson, Equivariant representable K-theory (arXiv:0710.1410)