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# K-theory

Topological K-theory is a generalised cohomology theory on the category of compact Hausdorff spaces. It classifies the vector bundles over a space $X$ up to stable equivalences. Equivalently, via the Serre-Swan theorem, it classifies the finitely generated projective modules over the $C^{*}$-algebra $C(X)$.

Let $A$ be a unital $C^{*}$-algebra over $\mathbb{C}$ and denote by $\mathord{\mathrm{M}_{{\infty}}(A)}$ the algebraic direct limit of matrix algebras $\mathord{\mathrm{M}_{{n}}(A)}$ under the embeddings $\mathord{\mathrm{M}_{{n}}(A)}\to\mathord{\mathrm{M}_{{n+1}}(A)}:a\mapsto\left(% \begin{array}[]{cc}a&0\\ 0&0\end{array}\right)$. Identify the completion of $\mathord{\mathrm{M}_{{\infty}}(A)}$ with the stable algebra $A\otimes\mathbb{K}$ (where $\mathbb{K}$ is the compact operators on $l_{2}(\mathbb{N})$), which we will continue to denote by $\mathord{\mathrm{M}_{{\infty}}(A)}$. The $K_{0}(A)$ group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the projections in $\mathord{\mathrm{M}_{{\infty}}(A)}$. Two projections $p$ and $q$ are homotopic if there exists a norm continuous path of projections from $p$ to $q$. Let $p\in\mathord{\mathrm{M}_{{m}}(A)}$ and $q\in\mathord{\mathrm{M}_{{n}}(A)}$ be two projections. The sum of their homotopy classes $[p]$ and $[q]$ is the homotopy class of their direct sum: $[p]+[q]=[p\oplus q]$ where $p\oplus q=\mathrm{diag}(p,q)\in\mathord{\mathrm{M}_{{m+n}}(A)}$. Alternatively, one can consider equivalence classes of projections up to unitary transformations. Unitary equivalence coincides with homotopy equivalence in $\mathord{\mathrm{M}_{{\infty}}(A)}$ (or $\mathord{\mathrm{M}_{{n}}(A)}$ for $n$ large enough).

Denote by $\mathrm{U}_{\infty}(A)$ the direct limit of unitary groups $\mathrm{U}_{n}(A)$ under the embeddings $\mathrm{U}_{n}(A)\to\mathrm{U}_{{n+1}}(A):u\mapsto\left(\begin{array}[]{cc}u&0% \\ 0&1\end{array}\right)$. Give $\mathrm{U}_{\infty}(A)$ the direct limit topology, i.e. a subset $U$ of $\mathrm{U}_{\infty}(A)$ is open if and only if $U\cap\mathrm{U}_{n}(A)$ is an open subset of $\mathrm{U}_{n}(A)$, for all $n$. The $K_{1}(A)$ group is the Grothendieck group (abelian group of formal differences) of the homotopy classes of the unitaries in $\mathrm{U}_{\infty}(A)$. Two unitaries $u$ and $v$ are homotopic if there exists a norm continuous path of unitaries from $u$ to $v$. Let $u\in\mathrm{U}_{m}(A)$ and $v\in\mathrm{U}_{n}(A)$ be two unitaries. The sum of their homotopy classes $[u]$ and $[v]$ is the homotopy class of their direct sum: $[u]+[v]=[u\oplus v]$ where $u\oplus v=\mathrm{diag}(u,v)\in\mathrm{U}_{{m+n}}(A)$. Equivalently, one can work with invertibles in $\mathrm{GL}_{\infty}(A)$ (an invertible $g$ is connected to the unitary $u=g|g|^{{-1}}$ via the homotopy $t\to g|g|^{{-t}}$).

Higher K-groups can be defined through repeated suspensions,

$K_{n}(A)=K_{0}(S^{n}A).$ | (1) |

But, the Bott periodicity theorem means that

$K_{1}(SA)\cong K_{0}(A).$ | (2) |

The main properties of $K_{i}$ are:

$\displaystyle K_{i}(A\oplus B)$ | $\displaystyle=$ | $\displaystyle K_{i}(A)\oplus K_{i}(B),$ | (3) | ||

$\displaystyle K_{i}(\mathord{\mathrm{M}_{{n}}(A)})$ | $\displaystyle=$ | $\displaystyle K_{i}(A)\quad\mbox{(Morita invariance)},$ | (4) | ||

$\displaystyle K_{i}(A\otimes\mathbb{K})$ | $\displaystyle=$ | $\displaystyle K_{i}(A)\quad\mbox{(stability)},$ | (5) | ||

$\displaystyle K_{{i+2}}(A)$ | $\displaystyle=$ | $\displaystyle K_{i}(A)\quad\mbox{(Bott periodicity)}.$ | (6) |

There are three flavours of topological K-theory to handle the cases of $A$ being complex (over $\mathbb{C}$), real (over $\mathbb{R}$) or Real (with a given real structure).

$\displaystyle K_{i}(C(X,\mathbb{C}))$ | $\displaystyle=$ | $\displaystyle\mathit{KU}^{{-i}}(X)\quad\mbox{(complex/unitary)},$ | (7) | ||

$\displaystyle K_{i}(C(X,\mathbb{R}))$ | $\displaystyle=$ | $\displaystyle\mathit{KO}^{{-i}}(X)\quad\mbox{(real/orthogonal)},$ | (8) | ||

$\displaystyle\mathit{KR}_{i}(C(X),J)$ | $\displaystyle=$ | $\displaystyle\mathit{KR}^{{-i}}(X,J)\quad\mbox{(Real)}.$ | (9) |

Real K-theory has a Bott period of 8, rather than 2.

# References

- 1 N. E. Wegge-Olsen, K-theory and $C^{*}$-algebras. Oxford science publications. Oxford University Press, 1993.
- 2 B. Blackadar, K-Theory for Operator Algebras. Cambridge University Press, 2nd ed., 1998.
- 3 M. Rørdam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for $C^{*}$-Algebras. Cambridge University Press, 2000.

## Mathematics Subject Classification

19-00*no label found*

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