User:Mct mht/K

K-theory for C*-algebras

unital case
Let A be a unital C*-algebra and Mn(A) be the C*-algebra of n &times; n matrices whose entries are elements A. Mn(A) can be embedded into Mn + 1(A) canonically, into the "upper left corner". Consider the algebraic direct limit


 * $$ M _{\infty} (A) = \varinjlim \cdots \rightarrow M_n(A) \rightarrow M_{n+1}(A) \rightarrow \cdots .$$

Denote the projections (self-adjoint idempotents) in this algebra by P(A). Two elements p and q are said to be Murray-von Neumann equivalent, denoted by p ~ q, if p = vv* and q = v*v for some partial isometry v in M∞(A). It is clear that ~ is an equivalence relation. Define a binary operation + on the set of equivalences P(A)/~ by


 * $$[p] + [q] = [p \oplus q]$$

where ⊕ is the orthogonal direct sum. This is well-defined and makes P(A)/~ into a semigroup, sometimes denoted by V(A).  K0(A) is then the abelian group obtained by performing the Grothendieck construction on V(A).

The semigroup V(A) need not have the cancellation property only if A is stably finite, in which case V(A) embedds into K0(A)

general case
For a general (possibly non-unital) C*-algebra A, denote its unitization by A+. By functoriality, the quotient map s: A+ &rarr; C induces a map s*: K0(A+) &rarr; Z. One then defines


 * $$K_0(A) = \mbox{Ker} (s_*).\; $$

One has the following standard picture of K0(A): an element &alpha; of K0(A) can be expressed as a formal difference [p] - [s(p)] for some projection p in P(A+). This can be shown as follows. Suppose &alpha; = [e] - [f]. One can arrange so that e ⊥ f and s(e) = s(f). Let p = (1n - f)⊕e, then s(p) = 1n, and [p] -  [s(p)]  = [e] - [f] = &alpha;.

Using the standard picture, one can prove the half-exactness of the K0 functor, that is, for a short exact sequence


 * $$0 \rightarrow J \rightarrow B \rightarrow A \rightarrow 0$$

of C*-algebras, the sequence


 * $$K_0(J) \rightarrow K_0(B) \rightarrow K_0(A) $$

is exact in the middle. Take &beta;= [p] - [s(p)] in Ker(&pi;* ), where p is a projection in P(B+). Then &pi;(p) ~ &pi;(s(p)) = s(p). One can then assume &pi;(p) = s(p) and conclude that p is in fact a projection in P(J+).

The sequence cannot be made into a short exact sequence by appending zero's at the ends. For example,

K_1
stable rank theory

The index map
Let


 * $$0 \rightarrow J \rightarrow B \rightarrow A \rightarrow 0$$

be a short exact sequence of of C*-algebras. The index map of K-theory expresses precisely the extent to which an element in K1(A) fails to lift to an element in K1(B). An example of this is given by the Fredholm index. A unitary in the Calkin algebra is liftable if and only it has index zero.

definition

Higher K-functor
suspensions and cones

contractibility

K_0(S(A)) and K_1 (A)
isomorphism given by index

e.g. Fredholm index, Toeplitz algebra