284c284
< Topoloogical $K_0$.
---
> Topological $K_0$.
288c288
< Topoloogical K-theory and K-homology.
---
> Topological K-theory and K-homology.
896a897,898
> 
> \vspace{2ex}
897a900,903
> In topology, this is more conventionally written
> using the relative group:
> $K^0 (X, Y) \longrightarrow K^0 (X) \longrightarrow K^0 (Y)$.
> 
919c925,926
< Long exact sequence: next slide.
---
> Long exact sequence: the slide after the next slide.
> 
943,944c950,953
< There are many different proofs of this,
< and also other formulations,
---
> There are many different proofs of this, some long but elementary,
> some short and slick, using machinery of \ca{s},
> and some of which say somewhat more.
> There also other formulations,
1151c1160
< This is {\emph{algebraiv Murray-von Neumann equivalence}}.
---
> This is {\emph{algebraic Murray-von Neumann equivalence}}.
1163a1173,1200
> 
> %
> \pause
> \pause
> \end{frame}
> 
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> 
> 
> \begin{frame}
> 
> \frametitle{$K_0 (L (l^2))$}
> 
> Addition in the concrete description of $V (A)$ is as follows.
> Let $e \in M_m (A)$ and $f \in M_n (A)$ be idempotents.
> Then $[e] + [f]$ is the class of
> %
> \begin{equation}\label{Eq_4117_DSum}
> e \oplus f = \left( \begin{matrix}
>   e     &  0        \\
>   0     &  f
> \end{matrix} \right)
> \in M_{m + n} (A).
> \end{equation}
> %
> (One checks that $(e \oplus f) A^{m + n} = e A^m \oplus f A^n$.)
1164a1202,1228
> \vspace{1ex}
> 
> Back to $K_0 (L (H))$, with $H = l^2$..
> First, $M_n (L (H)) \cong L (H^n)$, and $H^n \cong H$.
> Second, let's use \pj{s}.
> The \pj{s} $p \in L (H)$ are exactly the orthogonal \pj{s}
> onto closed subspaces of~$H$.
> Moreover, if $p H$ and $q H$ have the same dimension
> as Hilbert spaces, then $p H \cong q H$,
> which means that there is a unitary $s \in L (p H, q H)$,
> with inverse $s^*$.
> These can be regarded as operators in $L (H)$
> satisfying $s s^* = q$ and $s^* s = p$.
> Thus $p \sim q$.
> It is easy to see that if $p \sim q$ then $\dim (p H) = \dim (q H)$
> 
> \vspace{1ex}
> 
> Combining this with~(\ref{Eq_4117_DSum})
> % the description above of $[p] + [q]$,
> one sees that $V (L (H))$ is the semigroup of possible dimensions,
> as Hilbert spaces, of closed subspaces of $H^n$ for $n \in \Z$,
> with the ``obvious'' addition, that is,
> $V (L (H)) \cong \bigl\{ 0, 1, 2, \ldots, \I \bigr\}$.
> Since $\et + \I = \I$ for any $\et \in V (L (H))$,
> it is easy to see that $K_0 (L (H)) = 0$.
> 
1233,1235c1297,1299
< $V (A)$ is the set of Murray-von Neumann equivalence
< classes of idempotents in $\bigcup_{n = 1}^{\I} M_n (A)$,
< and $K_0 (A)$ results from forcibly making $V (A)$ into a group.
---
> $V (A)$ is the semigroup of Murray-von Neumann equivalence
> classes of idempotents in $M_{\I} (A)$,
> and $K_0 (A)$ is its universal enveloping group.
1237c1301
< \vspace{2ex}
---
> \vspace{1ex}
1241,1244c1305,1310
< be homotopic.
< Moreover, homotopic idempotents are Murray-von Neumann equivalent.
< (Given standard Banach algebra machinery, the proof is easier
< than for vector bundles.)
---
> be homotopic:
> there are \hm{s} $\ph_t \colon A \to B$ for $t \in [0, 1]$
> such that $t \mapsto \ph_t (a)$ is \ct{} for all $a \in A$.
> (With this definition, if $X$ and $Y$ are compact,
> then $h_0, h_1 \colon X \to Y$ are homotopic \ifo{}
> $C (h_0), C (h_1) \colon C (Y) \to C (X)$ are homotopic.)
1246c1312
< \vspace{2ex}
---
> \vspace{1ex}
1248c1314,1317
< Conclusion:
---
> Homotopic idempotents are Murray-von Neumann equivalent.
> (Given standard Banach algebra machinery, the proof is easier
> than for vector bundles.)
> Therefore,
1253c1322
< \vspace{2ex}
---
> \vspace{1ex}
1257c1326
< \vspace{2ex}
---
> \vspace{1ex}
1277a1347,1350
> It is: $S A$ is the algebra of \cfn{s} from $[0, 1]$ to $A$
> which vanish at $0$ and~$1$.
> (This can be identified as $C_0 ((0, 1), \, A)$,
> $A$-valued \cfn{s} vanishing at infinity.)
1287a1361
> with maps $\io \colon J \to A$ and $\pi \colon A \to B$,
1300,1301c1374
< \[
< 0 \longrightarrow I
---
> $0 \longrightarrow J
1304,1305c1377
<   \longrightarrow 0
< \]
---
>   \longrightarrow 0$
1353c1425
< There is a two variable version,
---
> For \ca{s}, there is a two variable version,
1355,1358c1427,1445
< is heuristically ``K-homolology of the first variable and K-theory
< of the second variable''.
< It has several quite difference constructions,
< and has a product
---
> is heuristically ``K-homolology in the first variable and K-theory
> in the second variable''.
> It has several quite different constructions.
> In the following, $K \otimes A$ is a norm completion of $M_{\I} (A)$.
> The constructions include: equivalence classes of extensions of \ca{s}
> \[
> 0 \longrightarrow K \otimes B
>   \longrightarrow E
>   \longrightarrow A
>   \longrightarrow 0
> \]
> (addition is not the same as for extensions of modules),
> homotopy classes of abstract elliptic operators,
> and homotopy classes of approximate (in a suitable sense) \hm{s}
> $K \otimes S A \to K \otimes S B$.
> 
> \vspace{2ex}
> 
> It has a product
1360d1446
< This theory plays a key role in classification theorems for \ca{s}.
1363a1450,1453
> These groups play a key role in classification theorems for \ca{s}.
> 
> \vspace{2ex}
> 
1372a1463,1639
> \begin{frame}
> 
> \frametitle{Algebraic $K_1$}
> 
> Recall that if $A$ is a unital Banach algebra, then
> $K_1 (A) = \varinjlim_n \inv (M_n (A)) / \inv_0 (M_n (A))$.
> If $A$ does not have a topology, then $\inv_0 (M_n (A))$ does not
> make sense.
> 
> \vspace{2ex}
> 
> $S A = C_0 ((0, 1), \, A)$ does not make sense,
> so defining $K_1 (A) = K_0 (S A)$ will not work,
> even independently of the problem of algebraic K-theory for nonunital
> algebras.
> 
> \vspace{2ex}
> 
> The replacement is to use the commutator subgroup instead of the
> identity component.
> If $G$ is a group, let $G'$ be the subgroup generated by all
> $g h g^{-1} h^{-1}$ for $g, h \in G$.
> For a unital ring~$A$, define
> \[
> K_1^{\mathrm{alg}} (A)
>  = \varinjlim_n \inv (M_n (A)) / \inv (M_n (A))'.
> \]
> To emphasize the difference, write $K_1^{\mathrm{top}} (A)$ for
> the group already defined.
> 
> %
> \pause
> \pause
> \end{frame}
> 
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> 
> \begin{frame}
> 
> \frametitle{$K_1^{\mathrm{alg}} (A)$ vs.\  $K_1^{\mathrm{top}} (A)$}
> 
> $G'$ is generated by $\bigl\{ g h g^{-1} h^{-1} \colon g, h \in G \bigr\}$.
> 
> \vspace{2ex}
> 
> $K_1^{\mathrm{alg}} (A)
>  = \varinjlim_n \inv (M_n (A)) / \inv (M_n (A))'$.
> 
> \vspace{2ex}
> 
> This is a functor in the same way that $K_1^{\mathrm{top}} (A)$ is.
> 
> \vspace{2ex}
> 
> For a Banach algebra, $K_1^{\mathrm{alg}} (A)$ and $K_1^{\mathrm{top}} (A)$
> need not be the same.
> 
> \vspace{2ex}
> 
> Example: $A = \C$.
> We have $\inv (M_n (\C)) = {\operatorname{GL}}_n (\C)$,
> which is connected for all~$n$.
> Therefore $K_1^{\mathrm{top}} (\C) - 0$.
> 
> \vspace{2ex}
> 
> However, all commutators have determinant~$1$.
> That is, $\inv (M_n (\C))' \S {\operatorname{SL}}_n (\C)$.
> In fact, these are equal, and the determinant defines an
> isomorphism from $K_1^{\mathrm{alg}} (\C)$ to the multiplicative
> group $\C \SM \{ 0 \}$.
> 
> \vspace{2ex}
> 
> It is not true that $\inv (M_n (\C))' \S \inv_0 (M_n (A))$,
> but this is true ``in the limit'', and one therefore gets a
> surjective comparison map
> $K_1^{\mathrm{alg}} (A) \to K_1^{\mathrm{top}} (A)$.
> As we have seen, it can be very far from injective.
> 
> %
> \pause
> \pause
> \end{frame}
> 
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> 
> \begin{frame}
> 
> \frametitle{Exact sequence; $K_2 (A)$}
> 
> For a unital ring $A$ and an ideal $J \S A$,
> there is an exact sequence (for which I say nothing about the
> definitions of the relative groups which appear):
> \[
> K_1 (A, J)
>   \longrightarrow K_1 (A)
>   \longrightarrow K_1 (A / J)
>   \longrightarrow K_0 (A, J)
>   \longrightarrow K_0 (A)
>   \longrightarrow K_0 (A / J).
> \]
> The last map is normally not surjective.
> 
> \vspace{2ex}
> 
> Milnor defined a group $K_2 (A)$ for a unital ring~$A$,
> in such a way that the exact sequence above at least extends to
> \[
> K_2 (A)
>   \longrightarrow K_2 (A / J)
>   \longrightarrow K_1 (A, J)
>   \longrightarrow K_1 (A)
>   \longrightarrow K_1 (A / J)
>   \longrightarrow \cdots.
> \]
> However, this construction didn't obviously generalize in an
> appropriate way to $K_n (A)$ for $n \geq 3$.
> 
> %
> \pause
> \pause
> \end{frame}
> 
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> 
> \begin{frame}
> 
> \frametitle{Higher algebraic K-theory}
> 
> Defining $K_n (A)$ for $n \geq 3$ had to wait for Quillen's
> work, for which he received a Fields Medal,
> using almost completely different ideas.
> Here is a very brief description.
> Start with the group $\varinjlim_n \inv (M_n (A))$.
> Form its classifying space.
> Attach additional cells; among other things, the attached cells
> change the fundamental group from $\varinjlim_n \inv (M_n (A))$
> to its abelianization, which is
> $\varinjlim_n \inv (M_n (A)) / \inv (M_n (A))' = K_1 (A)$.
> Then form the higher homotopy groups of the result.
> There is in general no periodicity.
> 
> \vspace{2ex}
> 
> There are by now variants, and algebraic K-groups $K_n (A)$
> for $n < 0$,
> about which I can say nothing here.
> There are even algebraic versions of KK-theory, about which I know
> very little.
> 
> \vspace{2ex}
> 
> There is always a comparison map
> $K_n^{\mathrm{alg}} (A) \to K_n^{\mathrm{top}} (A)$
> (in any degree, taking $n$ mod~$2$ on the right).
> For $n \neq 0$, rarely, but occasionally, these maps are
> isomorphismss.
> An example is if $A$ is a unital properly infinite \ca,
> such as a Cuntz algebra or $L (H)$ for an infinite dimensional
> Hilbert space~$H$.
> 
> %
> \pause
> \pause
> \end{frame}
> 
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>