\section{(6 March 2009)}
\subsection*{Cyclic cohomology and higher traces, cont.}
If \( \eta \) is a cyclic cocycle
\[ 
\langle \eta \, | \, [p]_{0} \rangle = c_{n} \eta 
\overset{n+1}{\overbrace{(p, \ldots ,p)}} \, , \quad c_{n} = c_{2k} = \frac{1}{(2 
\pi \mi)^{k}}\frac{1}{k!}
\]
is well-defined. 


Aside (exercise): if \( \eta = \mathtt{b} \tau \), then \( \langle \mathtt{b} 
\tau \, | \, [p]_{0} \rangle = 0 \).

So one has a bilinear map \( HC^{n} (\mathcal{A}) \times K_{0} ( 
\mathcal{A} ) \rightarrow \mathbb{C} \).

Ex.: \( HC^{n}(\mathbb{C}) = 
\begin{cases}
    \mathbb{C} & \text{if } n \text{ is even}  \\
    0 & \text{if } n \text{ is odd}
    \label{hcfield}
\end{cases}
 \)
 
 Smells like Bott periodicity.
 
 \begin{theorem*}
     If \( \mathcal{A} \) is a C*-algebra, then \( 
     HC^{n}(\mathcal{A}) = 
     \begin{cases}
         \text{bounded traces on } \mathcal{A} & \text{if } n \text{ 
	 is even}  \\
         0 & \text{if } n \text{ is odd}
         \label{hccstar}
     \end{cases}
      \)
 \end{theorem*}
 Nothing interesting for C*-algebras.
 
 So we need to include \underline{smoothness}.

Let \( \mathcal{A} = C(\mathcal{M}) \); \( \mathcal{M} \) an even 
dimensional, smooth, compact manifold without boundary.
\[ 
\mathcal{A}^{\infty} = C^{\infty}(\mathcal{M}) \quad \text{in its 
Fr\'{e}chet topology}
\]

\underline{Require} cyclic cocycles to be continuous w.r.t this 
topology.
\begin{lemma*}
    For any projection \( p \in \mathsf{M}_{n} ( C ( \mathcal{M} ) ) 
    \) there exists \( \tilde{p} \in \mathsf{M}_{n} ( C^{\infty} ( \mathcal{M} ) ) 
    \)  such that  \( p \sim \tilde{p} \).
\end{lemma*}

Then one could consider cyclic cocycles of \( C^{\infty} ( 
\mathcal{M} ) \) (in particular, \( C ^{\infty} \)-continuous) and 
extend the pairing with \( K_{0} ( C^{\infty} ( \mathcal{M} ) ) \) to 
a pairing with \( K_{0} ( C ( \mathcal{M} ) ) \)
 
 Purpose: to construct cyclic cocycles (continuous) for a dense 
 subalgebra \( \mathcal{A}^{\infty} \) of \( \mathcal{A} \) such that 
 the pairing extends from \( K_{0} ( \mathcal{A}^{\infty} ) \) to \( 
 K_{0} ( \mathcal{A} ) \).
 
 \underline{Cycles.} Look at \( (\Omega , d , \int) \), a \emph{cycle 
 over a Banach algebra} \( \mathcal{B} \) of 
 dimension \( n \):
 \begin{gather*}
     \Omega  \text{ a graded algebra, }  \Omega  = \bigoplus_{i \in 
     \mathbb{N}_{0}} \Omega^{i}  , \, \Omega^{i} \Omega^{j} \subset 
     \Omega^{i + j}  , \, \Omega^{n + p}  = 0 \,\, \forall \, p>0 \, ;  \\
     d  \text{ a differential of degree } +1 \, ,  d^{2} = 0 \, \, + 
     \text{ Leibniz rule;}  \\
     \int  \text{ is a \emph{graded} trace\footnotemark on } \Omega^{n} \, , 
     \text{ that is, }
     \int  \text{ is linear and }  \\ 
     \int \omega_{1} \omega_{2} = 
     (-1)^{k ( n - k )} \int \omega_{2} \omega_{1} \, , \, \omega_{1} 
     \in \Omega^{k} \, , \, \omega_{2} 
     \in \Omega^{n - k} \, ;   \\
     \text{ and lastly, } \mathcal{B} \text{ is a subalgebra of } 
     \Omega^{0} \, .
 \end{gather*}
 \footnotetext{In addition, \( \int \) is \emph{closed:} \( \int d \omega = 0 \) for \( \omega \in 
	\Omega^{n - 1} \).  See the notes for 9 March.}
 Ex.: \( \mathcal{B} = C^{\infty} ( \mathcal{M} ) \), \( \Omega = 
 \Omega ( \mathcal{M} ) \) the algebra of exterior forms over \( 
 \mathcal{M} \), \( d \) the exterior derivative, and \( \int \) the 
 integral of \( (\dmn \mathcal{M}) \)-forms over \( \mathcal{M} \). 
 \begin{proposition*}[Connes]
     Any cycle over \( \mathcal{B} \) of dimension \( n \) defines a 
     cyclic \( n \)-cocycle \( \eta \) (the character of the cycle):
     \[ 
     \eta ( A_{0} , \ldots , A_{n} ) = \int A_{0} d A_{1} d A_{2} 
     \cdots d A_{n}
     \]
     Conversely, any cyclic \( n \)-cocycle arises in this way.
 \end{proposition*}
 
 \begin{defn*}
     An \( n \)\emph{-trace} on a Banach algebra \( \mathcal{B} \) is the 
     character of a cycle of dimension \( n \), \( ( \Omega^{\prime} 
     , d , \int ) \), over a dense subalgebra \( \mathcal{B}^{\prime} 
     \) of \( \mathcal{B} \), such that \( \forall \, A_{1} , \ldots , 
     A_{n} \in \mathcal{B}^{\prime} \), \( \exists \, C( A_{1} , 
     \ldots , A_{n} ) \) for which
     \[ 
     \int (X_{1} d A_{1}) (X_{2} d A_{2}) \cdots (X_{n} d A_{n}) 
     \leq C \|X_{1}\| \|X_{2}\| \cdots \|X_{n}\| \, , \,
     \forall \, X_{i} \in \mathcal{B}^{\prime}
     \]
 \end{defn*}
 
 This means that \( \forall \, A_{1} , \ldots , A_{n} \in 
 \mathcal{B}^{\prime} \)
 \begin{align*}
     \underset{\text{n}}{\underbrace{\mathcal{B}^{\prime} \times 
     \cdots \times \mathcal{B}^{\prime}}} & \rightarrow \mathbb{C} 
     \quad n\text{-linear map}  \\
     ( X_{1} , \ldots , X_{n} ) & \mapsto \int (X_{1} d A_{1}) (X_{2} d A_{2}) \cdots (X_{n} d A_{n})
 \end{align*}
 is bounded with norm \( p( A_{1} , 
     \ldots , A_{n} ) :=  \) the smallest \( C( A_{1} , 
     \ldots , A_{n} ) \).
 \begin{theorem*}[Connes]
     Any \( n \)-trace on \( \mathcal{B} \) extends to an algebra \( 
     \mathcal{B}^{\prime\prime} \), \( \mathcal{B}^{\prime} 
     \subset \mathcal{B}^{\prime\prime} \subset \mathcal{B} \), such 
     that the inclusion \( \mathcal{B}^{\prime\prime} \overset{i}{\rightarrow} 
     \mathcal{B} \) induces an isomorphism \( i_{\ast} \) in 
     K-theory: \( i_{\ast} : K_{i} ( \mathcal{B}^{\prime\prime} ) 
     \underset{\cong}{\rightarrow} K_{i} ( \mathcal{B} ) \).
 \end{theorem*}
 
 Consequence:  An \( n \)-trace defines a functional on \( K_{i} ( 
 \mathcal{B} ) \); first on \( K_{i}( \mathcal{B}^{\prime\prime} ) \) 
 by continuous extension, and then by selecting for a dense-in-\( 
 \mathcal{B} \) representation of \( \mathcal{B}^{\prime\prime} \).
 
\begin{exa*}
    \begin{align*}
	\mathcal{A} & = \text{ a C*-algebra, } & \tau & =
	\text{ a trace, } & \delta & = \text{ a derivation} \\
	 \updownarrow & & \updownarrow & 
	 \rule[0em]{0em}{1em}^{\text{unbounded}} & \updownarrow &
	 \rule[0em]{0em}{1em}^{\text{unbounded}} \\
	 \Omega & = \mathcal{A} \otimes \bigwedge \mathbb{C}^{2} &
	 \int & & d &
    \end{align*}
\end{exa*}