| |
| | [No title] |
 | | This is the definition used in~\cite{6}; it agrees with the definition in~\cite{11} if we identify those asymptotic morphisms $\phi '$, $\phi ''$ which are {\em asymptotically equivalent,\/} in the sense that $\phi '_{t}(a)-\phi ''_{t}(a)\to 0$ as $t\to \infty $ (we shall write $\phi '_{t}(a)\sim \phi _{t}''(a)$). |
| | Two asymptotic morphisms from $A$ to $B$ are {\em homotopic\/} if there is an asymptotic morphism from $A$ to $B[0,1]$ (the $C^{*}$-algebra of continuous functions mapping the unit interval into $B$) from which the two may be recovered by evaluation at $0$ and $1$. |
| | If $V_{a}\subset V$ is a finite-dimensional affine subspace then the {\em Dirac operator\/} $D_{a}$, an unbounded operator on $\mathscr{H}$ with domain $\mathfrak{s}$, is defined by \begin{equation*}D_{a} \xi = \sum _{i=1}^{n} (-1)^{\deg (\xi)} {\frac{\partial \xi }{\partial x_{i}}}v_{i},\end{equation*} where $\{v_{1}, \dots,v_{n}\}$ is an orthonormal basis for $V_{a}^{0}$, and $\{x_{1},\dots, x_{n}\}$ are the dual coordinates to $\{v_{1},\dots, v_{n}\}$. |
| www.math.psu.edu /era-mirror/1997-01-022/1997-01-022.tex.html (3435 words) |
|
