\abstract
A CP-semigroup
is a semigroup $\phi=\{\phi_t: t\geq 0\}$
of normal completely positive linear maps on $\Cal B(H)$,
$H$ being a separable Hilbert space, which satisfies
$\phi_t(\bold 1)=\bold 1$ for all t and is
continuous in the natural sense.
Let dom(L) be the natural domain of the generator
$L$ of $\phi$, $\phi_t=\exp{tL}$.
Since the maps $\phi_t$ need not be multiplicative
dom(L) is typically an operator space, but not an algebra.
However, we show that the set of operators
$$
\Cal A=\{A\in dom(L): A^*A\in dom(L), AA^*\in dom(L) \}
$$
is a $*$-subalgebra of $\Cal B(H)$, indeed $\Cal A$ is the
largest self-adjoint algebra contained in $\Cal D$. Because $\Cal A$ is
a $*$-algebra one may consider its
$*$-bimodule of noncommutative $2$-forms
$\Omega^2(\Cal A)=\Omega^1(\Cal A)\otimes_\Cal A\Omega^1(\Cal A)$,
and any linear mapping $L:\Cal A \to\Cal B(H)$ has a
{\it symbol} $\sigma_L: \Omega^2(\Cal A)\to\Cal B(H)$, defined
as a linear map by
$$
\sigma_L(a\,dx\,dy)=aL(xy)-axL(y)-aL(x)y+axL(\bold 1)y, \qquad a,x,y\in\Cal A.
$$
The symbol is a homomorphism of $\Cal A$-bimodules for any
$*$-algebra $\Cal A\subseteq\Cal B(H)$ and any linear map
$L:\Cal A\to\Cal B(H)$.
When $L$ is the generator of a $CP$-semigroup with domain
algebra $\Cal A$ above, we show that the symbol
is negative in that
$\sigma_L(\omega^*\omega)\leq 0$ for every $\omega\in\Omega^1(\Cal A)$
($-\sigma_L$ is in fact completely positive).
Examples are given
for which the domain algebra
$\Cal A$ is, and is not, strongly dense in $\Cal B(H)$. We also
relate the generator of a $CP$-semigroup to its
commutative paradigm, the Laplacian of a Riemannian manifold.
\endabstract