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Operator Estimates for Fredholm Modules


Published:20000801
Printed: Aug 2000
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Abstract
We study estimates of the type
$$
\Vert \phi(D)  \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D  D_0
\Vert^{\alpha}, \quad \alpha = \frac12, 1
$$
where $\phi(t) = t(1 + t^2)^{1/2}$, $D_0 = D_0^*$ is an unbounded
linear operator affiliated with a semifinite von Neumann algebra
$\calM$, $D  D_0$ is a bounded selfadjoint linear operator from
$\calM$ and $(1 + D_0^2)^{1/2} \in \emt$, where $\emt$ is a symmetric
operator space associated with $\calM$. In particular, we prove that
$\phi(D)  \phi(D_0)$ belongs to the noncommutative $L_p$space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{1/2}$ belongs to the
noncommutative weak $L_r$space for some $r \in [1,p)$. In the case
$\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this
result continues to hold under the weaker assumption $(1 +
D_0^2)^{1/2} \in \calC_p$. This may be regarded as an odd
counterpart of A.~Connes' result for the case of even Fredholm
modules.
MSC Classifications: 
46L50, 46E30, 46L87, 47A55, 58B15 show english descriptions
unknown classification 46L50 Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Noncommutative differential geometry [See also 58B32, 58B34, 58J22] Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] Fredholm structures [See also 47A53]
46L50  unknown classification 46L50 46E30  Spaces of measurable functions ($L^p$spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46L87  Noncommutative differential geometry [See also 58B32, 58B34, 58J22] 47A55  Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 58B15  Fredholm structures [See also 47A53]
