  As is well known absolute convergence and unconditional convergence for
series are equivalent only in finite dimensional Banach spaces. Replacing the
classical notion of absolutely summing operators by the notion of 1 summing
operators \[ \summ_k || Tx_k || \leq c || \summ_k e_k \otimes x_k
||_{\ell_1\otimes_{min}E}\] in the category of operator spaces, it turns out
that there are quite different interesting examples of 1 summing operator
spaces. Moreover, the eigenvalues of a composition $TS$ decreases of order
$n^{\frac{1}{q}}$ for all operators $S$ factorizing completely through a
commutative $C^*$-algebra if and only if the 1 summing norm of the operator $T$
restricted to a $n$-dimensional subspace is not larger than $c
n^{1-\frac{1}{q}}$, provided $q>2$. This notion of 1 summing operators is
closely connected to the notion of minimal and maximal operator spaces.
