The authors continue their study of certain classical ideals in [Integral Equations Oper. Theory 79, No. 4, 507–532 (2014; Zbl 1322.46040)]. Therefore, to start with, we resume some definitions of the latter: Given two Banach spaces \(X\) and \(Y\), \(B(X,Y)\) (respectively, \(\mathcal{K}(X,Y)\), \(\mathcal{WK}(X,Y)\)) denotes the set of bounded (respectively, compact, weakly compact) linear operators from \(X\) to \(Y\). We say that \(T\in B(X,Y)\) fixes a copy of a subspace \(Z\subset X\) if \(Z\) is isomorphic to \(T(Z)\). \(T\) is called strictly singular (respectively, finitely strictly singular) if it does not fix any infinite-dimensional subspace (respectively, if given \(\varepsilon>0\) there is a natural number \(N\) such that each subspace of dimension at least \(N\) contains an element \(x\) with \(\|Tx\|<\varepsilon\|x\|\)). The strictly singular operators form a closed ideal \(\mathcal{SS}(X,Y)\subset B(X,Y)\) (in the “usual” sense: \(\mathcal{SS}(X,Y)\) contains the finite rank operators and if \(U\in B(X_0,X)\), \(V\in B(Y,Y_0)\), then \(VTU\in\mathcal{SS}(X_0,Y_0)\)), likewise for the finitely strictly singular operators \(\mathcal{FSS}(X,Y)\subset B(X,Y)\). The following inclusions hold and are in general strict: \[ \mathcal{K}(X,Y)\subset\mathcal{FSS}(X,Y)\subset\mathcal{SS}(X,Y)\subset\mathcal{IN}(X,Y), \]
\[ \mathcal{K}(X,Y)\subset\mathcal{SCS}(X,Y)\subset\mathcal{IN}(X,Y), \] where \(\mathcal{IN}\) denotes the closed ideal of inessential or Fredholm perturbation operators \(T\) defined by the fact that \(I+WT\) is Fredholm for every \(W\in B(Y,X)\) and where \(\mathcal{SCS}\) denotes the closed ideal of strictly cosingular operators \(T\) defined by the fact that \(Q_ZT\) is not surjective with \(Q_Z\) the quotient map \(Y\to Y/Z\). Also, the closed ideal \(\mathcal{DP}\) of Dunford-Pettis operators is considered: \(T\in B(X,Y)\) is called Dunford-Pettis if it takes weakly null sequences to norm null ones; a Banach space \(X\) is said to have the DPP (i.e., the Dunford-Pettis property) if \(\mathcal{WK}(X,Y)\subset\mathcal{DP}(X,Y)\) for any Banach space \(Y\).
The authors study the ideals above when \(X\) and/or \(Y\) are \(C^*\)-algebras, von Neumann algebras or their (pre)duals. The main topic is to find conditions (which often are necessary and sufficient) for inclusions or equivalences of the above ideals in terms of properties of the \(C^*\)-algebra.
For example, it is shown that if \(\mathcal{A}\) is a \(C^*\)-algebra and \(Y\) a Banach space, then \(T\in B(\mathcal{A},Y)\) is not strictly singular (if and) only if it fixes a copy of \(c_0\) or of \(\ell_2\); a natural improvement in case \(\mathcal{A}\) is a von Neumann algebra, is given, too. (The companion result in the authors’ above mentioned paper describes non strictly singular operators acting on certain non-commutative \(L_p\), \(1<p<\infty\), as those which fix \(\ell_p\) or \(\ell_2\).)
As ingredients for the proofs, the authors use, among others, the following known facts: If an operator \(T\in B(\mathcal{A},Y)\) (\(\mathcal{A}\) a \(C^*\)-algebra) is weakly compact, then it can be approximated by operators factoring through a Hilbert space, while if \(T\) is not weakly compact, then it fixes \(c_0\); a von Neumann algebra \(\mathcal{A}\) has the DPP iff it is of finite type I (which means that it can be written \(\mathcal{A}=(\sum N_k)_{\infty}\), where the \(N_k\) are von Neumann algebras of type I\(_{n_k}\) with \(\sup n_k<\infty\)). That the DPP enters the subject in a natural way comes from the observation that a weakly compact operator defined on a Banach space with the DPP is strictly singular.
In the following, we list some more results.

(1)

A von Neumann algebra \(\mathcal{A}\) is of finite type I iff \(\mathcal{FSS}(\mathcal{A})=\mathcal{SS}(\mathcal{A})=\mathcal{IN}(\mathcal{A})=\mathcal{WK}(\mathcal{A})\), while all these ideals are different if \(\mathcal{A}\) is not of finite type I.

(2)

For a \(C^*\)-algebra algebra \(\mathcal{A}\), the following are equivalent: (i) \(\mathcal{A}\) has the DPP, (ii) \(\mathcal{A}\) does not contain complemented copies of \(\ell_2\), (iii) \(\mathcal{WK}(\mathcal{A},Y)=\mathcal{SS}(\mathcal{A},Y)\) for any Banach space \(Y\), (iv) \(\mathcal{SS}(\mathcal{A},Y)=\mathcal{D}P(\mathcal{A},Y)\) for any Banach space \(Y\), (v) \(B(\mathcal{A},H)=\mathcal{SS}(\mathcal{A},H)\) for any Hilbert space \(H\).

(3)

The predual of a von Neumann algebra has the DPP iff it does not contain complemented copies of \(\ell_2\).

(4)

Let \(\mathcal{A}\) be a compact \(C^*\)-algebra (meaning that, for each \(a\in\mathcal{A}\), the operator \(\mathcal{A}\ni b\mapsto aba\in\mathcal{A}\) is compact). Then \(\mathcal{SS}(\mathcal{A})=\mathcal{SCS}(\mathcal{A})=\mathcal{IN}(\mathcal{A})\) and \(\mathcal{SS}(\mathcal{A})=\mathcal{SS}^*(\mathcal{A})\), where the adjoint ideal \(\mathcal{SS}^*(\mathcal{A})\) is defined by \(\{T\in B(\mathcal{A}): T^*\in\mathcal{SS}(\mathcal{A}^*)\}\). Furthermore (with “\(+\)” having the obvious meaning), \[ \mathcal{IN}(\mathcal{A})_+=\mathcal{SS}(\mathcal{A})_+=\mathcal{SCS}(\mathcal{A})_+=\mathcal{WK}(\mathcal{A})_+= \mathcal{K}(\mathcal{A})_+\tag{1} \] while there are non-compact \(\mathcal{A}\) for which (1) does not hold.

(5)

A von Neumann algebra is atomic (i.e., each nonzero projection majorizes a nonzero minimal projection) iff (1) holds with \(\mathcal{A}_*\) instead of \(\mathcal{A}\).

(6)

A von Neumann algebra \(\mathcal{A}\) is of finite type I iff the product of any two strictly singular operators on \(\mathcal{A}\) is compact.