Quasi-Banach operator ideals with a very strange trace (Q378896)

The paper presents quasi-Banach operator ideals \(\mathfrak{A}\) with surprising properties: (1) \(\mathfrak{A}\) supports a continuous trace \(\tau\) that vanishes at all finite rank operators. (2) \(\mathfrak{A}\) contains the identity map \(I_{Z}\) of an infinite-dimensional Banach space \(Z\) and \(\tau( I_{Z}) =1.\) So, there exist operators \(T\in\mathfrak{A}\left( Z\right) \) such that \(\tau( T^{n}) =1\) for all positive integers \(n\), and this is not possible for singular traces in the framework of Hilbert spaces.