Symmetric functionals on simply generated symmetric spaces

In the present paper we suggest a construction of symmetric functionals on a large class of symmetric spaces over a semifinite von Neumann algebra. This approach establishes an isometric isomorphism between the symmetric functionals on simply generated symmetric spaces and shift-invariant functionals on the space of bounded sequences. It allows to obtain an isometric isomorphism between the classes of all continuous symmetric functionals on different symmetric spaces. Notably, we show that this mapping is not bijective on the class of all Dixmier traces. As an application of our results we prove an extension of the Connes trace formula for wide classes of operators and symmetric functionals.