Will an axial compressor work with an ideal fluid?
First of all, incompressible flow is subsonic, however subsonic flows are not all incompressible. An ideal gas flow may be "incompressible" until it meets the compressor. Even if the vehicle is travelling at supersonic speed, the flow is reduced to subsonic ("incompressible") flow by the intake system before reaching the compressor. At this point the subsonic flow gets compressed. Presumably this is what is meant here.

The compressor stages each reduce the cross-sectional area, while the fans draw the gas backwards to maintain velocity. Thus the gas, ideal or not, must compress to fit the reduced volume available to it. This compression creates heat which raises the gas temperature, further raising its pressure.

Viscosity is necessary to establish a circulatory flow component around each airfoil-shaped blade (the Kutta condition). It is this circulatory component which creates the velocity differentials needed for the Bernoulli principle to create the dynamic pressure differentials which make the blades do their work. Superfluid helium (close to absolute zero) has no viscosity and has been found to exert neither lift nor drag on an immersed body.