If the front fan on a gas turbine had a blade, or some of its blades extending into the surrounding housing, where there was a maglev ring of electromagnets, could fuel use be reduced by the engine? While still providing similar levels of thrust? Perhaps at the expense of electricity in the cabin.
There's a few layers to this question, each with its own answer. However, fortunately, the answers for each layer are the same: no. You cannot improve your fuel economy by adding a maglev system to an engine.
The first layer is easy -- conservation of energy. Any energy that you put into the compressor blades from the maglev system must come from somewhere. That "somewhere" is another engine, such as the APU. All you do in this case is rob Peter to pay Paul. However, it's even worse in this case. Jet engines are tremendously optimized beasts. A 1/10th of a percent improvement in fuel economy translates to somewhere around $500 in savings per plane, per flight! I hand waved the distances there... math is based on a 10 hour flight and 36,000 gallons of gas consumed. You should get the idea for just how much money there is in this business.
What does this mean? Well, it means that the engines are already optimized to do exactly what they need to do. Using an APU and bulky maglev system would consume far more energy than it cost to keep the compressors spinning the normal way. Yes, your main engine may be slightly more efficient, but you more than pay for those enhancements by running a less efficient engine.
Worse, you might not get any benefit at all! Maglev equipment doesn't exactly come without costs. It takes space to put that equipment into the engine space, and it brings mass. You may not be able to use the most ideal materials for compressor blades because they need to have good magnetic properties now... that can decrease efficiency. You may find that not only does the maglev equipment require a separate power source, but that it actually decreases the efficiency of the engine by applying new requirements to the design!
I hate to use an appeal to authority on Physics.SE, but truthfully, if there was a solution like this, the engineers who pour hundreds of thousands of hours into these devices would probably have found it. Airplane manufacturing is a multi-billion dollar industry feeding an even more massive air travel industry. There's a lot of eyes on efficiency of planes. Lots of sharp eyes.
No is the quick answer
If the front fan on a gas turbine had a blade, or some of its blades extending into the surrounding housing, where there was a maglev ring of electromagnets,
This is an engineering question. The thrust from a Rolls-Royce_Trent, for example, is 420 kN and a lot of that comes from the high by pass ratio. We know exactly how fan blades thrust air backwards and that fan housing is vital to performance.
It's a double whammy, you take power from the engine core to power the maglev, (if they can provide it), then you block off the very point of the high pass systems but without giving an explanation for how magnets would have sufficient power to drive enough air to rip off the top layer of a badly maintained runway, which a modern turbine engine is perfectly capable of doing, if the aircraft rotates too quickly on takeoff.
The airplane was positioned for the run with asphalt extending from close to the wing trailing edges to beyond the empennage. During the high-power part of the run, asphalt lifted from behind the left engine and broke into pieces, sending large fragments into the aft fuselage and outboard horizontal stabilizer.
Or am I misunderstanding you, if I am, my sincere apologies.