Let's say your house is worth 480k.
Your girlfriend must pay you €240k to buy half of the house.
If, for example, she had €240k cash, then she could give you this money and she would own half the house but none of the mortgage.
If she gives you €140k cash, she will owe €100k on the mortgage.
Your mortgage is €280k. For simplicity, you could pay just €80k of the mortgage off and invest the balance of €60k in your own name.
Now
She owns half the house.
She "owns" half the mortgage - i.e. 100k of a €200k mortgage.
Alternatively, reduce the mortgage to €140k. She still owes €100k of this, so she would be responsible for 100/140ths of every repayment.
Brendan