What is the ruling on dealing with Islamic finance branches and Islamic windows, and with some commercial banks engaging in Islamic activities in some of their operations, considering the source of their funds and the fact that it is an investment for commercial banks?

If commercial banks open investment windows that are subject to Sharia (definition) provisions, there is no impediment to participating in them. This is because dealing with the bank is prohibited in matters related to usury (riba), but if it engages in a permissible activity, then dealing with it in that regard is not prohibited. It does not matter that the funds for these windows come from mixed funds of commercial banks, for dealing with an owner of mixed wealth is not forbidden; rather, it is merely dislikable (makruh (definition) tanzihi). Refraining from participating in them would be out of preference and recommendation.