Is it permissible, Islamically, to open a shop for selling women's underwear managed by a woman, where the owner's role is limited to checking accounts at the end of the day?

Practicing this type of trade is permissible, as a man's dealings with a foreign woman, whether through sale, lease, or the like, are permissible when there is a need, provided the rules of Sharia (definition) are observed.

Therefore, there is no harm in a man entering a female worker's place of business to close her account or perform other necessary tasks, provided that it does not lead to a forbidden act such as khalwa (definition) (private seclusion) or unlawful gazing.

This is achieved by ensuring that he does not enter upon her unless another person is present, thereby negating khalwa; that the woman is wearing complete Shari'i attire; and that she avoids speaking with him about anything unnecessary, and that her speech is decisive, without softness or elaborateness.

If these controls are adhered to, there is nothing to prevent engaging in this commercial activity.