Praise be to Allah.
If this bank is calling itself “Islamic” in order to deceive people, and it is dealing in riba or other haraam transactions, then it is not an Islamic bank, and it is not permissible to buy shares in it, or to buy and sell its shares, because that is cooperating in doing something haraam. Allaah says (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression”
But if this bank is really Islamic, and organizes its dealings in accordance with the Qur’aan and Sunnah, and does not engage in haraam transactions, then there is nothing wrong with buying shares in it and trading in its shares. Rather that should be done, because it will encourage it and help it to deal with the challenges it faces as a result of the spread of riba and haraam transactions.
In the answer to question no. 47651 we have explained the characteristics of an Islamic bank.
In Fataawa al-Lajnah al-Daa’imah (13/506) it says:
If banks are based on riba and deal in riba, then it is not permissible to buy shares in them, because this is cooperating in sin and transgression, which Allaah has forbidden when He said (interpretation of the meaning):
“but do not help one another in sin and transgression”
[al-Maa'idah 5:2]. End quote.
It also says (13/507):
It is permissible to buy shares in banks that do not deal with riba, and to take the profits that come from buying shares in the bank, because this is the result of a transaction that is not haraam and there is nothing wrong with it, so it is halaal. End quote.
It also says (13/508):
Buying shares in banks or companies that deal with riba is not permitted. If a subscriber wants to get rid of his riba-based shares, he should sell his shares at the current market value but only take his capital; the rest should be donated to charity. It is not permissible for him to take any of the interest of his shares or riba-based profits. But if the shares are in a company that does not deal with riba, then their profits are halaal. End quote.
And Allaah knows best.