I want to apologize to all of you and thank you for your help and collaboration.
The truth is that the problem was that I did not know that WordPress (and therefore bbpress) uses the “Display Name Publicly As” field to show him as the author of the publications. It is an error due to my ignorance.
Thanks to your clarifications and after reviewing the plugins suggested by Robin, I have been able to install the “Display Name Publicly As” plugin and correctly configure how bbpress displays the author of the content.
Thanks a lot again for everything.
Thank you for the answer.
And would it be possible to show the name instead of the user in the topic index view (‘started by…’)? That is, do not internally modify the management of bbpress with the user, but only show the name corresponding to the user who makes a publication.