Is there a way like in on-premise to use a custom trusted identity provider to perform claims based authentication and search in SharePoint Online? If so could anyone point to any links on how to achieve this? I looked at Office 365 admin center as well as SharePoint admin center and could not find any configuration setting to achieve this (Just like how we can register TIP and use it to assign permissions for sites and perform various operations). Any help is really appreciated.
From https://technet.microsoft.com/en-us/library/jj819267.aspx or https://technet.microsoft.com/en-us/library/b6db338b-522b-44bf-afb7-1de7827691d0#bkmk_ClaimsBasedAuthenticationSupport or https://technet.microsoft.com/en-us/library/b6db338b-522b-44bf-afb7-1de7827691d0#bkmk_AlternateAccessMapping it looks like Alternate Access Mapping (AAM) and Claims-Based authentication is not supported in Office 365 (SharePoint online).
Alternate Access Mapping (AAM)Not available to SharePoint Online customers. SharePoint Server 2013 customers can configure alternate access mapping to map requests where the URL of a Web request received by Internet Information Services (IIS) differs from the URL that was typed by a user.
Claims-Based Authentication SupportNot available to SharePoint Online customers. SharePoint Server 2013 customers configure claims-based authentication for web applications that support SharePoint Server 2013 server-to-server authentication and app authentication.