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Could you provide more info here? What provider version, some code, before and after state, what operations you ran etc. all of these would be helpful...

(not from Pulumi) Do you have CloudTrail logs enabled? If so, it should be pretty quick to find what process deleted the container

Sorry. My fault. We use an Azure BlockBlobStorage. Today we added a new container using Pulumi script. After that, the container was no longer in the container list. When calling from the ID via the browser url, the view for the container appears but it loads infinitely. We are sure that we have not overwritten any container or the container was deleted by us. Also Pulumi shows it in the output and says that we have added a resource. In the resource view of Pulumi the container is still displayed.

Sorry trying to parse this.. So

az storage container list

doesn't list the container? Does a pulumi refresh detect a "deletion" of the container since I believe you mention it is still visible in the state? Do you have soft delete enabled? If so

az storage container list --include-deleted

should show it if it was deleted. I am sure you have done this but double check you are working with the same subscription on the CLI/web console as the pulumi environment (I have done this more often than I should admit)...