How to deal with incommunicable co-author The background: In the grand area of Mathematics in which I do research, there is a famous Professor, let us call him 'A', who started a joint project with another younger colleague, let
us call him 'B'. 'A' gave the main ideas and they produced an initial draft. However, after a while 'A' perhaps lost interest or was busy with other things and stopped communicating with 'B'.
'B' could not manage to move the project forward by himself, and it stayed put. After many years,
on the initiative of a third colleague, let us call him say 'C', and with both the approval
of 'A' and 'B', I joined them on this project and started working with 'B'. After, say, two years, 'B' and I  managed to have a paper, which we think is quite good.
We have sent it to 'A' to hear his opinion. However, he never
returns our email messages. We have also contacted his secretary, and she has been very helpful, but we don't seem to get any reply from 'A'. Because of the pandemic, travelling and meeting 'A'
in person is not an option.
'B' and I would like to distribute and eventually publish this work. We have spent quite some
time and energy on it. Moreover, the results are interesting and could be used by ourselves in other projects, as well as by other researchers. If we wait too much, there is even the risk
of our paper becoming old and superseded by other people's work.
Of course we cannot post this work publically without 'A's name, as he was who gave the key ideas. On the other hand, posting this work with his name could also upset 'A', say, he
could think there are mistakes in the paper and he would not want to have his name associated with it.
What is the ethical way to solve this?
Edit: Thanks for all answers! They really add to the discussion. In view of your comments, I feel like adding some more details to the story and making more clear our position.
It is not that A just gave some vague ideas. He really contributed technical stuff during the first year or so of his collaboration with B.
What I did with B, in the past two years, was cleaning up the Mathematics
and adding some interesting applications. I think we have really turned the draft into a (quasi?) paper. I believe our manuscript is polished.
A is well known for his irresponsiveness in the community of researchers in the field. Indeed this is the reason why there has been the hiatus in the work on the paper, until I joined forces with B.
B and I met A in conferences in the past two years. He was nice and seemed interested in finishing the work. However, on those occasions there was
little time to collaborate with him, as he is a busy person.
B and I are fine with whatever course of action A thinks it is appropriate. We do not require him to check our calculations. We are fine to include him as an author, indeed we would like very much to have him as  a co-author. We are fine to leave his name out, if he prefers so. But I have the impression he wants to be included.
Oh, yes, we certainly want to keep a collegial relation with A!
Anyway, we cannot proceed without his response!
When dealing with extremely busy people, you want to formulate your emails in a way that makes it as easy as possible for them to give you what you need.
Providing one's opinion on a paper is a complex, open-ended process. Even more so if you are a coauthor - you need to not only read and think about what is written, but also consider what else could have been written to make the paper busy. In a situation as you describe, I'd consider it perfectly normal for the professor to take a few months to get back you even they seriously intend to do so. Thus, your request is not a good choice.
Instead provide them with a straight-forward way out that works for you. Thus, complete the draft and makes sure it is completely polished. Make a plan regarding what you want to do with it. Then send it to the professor, pointing out that you and B agree that the paper is finished, and that you want to submit it to the arXiv/Journal X, and that you just ask for their consent to proceed. If the contribution of the professor is very limited, you can consider also stating merely acknowleding them as an option. If C or anyone else respected by A has read the draft and also considers it ready, include this information. Overall, you want to make it very easy for A to just respond &quot;its fine, go ahead&quot;.
If this doesn't work, there is one way to escalate further, but I'd consider this borderline from an ethical perspective. That is to let A know that if they don't object within eg 2 weeks, you will consider this as permission to go ahead with the plan. Only ever think about this if your draft is really, really done.