Who wrote the book of Hebrews and is the book complete? I was recently reading in the book of Hebrews when I realized that I didn't know who was speaking, and I haven't been able to figure it out.

I was just reading a commentary that stated it was James, the brother of Jesus.  And another commentary that thought it was Paul, but, I've noticed that Paul always seems to introduce himself.  Which made me wonder something else - is the first page missing?  Because it seems as if the chapter starts in the middle of a conversation.  Have there been any studies on this?

What is an overview of scholarly research on authorship of Hebrews?
We do not know who wrote the book now known as the Epistle to the Hebrews, although at one stage it was thought that Paul was the author.  Hebrews was never seriously attributed to James, brother of Jesus.

Burton L. Mack says, in Who Wrote the New Testament, page 188, the author of Hebrews was male, because of a masculine gender self-reference (Heb ll:32). He  may not have had as quick and sharp a mind as Paul, or as personal and passionate an approach to public debate and theological argumentation, but he was far superior to Paul in learning, analytical capacity, and systematic thinking.  In Mack's view, he was capable of keeping in mind large quantities of conceptual detail and working with multiple themes as he wove concepts in and out of a vast Platonic world of ideas. 

Mack says that as far as scholars have been able to tell, the author and the Christian congregation he had in mind might have been located anywhere in the Eastern Mediterranean area. Paul Barnett believes he can narrow the location of the author down to Alexandria in Egypt, although this still does not give us a name.  He says, in The Birth of Christianity: The First Twenty Years, page 108, that the Letter is written in superior koine Greek, with an elaborate use of allegory consistent with the Hellenistic Judaism of Alexandria.

Hebrews would appear to be incomplete if it had been an epistle, as it was normal practice, followed by Paul and others, to open with a greeting. However, there is nothing - other than the salutations at the end of the book - to suggest that, in its original form, Hebrews was really an epistle.