Division algebras of Gelfand-Kirillov dimension two

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Let A be a finitely generated K-algebra that is a domain of GK dimension less than 3, and let Q(A) denote the quotient division algebra of A. We show that if D is a division subalgebra of Q(A) of GK dimension at least 2 then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dimension less than 3 over an algebraically closed field K then any division subalgebra D of Q(A) is either a finitely generated field extension of K of transcendence degree at most one, or Q(A) is finite dimensional as a left D-vector space.