Are measurable homomorphisms $ (\Bbb{C},+) \to (\Bbb{C},+) $ or $ (\Bbb{C},+) \to (\Bbb{C},*) $ continuous, and do they admit an explicit description?

Question

I am interested in generalizations of the following fact (known as automatic continuity, as pointed out below). I am especially looking for references to papers dating back to 1920’s. I feel that question like these have been well-studied when people were still interested in set-theoretic aspects of analysis...

(Cauchy) Any measurable automorphism $ (\Bbb{R},+) \to (\Bbb{R},+) $ is necessarily a linear function, and any measurable homomorphism $ (\Bbb{R},+) \to (\Bbb{R}^{\times},\times) $ is necessarily an exponential function $ x \mapsto e^{a x} $.

Is something similar true for homomorphisms $ f: (\Bbb{C},+) \to (\Bbb{C},+) $ or $ f: (\Bbb{C},+) \to (\Bbb{C}^{\times},\times) $ (assuming that $ \ker(f) = \Bbb{Z} $ in the latter)? (Yes, see answers below.)

That is, I am interested in facts which follow the following rough pattern:

If a map is not set-theoretically wild, e.g. it is measurable or Borel, and satisfies some identities, then it is in fact continuous, and, furthermore, it can be given by an explicit formula.

Answer 1

The general phenomenon you're looking for is called automatic continuity. This is the line of research that stemmed from results regarding the continuity of homomorphisms from (R,+) to (R,+). The two principal general results illustrating the phenomenon of automatic continuity are the following.

Theorem (S. Banach). Any Baire measurable homomorphism between Polish groups is continuous.

Theorem (A. Weil). Any Haar measurable homomorphism from a locally compact Polish group into a Polish group is continuous.

Both of these theorems apply to the locally compact Polish group (C,+).

Automatic continuity is still an active area of research, see this recent survey by Christian Rosendal.

Answer 2

On page 23 of his 1932 book Sur la théorie des opérations linéaires Banach proves:

Theorem. A Baire measurable homomorphism between Polish groups is continuous.

Note that Banach writes $+$ for the composition but neither does he assume nor use that a group is abelian. He proves the result first for Borel measurable homomorphisms and remarks immediately afterwards that the same argument even shows that a Baire measurable homomorphism between Polish groups is continuous, a fact usually attributed to Pettis for reasons that are unclear to me.

Recall that a topological group is Polish if its topology is second countable and metrizable with respect to a complete metric. The Baire $\sigma$-algebra is the $\sigma$-algebra generated by the Borel sets and the meager sets (beware that there are other uses of the term "Baire measurable" in the literature -- there are even published papers whose authors fell for this trap).

In particular, the specific question you ask about the real and complex numbers has a positive answer in all cases.

There is an entire industry, called automatic continuity, asking the question whether homomorphisms/derivations etc are continuous only due to their algebraic property. The easiest result in that direction states that a $*$-homomorphism between $C^{\ast}$-algebras is a linear contraction. One of the major players of that topic, H.G. Dales, has recently written a voluminous book of the same title, containing many results of that spirit and many historical remarks.

Answer 3

I just thought that I should mention a result that considerably strengthens André Weil’s result mentioned by François in his answer above.

Adam Kleppner, in his 1989 paper Measurable Homomorphisms of Locally Compact Groups, proved that any Borel-measurable homomorphism between locally compact (Hausdorff) groups is continuous. It does not matter whether or not any one of the groups is Polish.