# Properties of the spectrum of the Koopman representation

Let $$G$$ be a discrete countable infinite group acting on a compact metric space $$X$$ via homeomorphisms preserving a probability measure $$\mu$$.

A function $$\lambda\colon G\to \mathbb C$$ is an eigenvalue of the action of $$G$$ if there exists a function $$f\in L^2(X,\mu)$$ such that for every $$g\in G$$ one has $$\lambda(g)\cdot f=f\circ g$$.

In this paper: Ergodicity of the Cartesian product by E. Flytzanis, Trans. Amer. Math. Soc. 186 (1973), 171-176 (freely available link at AMS site), there are two results concering the product of two dynamical systems given by $$\mathbf Z$$-action:

• a sufficient condition for the ergodicity of the product,
• a description of the spectrum of their cartesian product.

It is also written there that the above results hold also for $$G$$-action, but it is not specified what does it exactly mean. Am I right that:

• A product of $$(X,G)$$ and $$(Y,G)$$ is ergodic if the function constantly equal to 1 is the only common eigenvalue for $$(X,G)$$ and $$(Y,G)$$?
• The set of eigenvalues of $$X\times Y$$ equals the set of all functions of the form $$f\cdot g$$ (pointwise multiplication), where $$f$$ is an eigenvalue for $$X$$ and $$g$$ is an eigenvalue for $$Y$$?

I need also two other properties, but I could not find the appropriate references for them (maybe I am wrong that they are true?):

• The set of eigenvalues of a factor of a dynamical system $$(X,G, \mu)$$ is contained in the set of eigenvalues of $$(X,G,\mu)$$?
• Every eigenvalue of an ergodic dynamical system is simple (beware: I do not assume that $$G$$ is abelian).
• For your first question the answer is "no". This is already seen when considering the regular representation of a finite group with trivial abelinzation. As a general rule of thumb, it is convenient to replace "eigenvalues" with "finite dimensional subrepresentations" when considering non-commutative groups. Upon such a replacement you'd get a positive answer. Your second question is badly formulated. Are you looking for eigenvalues or eigenvectors? May 19, 2020 at 9:44
• The last two properties that you ask about are correct. For the first, a representation of a factor is a subrepresentation, and for the second, in the particular case of a compact group, every ergodic representation is a factor of the regular. Note that when asking about eigenvalues, the question always reduces to the case of a compact group. May 19, 2020 at 9:45
• Thank you, Uri Bader! Your answer is really helpful. For the first question - could you recommend some reference for this topic? In the second question I meant eigenvalues. If I understand correctly, an eigenvalue is a function from $G$ to $\mathbb C$ so I can consider pointwise multiplication (that is $f\cdot g(x)=f(x)\cdot g(x)$). May 20, 2020 at 10:46
• I am sorry for the confusion - I used $f$ and $g$ for eigenvalues which is probably not standard. May 20, 2020 at 10:58
• oh... this is a horrible notation you adopted there. The answer is negative here as well. I'll write it in an answer format. May 21, 2020 at 7:58

Take $$G=S_3$$, the symmetric group of the set $$X=\{1,2,3\}$$. Then $$L^2(X)$$ is decomposed to the trivial representation and a another two dimensional irreducible representation. As a $$G$$-space, $$X\times X\simeq X \cup G$$ where $$X\subset X\times X$$ is the diagonal and $$G$$ corresponds to the rest. This shows that $$X\times X$$ is not ergodic, providing a counter example to the first question. It also follows that $$L^2(X\times X)$$ contains a sub-representation isomorphic to the regular representation $$L^2(G)$$ and in particular it contains the one dimensional sign representation. This provides a counter example to the second question.