Factors, roots and embeddability of measures on lie groups

Convolution roots and embeddings of probability measures on locally compact groups

“…Now let G be an almost algebraic group and μ ∈ P (G). In this case it has been known [3] that F (μ)/Z(μ) is compact. Additionally, it was shown in [5] (see also [6]) that if μ is infinitely divisible in G then there exists an almost algebraic subgroup H of G containing supp μ such that μ is infinitely divisible on H and Z(μ) ∩ H, which is the analogue of Z(μ) within H, contains a simply connected nilpotent subgroup N such that Z(μ)/N is compact.…”

“…We recall that for any root ρ of μ the support of ρ is contained in N (μ); this holds more generally for all two-sided factors of μ (see [3], Proposition 1.1, or [1], Lemma 5.1). Furthermore, if ρ ∈ R n (μ) for n ∈ N, then there exists x ∈ N (μ) such that ρ ∈ P (G(μ)x), and for any such x we have x n ∈ G(μ) (this is straightforward to deduce from the fact that the image of μ in the quotient N (μ)/G(μ) is the point mass at identity; for details see [5], Proposition 2.7, or [1], Lemma 5.2).…”

Affinely infinitely divisible distributions and the embedding problem

“…We note that any compact connected Lie group of rank 1 is of type C, since in this case any maximal torus is a circle group and all maximal tori are conjugate. The special linear group SL(2, IR) is also of type C, so are its quotient P SL (2,IR) and all the covering groups with finite fibers. The semidirect product of T , the circle group, with a vector group V under an action of T on V can also be seen to be a group of type C. We now extend the result as in Corollary 4.2 for T -actions to actions of these groups, under the condition that the measure µ is invariant under the action.…”

“…Let B be the smallest closed subgroup of G containing the support of µ. We recall that any root ν of µ on G is supported on the normaliser of B in G (see [2], Proposition 1.1). Therefore replacing G by the normaliser of B we may without loss of generality assume that B is normal in G.…”

An introduction to the embedding problem for probabilities on locally compact groups

“…Clearly C = [2, ∞)SL (2, R) is path connected, and for any A ∈ [1, 2] D, the path [1,2] A connects A to a point in C. Hence S is (path) connected. However, −I is not continuously embedded on S, for if θ : R + → S is a continuous homomorphism such that θ (1) = −I, then t → | θ (t) | is a continuous homomorphism of (R + , +) into (R + , ×) such that θ (1) = 1, hence | θ (t) | = 1 for allt ∈ R + .…”