Boundedness and absoluteness of some dynamical invariants in model theory

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any (possibly infinite) tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text] (so a tuple of unbounded length). By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see text] are naturally [Formula: see text]-flows (even [Formula: see text]-ambits). We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of [Formula: see text]), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend (as groups equipped with the so-called [Formula: see text]-topology) on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal (equivalently, of all the minimal left ideals) is an absolute property (i.e. it does not depend on the choice of [Formula: see text]) and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations (in various terms) of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 (2018) 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism (described in [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism (in particular, [Formula: see text] is G-compact). We also obtain some variants of these results, formulate some questions, and explain differences (providing a few counterexamples) which occur when the flow [Formula: see text] is replaced by [Formula: see text].

Spatiality of Derivations of Operator Algebras in Banach Spaces

Suppose thatAis a transitive subalgebra ofB(X)and its norm closureA¯contains a nonzero minimal left idealI. It is shown that ifδis a bounded reflexive transitive derivation fromAintoB(X), thenδis spatial and implemented uniquely; that is, there existsT∈B(X)such thatδ(A)=TA−ATfor eachA∈A, and the implementationTofδis unique only up to an additive constant. This extends a result of E. Kissin that “ifA¯contains the idealC(H)of all compact operators inB(H), then a bounded reflexive transitive derivation fromAintoB(H)is spatial and implemented uniquely.” in an algebraic direction and provides an alternative proof of it. It is also shown that a bounded reflexive transitive derivation fromAintoB(X)is spatial and implemented uniquely, ifXis a reflexive Banach space andA¯contains a nonzero minimal right idealI.

REWRITING A SEMIGROUP PRESENTATION

Let [Formula: see text] be a finitely presented semigroup having a minimal left ideal L and a minimal right ideal R. The main result gives a presentation for the group R∩L. It is obtained by rewriting the relations of [Formula: see text], using the actions of [Formula: see text] on its minimal left and minimal right ideals. This allows the structure of the minimal two-sided ideal of [Formula: see text] to be described explicitly in terms of a Rees matrix semigroup. These results are applied to the Fibonacci semigroups, proving the conjecture that S(r, n, k) is infinite if g.c.d.(n, k)>1 and g.c.d.(n, r+k−1)>1. Two enumeration procedures, related to rewriting the presentation of [Formula: see text] into a presentation for R∩L, are described. The first enumerates the minimal left and minimal right ideals of [Formula: see text], and gives the actions of [Formula: see text] on these ideals. The second enumerates the idempotents of the minimal two-sided ideal of [Formula: see text].

Footnote to a paper of Baker and Mines

Let S be a semigroup with a compact topology in which multiplication is continuous on the left (i.e. xi→x implies xiy→xy for each y in S). Then S has a minimal left ideal L which is compact; each idempotent e in L is a right identity for L (xe = xfor each x ∈ L)and L = Se; Ge = eL is a group and L is the union of all such groups; and if f is a second idempotent in L, the canonical map x ↦ fx of Ge to Gf is an algebraic isomorphism (see Ruppert(2) for these facts). Baker and Milnes(1), §4(A), have observed that, in the case in which S is the Stone–Cech compactification of a discrete abelian group, the canonical map from Ge to Gf may not be a homeomorphism. (This contrasts with the situation in compact semigroups with separately continuous multiplication.) We present a simple proof of a more definitive result.

On the equivalence of invariant integrals and minimal ideals in semigroups

Let S be a Hausdorff topological semigroup and Cb,(S), Cc (S), the spaces of real valued continuous functions on S which are respectively bounded and have compact support. A regular measure m on S is r*-invarient if m(B) = for every Borel B ⊂ S and every x ∈ S, where tx: s → sx is the right translation by x. The following theorem is proved: Let S be locally compact metric with the tx's closed. Then the following statements are equivalent: (i) S admits a right invariant integral on Cc (S). (ii) S admits an r*–invariant measure, (iii) S has a unique minimal left ideal. The above equivalence is considered also for normal semigroups and analogous results are obtained for finitely additive r*–invariant measures. Also in the case when S is a complete separable metric semigroup with the tx's closed, the following statements are equivalent: (i) S admits a right invariant integral I on Cb(S) such that I(1) = 1 and satisfying Daniel's condition. (ii) S admits an r*–invariant probability measure. (iii) S has a right ideal which is a compact group and which is contained in a unique minimal left ideal. Finally, in order that a locally compact S admit a right invariant measure, it suffices that S contain a right ideal F which is a left group such that (B ∩ F)x = BX ∩ Fx for all Borel B ⊂ S.