On Maximal Ideals of Compact Connected Topological Semigroups

Several results concerning ideals of a compact topological semigroup with can be found in the literature. In this paper, we further investigate in a compact connected topological semigroup how the conditions and affect the structure of ideals of , especially the maximal ideals.

Continuity of the Inverse in Pseudocompact Paratopological Groups

By a celebrated theorem of Numakura, a Hausdorff compact topological semigroup with two-sided cancellation is a group which has inverse continuous, i.e., it is a topological group. We improve Numakura's Theorem in the realm of non-Hausdorff topological semigroups. This improvement together with some properties of pseudocompact nature in the field of bitopological spaces is used in order to prove that a T0 paratopological group (G,τ) is a (Hausdorff) pseudocompact topological group if and only if (G, τ ∨ τ-1) is pseudocompact or, equivalently, G is Gδ-dense in the Stone–Čech bicompactification [Formula: see text] of (G, τ, τ-1). We also present a version for paratopological groups of the renowned Comfort–Ross Theorem stating that a topological group is pseudocompact if and only if its Stone–Čech compactification is a topological group.

Multipliers onL(S),L(S)**, andLUC(S)*for a locally compact topological semigroup

We study compact and weakly compact multipliers onL(S),L(S)**, andLUC(S)*, where the latter is the dual ofLUC(S). We show that a left cancellative semigroupSis left amenable if and only if there is a nonzero compact (or weakly compact) multiplier onL(S)**. We also prove thatSis left amenable if and only if there is a nonzero compact (or weakly compact) multiplier onLUC(S)*.

A note on the support of right invariant measures

A regular measureμon a locally compact topological semigroup is called right invariant ifμ(Kx)=μ(K)for every compactKandxin its support. It is shown that this condition implies a property reminiscent of the right cancellation law. This is used to generalize a theorem of A. Mukherjea and the author (with a new proof) to the effect that the support of anr*-invariant measure is a left group iff the measure is right invariant on its support.

Products of Locally Compact Groups with Zero and Their Actions

In [4], Hofmann defines a locally compact group with zero as a Hausdorff locally compact topological semigroup, S, with a non-isolated point, 0, such that G = S — {0} is a group. He shows there that 0 is indeed a zero for 5, G is a locally compact topological group, and the identity of G is the identity of S. The author has investigated actions of such semigroups on locally compact spaces in [1; 2]. In this paper, we are investigating direct products of semigroups of the above type and actions of these products; for a special case of this, the reader is referred to [3].

A note on countably compact semigroups

It is well known that every compact topological semigroup has an idempotent and every compact bicancellative semigroup is a topological group. Also every locally compact semigroup which is algebraically a group, is a topological group. In this note we extend these results to the case of countably compact semigroups satisfying the Ist axiom of countability. Some of our results are valid under the weaker condition of sequential compactness.

A universal semigroup

In [4] S. Ulam asks the following question. ‘Does there exist a universal compact semigroup; i.e., a semigroup U such that every compact topological semigroup is continuously isomorphic to a subsemigroup of it?’ The author has not been able to answer this question. However, in this paper, a proof is given for the following related result.Let Q denote the Hilbert cube of countably infinite dimension and C(Q) the Banach space of continuous real-valued functions on Q with the usual norm. Let U denote the semigroup consisting of all bounded linear operators T: C(Q)→ C(Q) with ∥T∥ ≦ 1 and let U be endowed with the strong topology. Then, for every compact metric semigroup S with the property: (1.1) for all x, y ∈ S, with x ≠ y, there exists a z ∈ S, such that xz ≠ yz or zx ≠ zy;there exists a 1 − 1 mapping φ of S into U such that φ is both a semigroup isomorphism and a homeomorphism.U is metrizable, but is not compact; hence it does not provide an answer to the question of Ulam. The proof of the above statement leans heavily on a result of S. Kakutani [1].

The topological structure of 𝒟–classes

Let S be a compact, topological semigroup with identity. Suppose D, L and R are the D, L and R classes of some x ∈ S. Let (L, α., L/H), (R, β, R/H), (D, γ, D/H) and (D, δ, D/R) by the fibre spaces gotten where α, β γ an δ are the natural maps. It is shown that (D, γ, D/H) has topologically the same structure as the fibre space associated with (L, α, L/H) by R. Also if (L, α, L/H) is locally trivial (locally a cartesian product) then so is (D, δ, D/R) and if both (L, α, L/H) and (R, β, R/H) are locally trivial then so is (D, γ, D/H).