The Characterizations of Extreme Amenability of Locally Compact Semigroups

Abstract

We demonstrate that the characterizations of topological extreme amenability. In particular, we prove that for every locally compact semigroup 𝑆 with a right identity, the condition 𝜇⊙(𝐹×𝐺)=(𝜇⊙𝐹)×(𝜇⊙𝐺), for 𝐹, 𝐺 in 𝑀(𝑆)^∗, and 0<𝜇∈𝑀(𝑆), implies that 𝜇=𝜀_𝑎, for some 𝑎∈𝑆 (𝜀_𝑎 is a Dirac measure). We also obtain the conditions for which 𝑀(𝑆)^∗ is topologically extremely left amenable.

1. Introduction

Let 𝑆 be a locally compact (Hausdorff) semigroup such that its multiplication is separately continuous. We denote by 𝑚(𝑆) the Banach algebra under the supremum norm of all bounded real-valued functions on 𝑆. For a topological semigroup 𝑆, let BM(𝑆) and CB(𝑆) be the closed subalgebras of 𝑚(𝑆) consisting of all Borel measurable functions and all continuous functions on 𝑆, respectively. Let 𝐶₀(𝑆) be the subalgebra of CB(𝑆) consisting of the functions which vanish at infinity. Let 𝑀(𝑆) be the Banach space of all bounded regular Borel (signed) measures on 𝑆 with a total variation norm. Let 𝑀₀(𝑆)={𝜇∈𝑀(𝑆)∶𝜇≥0and‖𝜇‖=1} be the set of all probability measures in 𝑀(𝑆).

It is known that 𝑀(𝑆)≃𝐶₀(𝑆)^∗ via the correspondence 𝜇→𝜇, where 𝜇(𝑓)=∫𝑓𝑑𝜇 for any 𝑓 in 𝐶₀(𝑆) [1, Section 14]. Consider the continuous dual 𝑀(𝑆)^∗ of 𝑀(𝑆). An element 𝑀 in 𝑀(𝑆)^∗∗ is called a mean on 𝑀(𝑆), if 𝑀(1)=1 and 𝑀(𝐹)≥0, whenever 𝐹≥0. An equivalent definition for a mean is thatinf{𝐹(𝜇)∶𝜇∈𝑀₀(𝑆)}≤𝑀(𝐹)≤sup{𝐹(𝜇)∶𝜇∈𝑀₀(𝑆)}, (1.1)for any 𝐹 in 𝑀(𝑆)^∗. We also note that 𝑀∈𝑀(𝑆)^∗∗ is a mean if and only if ‖𝑀‖=𝑀(1)=1. Each probability measure 𝜇 in 𝑀₀(𝑆) is a mean on 𝑀(𝑆)^∗∗ if we put 𝜇(𝐹)=𝐹(𝜇), for any 𝐹 in 𝑀(𝑆)^∗. An application of Hahn-Banach separation theorem shows that 𝑀₀(𝑆) is weak∗ dense in the set of all means on 𝑀(𝑆)^∗.

Under point-wise operations and the supremum norm, 𝐶₀(𝑆) becomes a Banach algebra. Arens product can thus be defined in 𝐶₀(𝑆)^∗∗. In particular, we have the following defining formulas for any 𝑓,𝑔 in 𝐶₀(𝑆), 𝑚 in 𝐶₀(𝑆)^∗, and 𝜃,𝜙 in 𝐶₀(𝑆)^∗∗:(𝑚⊙𝑓)(𝑔)=𝑚(𝑓𝑔),(𝜙⊙𝑚)(𝑓)=𝜙(𝑚⊙𝑓),(𝜃⊙𝜙)(𝑚)=𝜃(𝜙⊙𝑚). (1.2)

This product induces a multiplication in 𝑀(𝑆)^∗ via the identification 𝑀(𝑆)≅𝐶₀(𝑆)^∗. Since 𝑀(𝑆) is a set of measures beside being the continuous dual of 𝐶₀(𝑆), this multiplication in 𝑀(𝑆)^∗ is richer in content than just a generic Arens product in the second dual of a Banach algebra, and it is different from the point-wise multiplication in 𝐶₀(𝑆).

For 𝐹,𝐺 in 𝑀(𝑆)^∗, we denote the multiplication of 𝐹 and 𝐺 by 𝐹×𝐺. In [2], it is shown that 𝐹×𝐺 is defined via the following three steps.(a) For any 𝜇∈𝑀(𝑆) and 𝑓∈𝐶₀(𝑆), the measure 𝜇_𝑓∈𝑀(𝑆) is defined by𝑔𝑑𝜇_𝑓=𝑔𝑓𝑑𝜇,∀𝑔∈𝐶₀(𝑆). (1.3)(b) For any 𝜇∈𝑀(𝑆) and 𝐹∈𝑀(𝑆)^∗, the measure 𝐹×𝜇∈𝑀(𝑆) is defined by𝑓𝑑(𝐹×𝜇)=𝐹(𝜇_𝑓),∀𝑓∈𝐶₀(𝑆). (1.4)(c) For any 𝐹,𝐺∈𝑀(𝑆)^∗, the functional 𝐹×𝐺∈𝑀(𝑆)^∗ is defined by(𝐹×𝐺)(𝜇)=𝐹(𝐺×𝜇),∀𝜇∈𝑀(𝑆). (1.5) We can conclude that 𝑀(𝑆)^∗ becomes a commutative Banach algebra with an identity [2].

For the topological semigroup 𝑆, we define(𝑟_𝑠𝑓)(𝑡)=𝑓(𝑡𝑠),(𝑙_𝑠𝑓)(𝑡)=𝑓(𝑠𝑡), (1.6)where 𝑠∈𝑆,𝑡∈𝑆, and 𝑓∈𝑚(𝑆). Hence, 𝑟_𝑠 and 𝑙_𝑠 are the operators defined on 𝑚(𝑆) onto 𝑚(𝑆). A subset 𝑋 of 𝑚(𝑆) is called a left (right) translation invariant, if 𝑙_𝑠𝑋⊆𝑋(𝑟_𝑠𝑋⊆𝑋) for all 𝑠∈𝑆. It is well known that both BM(𝑆) and CB(𝑆) are left and right translation invariants [1]. Let 𝑆 be a topological semigroup, which for each compact subset 𝐸 of 𝑆 and 𝑎∈𝑆,𝑎⁻¹𝐸={𝑠∈𝑆∶𝑎𝑠∈𝐸},(𝐸𝑎⁻¹={𝑠∈𝑆∶𝑠𝑎∈𝐸}) (1.7)is compact. Then, 𝐶₀(𝑆) is a left (right) translation invariant.

Let 𝑋 be a left (right) translation invariant subspace of 𝑚(𝑆) containing the constant function 1. A mean 𝑚 on 𝑋 is called a left (right) invariant, if𝑚(𝑙_𝑠𝑓)=𝑚(𝑓)(𝑚(𝑟_𝑠𝑓)=𝑚(𝑓)), (1.8)for every 𝑠∈𝑆 and 𝑓∈𝑋 [1]. If 𝑚(𝑆) has a left invariant mean, then 𝑆 is said to be left amenable [3]. If 𝑚(𝑆) has a multiplicative left-invariant mean, then 𝑆 is said to be extremely left amenable (see [4, 5] for more details).

Suppose that 𝑆 is a locally compact semigroup, then for 𝜇,𝜈∈𝑀(𝑆), the convolution 𝜇∗𝜈 is defined by𝑓𝑑𝜇∗𝜈=𝑓(𝑠𝑡)𝑑𝜇(𝑠)𝑑𝜈(𝑡)=𝑓(𝑠𝑡)𝑑𝜈(𝑡)𝑑𝜇(𝑠), (1.9)where 𝑓∈𝐶₀(𝑆). Hence, 𝑀(𝑆) with a convolution 𝜇∗𝜈 as multiplication is a Banach algebra.

Now, for 𝐹∈𝑀(𝑆)^∗ and 𝜇∈𝑀(𝑆), the linear functional 𝑙_𝜇∶𝑀(𝑆)^∗→𝑀(𝑆)^∗ is defined by(𝑙_𝜇𝐹)(𝜈)=𝐹(𝜇∗𝜈),∀𝜈∈𝑀(𝑆). (1.10)We denote 𝑙_𝜇𝐹 by 𝜇⊙𝐹. Similarly, the 𝑟_𝜇𝐹=𝐹⊙𝜇 is defined by(𝑟_𝜇𝐹)(𝜈)=(𝐹⊙𝜇)(𝜈)=𝐹(𝜈∗𝜇),∀𝜈∈𝑀(𝑆). (1.11)

A mean 𝑀 on 𝑀(𝑆)^∗ is called left invariant (LIM) if 𝑀(𝑙_𝜀𝑎𝐹)=𝑀(𝜀_𝑎⊙𝐹)=𝑀(𝐹) for all 𝐹∈𝑀(𝑆)^∗ and for all 𝑎∈𝑆. Also, a mean 𝑀 on 𝑀(𝑆)^∗ is called topological left invariant (TLIM) if 𝑀(𝑙_𝜇𝐹)=𝑀(𝜇⊙𝐹)=𝑀(𝐹) for all 𝐹∈𝑀(𝑆)^∗ and for all 𝜇∈𝑀₀(𝑆). A topological left invariant mean 𝑀 on 𝑀(𝑆)^∗ is called a multiplicative topological left-invariant mean (MTLIM) if 𝑀(𝐹×𝐺)=𝑀(𝐹)𝑀(𝐺) for all 𝐹,𝐺∈𝑀(𝑆)^∗. If there is an MTLIM on 𝑀(𝑆)^∗, we say that 𝑆 is extremely topological left amenable (ETLA). For results concerning ETLA and ELA semigroups, see [1, 6].

In this paper, we demonstrate that the Arens product and multiplication on 𝑀(𝑆)^∗ defined by (1.3), (1.4), and (1.5) are associative (see Lemma 2.4). In [7], it is proved that(𝐹×𝐺)⊙𝜇=(𝐹⊙𝜇)×(𝐺⊙𝜇) (1.12)is valid, for all 𝜇∈𝑀₀(𝑆) and 𝐹,𝐺∈𝑀(𝑆)^∗. This means that Arens product ⊙ distributes over multiplication × on 𝑀(𝑆)^∗ from right, for all 𝜇∈𝑀₀(𝑆) and 𝐹,𝐺∈𝑀(𝑆)^∗. Note that the multiplication in 𝑀(𝑆)^∗ is different from the point-wise multiplication in 𝐶₀(𝑆). We show that Arens product ⊙ distributes over multiplication × on 𝑀(𝑆)^∗ from left, when 𝐹,𝐺∈𝑀(𝑆)^∗, and 𝜇 is a Dirac measure (see Lemma 2.2(ii)). Also, it is shown that if 𝑆 is a locally compact semigroup with a right identity and Arens product ⊙ distributes from left over multiplication ×, then 𝜇 must be a Dirac measure (see Theorem 3.1). In the rest of this paper, we give some characterizations on ETLA of locally compact semigroups.

2. Preliminaries

In this section, we offer some results which are useful in the sequel. For more details, refer to [1, 2, 8]. Let |𝜇| be the total variation of 𝜇, where 𝜇∈𝑀(𝑆) and 𝐿_∞(𝑆,|𝜇|)=BM(𝑆)/𝑁(𝜇) are the quotient Banach algebra with a quotient norm ‖⋅‖_𝜇,∞ ; 𝑁(𝜇) is the closed ideal of BM(𝑆) consisting of all locally |𝜇|-null functions. Consider the product linear space{𝐿_∞(𝑆,|𝜇|)∶𝜇∈𝑀(𝑆)}. (2.1)

An element 𝑓={𝑓_𝜇}_{𝜇∈𝑀(𝑆)} is called a generalized function on 𝑆, if the following conditions are satisfied:(a)‖𝑓‖=sup{‖𝑓_𝜇‖_𝜇,∞∶𝜇∈𝑀(𝑆)}<∞,(b)for 𝜇,𝜈 in 𝑀(𝑆) with 𝜇≪𝜈, we have 𝑓_𝜇=𝑓_𝜈|𝜇|-a.e. Let GL(𝑆) be the linear space of all generalized functions on 𝑆. It is known that GL(𝑆) is a Banach space with the norm defined by the formula (a) and that GL(𝑆)≅𝑀(𝑆)^∗ via the isometric Banach space isomorphism, 𝜑∶GL(𝑆)→𝑀(𝑆)^∗, where (𝜑𝑓)(𝜇)=∫𝑓_𝜇𝑑𝜇 for any 𝜇 in 𝑀(𝑆) and 𝑓 in GL(𝑆) (see [8, 9]). A function 𝑓 in BM(𝑆) can be treated as an element in GL(𝑆) with 𝑓_𝜇=𝑓 for all 𝜇 in 𝑀(𝑆). The space BM(𝑆) is thus a subspace of GL(𝑆).

For 𝑓∈BM(𝑆),𝜇∈𝑀(𝑆), and 𝜈∈𝑀(𝑆), we define 𝑓⊙𝜇 and 𝜇⊙𝑓 in 𝐿_∞(𝑆,|𝜈|) by𝑓⊙𝜇(𝑠)=𝑓(𝑠𝑡)𝑑𝜇(𝑡)=𝑙_𝑠𝑓𝑑𝜇|𝜈|-a.e.,𝜇⊙𝑓(𝑠)=𝑓(𝑡𝑠)𝑑𝜇(𝑡)=𝑟_𝑠𝑓𝑑𝜇|𝜈|-a.e. (2.2)

It is shown that 𝑓⊙𝜇,𝜇⊙𝑓∈GL(𝑆) [8]. If 𝑓∈CB(𝑆), then the above equalities hold everywhere, and 𝑓⊙𝜇 and 𝜇⊙𝑓 are in CB(𝑆). Also, if 𝑓∈BM(𝑆) and 𝑎∈𝑆, then𝑓⊙𝜀_𝑎(𝑠)=𝑓(𝑠𝑡)𝑑𝜀_𝑎(𝑡)=𝑓(𝑠𝑎)=(𝑟_𝑎𝜀𝑓)(𝑠),_𝑎⊙𝑓(𝑠)=𝑓(𝑡𝑠)𝑑𝜀_𝑎(𝑡)=𝑓(𝑎𝑠)=(𝑙_𝑎𝑓)(𝑠). (2.3)(2.4)Hence, 𝑓⊙𝜀_𝑎 and 𝜀_𝑎⊙𝑓 belong to BM(𝑆).

Lemma 2.1. The map 𝜑∶GL(𝑆)→𝑀(𝑆)^∗ defined by (𝜑𝑓)(𝜇)=∫𝑓_𝜇𝑑𝜇 for any 𝜇 in 𝑀(𝑆) and 𝑓 in GL(𝑆) satisfies the following statements.(i) For any 𝑓∈BM(𝑆) and 𝜇∈𝑀(𝑆),𝜑(𝜇⊙𝑓)=𝜇⊙𝜑𝑓,𝜑(𝑓⊙𝜇)=𝜑𝑓⊙𝜇. (2.5)(ii) For any 𝑎∈𝑆 and 𝑓∈BM(𝑆),𝜑(𝑙_𝑎𝑓)=𝜀_𝑎⊙𝜑𝑓,𝜑(𝑟_𝑎𝑓)=𝜑𝑓⊙𝜀_𝑎. (2.6)

Proof. We have (𝜇⊙𝑓)_𝜈=𝜇⊙𝑓_𝜇∗𝜈 for any 𝜈∈𝑀(𝑆) [8]. Hence,𝜑(𝜇⊙𝑓)(𝜈)=(𝜇⊙𝑓)_𝜈𝑑𝜈=(𝜇⊙𝑓_𝜇∗𝜈=𝑓)(𝑠)𝑑𝜈(𝑠)_𝜇∗𝜈=(𝑡𝑠)𝑑𝜇(𝑡)𝑑𝜈(𝑠)𝑓_𝜇∗𝜈𝑑𝜇∗𝜈=𝜑𝑓(𝜇∗𝜈)=(𝜇⊙𝜑𝑓)(𝜈). (2.7)Thus, 𝜑(𝜇⊙𝑓)=𝜇⊙𝜑𝑓. Similarly, 𝜑(𝑓⊙𝜇)=𝜑𝑓⊙𝜇. This proves (i). From (i) and (2.4), part of (ii) is trivial.

Lemma 2.2. For each 𝑎∈𝑆 and 𝐹,𝐺∈𝑀(𝑆)^∗, we have(i)(𝐹×𝐺)⊙𝜀_𝑎=(𝐹⊙𝜀_𝑎)×(𝐺⊙𝜀_𝑎),(ii)𝜀_𝑎⊙(𝐹×𝐺)=(𝜀_𝑎⊙𝐹)×(𝜀_𝑎⊙𝐺).

Proof. (i) For each 𝜇∈𝑀(𝑆), from (2.3), we have((𝐹×𝐺)⊙𝜀_𝑎)(𝜇)=𝑟_𝜀𝑎(𝐹×𝐺)(𝜇)=(𝐹×𝐺)(𝜇∗𝜀_𝑎)=𝐹(𝐺×(𝜇∗𝜀_𝑎)). (2.8)But, for 𝑔∈𝐶₀(𝑆), from (1.4) and (1.9), we have𝑔𝑑(𝐺×(𝜇∗𝜀_𝑎))=𝐺((𝜇∗𝜀_𝑎)_𝑔)(by(1.4))=𝐺(𝜇_𝑟𝑎𝑔∗𝜀_𝑎)=(𝐺⊙𝜀_𝑎)(𝜇_𝑟𝑎𝑔)=(𝑟_𝑎𝑔)(𝑦)𝑑((𝐺⊙𝜀_𝑎)×𝜇)(𝑦)(by(1.4))=𝑔(𝑦𝑎)𝑑((𝐺⊙𝜀_𝑎=)×𝜇)(𝑦)𝑔(𝑦𝑥)𝑑((𝐺⊙𝜀_𝑎)×𝜇)(𝑦)𝑑𝜀_𝑎=(𝑥)(by(1.9))𝑔𝑑(((𝐺⊙𝜀_𝑎)×𝜇)∗𝜀_𝑎). (2.9) Hence, by the Riesz representation theorem, 𝐺×(𝜇∗𝜀_𝑎)=((𝐺⊙𝜀_𝑎)×𝜇)∗𝜀_𝑎. Thus,((𝐹×𝐺)⊙𝜀_𝑎)(𝜇)=𝐹(𝐺×(𝜇∗𝜀_𝑎))=𝐹(((𝐺⊙𝜀_𝑎)×𝜇)∗𝜀_𝑎)=(𝐹⊙𝜀_𝑎)((𝐺⊙𝜀_𝑎)×𝜇)=((𝐹⊙𝜀_𝑎)×(𝐺⊙𝜀_𝑎))(𝜇). (2.10)Therefore, (𝐹×𝐺)⊙𝜀_𝑎=(𝐹⊙𝜀_𝑎)×(𝐺⊙𝜀_𝑎).
(ii) For each 𝜇∈𝑀(𝑆), equality (2.4) implies that(𝜀_𝑎⊙(𝐹×𝐺))=𝑙_𝜀𝑎(𝐹×𝐺)(𝜇)=(𝐹×𝐺)(𝜀_𝑎∗𝜇)=𝐹(𝐺×(𝜀_𝑎∗𝜇)). (2.11)Now, for 𝑔∈𝐶₀(𝑆), from (1.4) and (1.9), we obtain𝑔𝑑(𝐺×(𝜀_𝑎∗𝜇))=𝐺((𝜀_𝑎∗𝜇)_𝑔)=𝐺(𝜀_𝑎∗𝜇_𝑙𝑎𝑔)=(𝜀_𝑎⊙𝐺)(𝜇_𝑙𝑎𝑔)=(𝑙_𝑎𝑔)(𝑦)𝑑((𝜀_𝑎⊙𝐺)×𝜇)(𝑦)=𝑔(𝑎𝑦)𝑑((𝜀_𝑎=⊙𝐺)×𝜇)(𝑦)𝑔(𝑥𝑦)𝑑𝜀_𝑎(𝑥)𝑑((𝜀_𝑎=⊙𝐺)×𝜇)(𝑦)𝑔𝑑(((𝜀_𝑎⊙𝐺)×𝜇)∗𝜀_𝑎). (2.12) Hence, by the Riesz representation theorem, 𝐺×(𝜀_𝑎∗𝜇)=((𝜀_𝑎⊙𝐺)×𝜇)∗𝜀_𝑎. Thus(𝜀_𝑎⊙(𝐹×𝐺))(𝜇)=𝐹(𝐺×(𝜀_𝑎∗𝜇))=𝐹(((𝜀_𝑎⊙𝐺)×𝜇)∗𝜀_𝑎)=(𝜀_𝑎⊙𝐹)((𝜀_𝑎⊙𝐺)×𝜇)=((𝜀_𝑎⊙𝐹)×(𝜀_𝑎⊙𝐺))(𝜇). (2.13)Therefore, 𝜀_𝑎⊙(𝐹×𝐺)=(𝜀_𝑎⊙𝐹)×(𝜀_𝑎⊙𝐺).

Remark 2.3. (a) In the proof of Lemma 2.2, we use the equalities (𝜇∗𝜀_𝑎)_𝑔=𝜇_𝑟𝑎𝑔∗𝜀_𝑎 and (𝜀_𝑎∗𝜇)_𝑔=𝜀_𝑎∗𝜇_𝑙𝑎𝑔. For 𝑓∈𝐶₀(𝑆), we have 𝑓𝑑(𝜇∗𝜀_𝑎)_𝑔=𝑔𝑓𝑑(𝜇∗𝜀_𝑎)(by(1.3))=(𝑔𝑓)(𝑥𝑦)𝑑𝜇(𝑥)𝑑𝜀_𝑎===(𝑦)(by(1.9))(𝑔𝑓)(𝑥𝑎)𝑑𝜇(𝑥)𝑔(𝑥𝑎)𝑓(𝑥𝑎)𝑑𝜇(𝑥)(𝑟_𝑎𝑔)(𝑟_𝑎𝑓)𝑑𝜇(by(1.6))=(𝑟_𝑎𝑓)(𝑥)𝑑𝜇_𝑟𝑎𝑔=(𝑥)(by(1.3))𝑓(𝑥𝑎)𝑑𝜇_𝑟𝑎𝑔=(𝑥)(by(1.9))𝑓(𝑥𝑦)𝑑𝜇_𝑟𝑎𝑔(𝑥)𝑑𝜀_𝑎=(𝑦)(by(1.6))𝑓𝑑(𝜇_𝑟𝑎𝑔∗𝜀_𝑎). (2.14)Hence, by the Riesz representation theorem, (𝜇∗𝜀_𝑎)_𝑔=𝜇_𝑟𝑎𝑔∗𝜀_𝑎. Similarly, (𝜀_𝑎∗𝜇)_𝑔=𝜀_𝑎∗𝜇_𝑙𝑎𝑔.
(b) The statement (i) of Lemma 2.2 has a general form as replacing a Dirac measure by 𝜇∈𝑀₀(𝑆) [7]. It is natural to ask for which 𝜇 in 𝑀₀(𝑆), the equality 𝜇⊙(𝐹×𝐺)=(𝜇⊙𝐹)×(𝜇⊙𝐺),∀𝐹,𝐺∈𝑀(𝑆)^∗ (2.15) is valid?
Now, we demonstrate that the multiplication on 𝑀(𝑆)^∗ defined by (1.3), (1.4), and (1.5) is associative.

Lemma 2.4. The multiplication × defined by (1.3), (1.4), and (1.5) on 𝑀(𝑆)^∗ is associative.

Proof. We know that the Arens product ⊙ is associative [3, Lemma 1, page 527]. Let 𝜋∶𝐶₀(𝑆)^∗→𝑀(𝑆) be isometric order-preserving linear space isomorphism in [1, Theorem 14.10, page 170], namely, for any 𝑚∈𝐶₀(𝑆)^∗ and 𝑓∈𝐶₀(𝑆),𝑓𝑑𝜋(𝑚)=𝑚(𝑓). (2.16)Now, let 𝑓,𝑔∈𝐶₀(𝑆)^∗, 𝑚∈𝐶₀(𝑆)^∗, and 𝐹,𝐺∈𝑀(𝑆)^∗, then (1.3) implies that𝑔𝑑𝜋(𝑚)_𝑓==𝑓𝑔𝑑𝜋(𝑚)=𝑚(𝑓𝑔)=(𝑚⊙𝑓)(𝑔)𝑔𝑑𝜋(𝑚⊙𝑓). (2.17)Thus, 𝜋(𝑚)_𝑓=𝜋(𝑚⊙𝑓). Also, from (1.4), we have𝑓𝑑(𝐹×𝜋(𝑚))=𝐹(𝜋(𝑚)_𝑓)=𝐹(𝜋(𝑚⊙𝑓))=𝜋^∗(𝐹)(𝑚⊙𝑓)=(𝜋^∗=(𝐹)⊙𝑚)(𝑓)𝑓𝑑𝜋(𝜋^∗(𝐹)⊙𝑚). (2.18)Hence, 𝐹×𝜋(𝑚)=𝜋(𝜋^∗(𝐹)⊙𝑚). Also, from (1.5), we have(𝜋^∗(𝐹×𝐺))(𝑚)=(𝐹×𝐺)(𝜋(𝑚))=𝐹(𝐺×𝜋(𝑚))=𝐹(𝜋(𝜋^∗(𝐺)⊙𝑚))=(𝜋^∗(𝐹))(𝜋^∗(𝐺)⊙𝑚)=(𝜋^∗(𝐹)⊙𝜋^∗(𝐺))(𝑚). (2.19)Therefore, 𝜋^∗(𝐹×𝐺)=𝜋^∗(𝐹)⊙𝜋^∗(𝐺). Now, for any 𝐹,𝐺,𝐻∈𝑀(𝑆)^∗,𝜋^∗((𝐹×𝐺)×𝐻)=𝜋^∗((𝐹×𝐺)⊙𝜋^∗(𝐻))=(𝜋^∗(𝐹)⊙𝜋^∗(𝐺))⊙𝜋^∗(𝐻)=𝜋^∗(𝐹)⊙(𝜋^∗(𝐺)⊙𝜋^∗(𝐻))=𝜋^∗(𝐹)⊙(𝜋^∗(𝐺×𝐻))=𝜋^∗(𝐹×(𝐺×𝐻)). (2.20)So, (𝐹×𝐺)×𝐻=𝐹×(𝐺×𝐻), and thus the multiplication of × is associative.  

Remark 2.5. We note that one can go through a process analogous to Day's proof [3] and establish the associativity of × via the demonstration of the following identities one by one.(i) For any 𝜇∈𝑀(𝑆) and 𝑓,𝑔∈𝐶₀(𝑆),(𝜇_𝑓)_𝑔=𝜇_𝑓𝑔. (2.21)(ii) For any 𝐹∈𝑀(𝑆)^∗, 𝜇∈𝑀(𝑆) and 𝑓∈𝐶₀(𝑆),𝐹×(𝜇_𝑓)=(𝐹×𝜇)_𝑓. (2.22)(iii) For any 𝐹,𝐺∈𝑀(𝑆)^∗ and 𝜇∈𝑀(𝑆),(𝐹×𝐺)×𝜇=𝐹×(𝐺×𝜇). (2.23)(iv) For any 𝐹,𝐺,𝐻∈𝑀(𝑆)^∗,𝐹×(𝐺×𝐻)=(𝐹×𝐺)×𝐻. (2.24)

The proofs of (i), (ii), and (iii) use the Riesz representation theorem and the relations (1.3), (1.4), and (1.5). The proof of (iv) follows from (iii) using definition.

3. Main Results

Each probability measure 𝜇 in 𝑀₀(𝑆) is a mean on 𝑀(𝑆)^∗, if we put 𝜇(𝐹)=𝐹(𝜇) for any 𝐹 in 𝑋. We give a partial answer to the question: For which 𝜇∈𝑀₀(𝑆), is the equality 𝜇⊙(𝐹×𝐺)=(𝜇⊙𝐹)×(𝜇⊙𝐺),∀𝐹,𝐺∈𝑀(𝑆)^∗ (3.1) valid?

Let 𝑓∈GL(𝑆), from the isometric Banach space isomorphism 𝜑∶GL(𝑆)→𝑀(𝑆)^∗, we have 𝜑𝑓=𝐹 which is in 𝑀(𝑆)^∗, where 𝐹(𝜇)=(𝜑𝑓)(𝜇)=∫𝑓_𝜇𝑑𝜇, for any 𝜇 in 𝑀(𝑆). For 𝜇∈𝑀(𝑆) and 𝑔∈𝐶₀(𝑆), we have𝑔𝑑(𝐹×𝜇)=𝐹(𝜇_𝑔)=(𝜑𝑓)(𝜇_𝑔)=𝑓_𝜇𝑔𝑑𝜇_𝑔=𝑓_𝜇𝑑𝜇_𝑔=𝑓_𝜇=𝑔𝑑𝜇𝑔𝑑𝜇_𝑓𝜇. (3.2)Therefore, 𝐹×𝜇=𝜑𝑓×𝜇=𝜇_𝑓𝜇. In particular, 𝜑𝑓×𝜇≪𝜇 and so 1×𝜇₁=𝜇₁=𝜇. In view of (1.4), if 𝐹∈𝑀(𝑆)^∗ and 𝜇∈𝑀(𝑆), then(𝐹×𝜇)(𝑆)=𝑑(𝐹×𝜇)=𝐹(𝜇₁)=𝐹(𝜇). (3.3)Hence, if 𝐹≥0 and 𝜇≥0, then‖𝐹×𝜇‖=(𝐹×𝜇)(𝑆)=𝐹(𝜇). (3.4)Also, since ‖𝐺‖1−𝐺≥0, we have(‖𝐺‖1−𝐺)×𝜇≥0, (3.5)and hence, 𝐺×𝜇≤‖𝐺‖1×𝜇=‖𝐺‖𝜇 whenever 𝜇≥0.

Theorem 3.1. Let 𝑆 be a locally compact semigroup with a right identity and that 0<𝜇∈𝑀(𝑆). If 𝜇⊙(𝐹×𝐺)=(𝜇⊙𝐹)×(𝜇⊙𝐺) for any 𝐹,𝐺∈𝜑(𝐶₀(𝑆)), then 𝜇 is a Dirac measure.

Proof. For 𝑓,𝑔∈𝐶₀(𝑆), we have𝜑(𝜇⊙(𝑓𝑔))=𝜇⊙𝜑(𝑓𝑔)(byLemma2.1(i))=𝜇⊙(𝜑𝑓×𝜑𝑔)=(𝜇⊙𝜑𝑓)×(𝜇⊙𝜑𝑔)=𝜑(𝜇⊙𝑓)×𝜑(𝜇⊙𝑔)(byLemma2.1(i))=𝜑((𝜇⊙𝑓)(𝜇⊙𝑔)). (3.6)Thus, for any 𝑓,𝑔∈𝐶₀(𝑆), from (3.6), we have𝜇⊙(𝑓𝑔)=(𝜇⊙𝑓)(𝜇⊙𝑔). (3.7)Now, let 𝑒_𝑟 be a right identity of 𝑆, that is, 𝑠𝑒_𝑟=𝑠 for any 𝑠∈𝑆, then for any 𝑓∈𝐶₀(𝑆), we have(𝜇⊙𝑓)(𝑒_𝑟)=𝑓(𝑡𝑒_𝑟)𝑑𝜇(𝑡)=𝑓(𝑡)𝑑𝜇(𝑡). (3.8)Hence, for each 𝑓,𝑔∈𝐶₀(𝑆),𝑓𝑔𝑑𝜇=(𝜇⊙(𝑓𝑔))(𝑒_𝑟)=(𝜇⊙𝑓)(𝑒_𝑟)(𝜇⊙𝑔)(𝑒_𝑟)=(𝑓𝑑𝜇)(𝑔𝑑𝜇). (3.9)In (3.9), we put 𝑓=𝑔, then for any 𝑓∈𝐶₀(𝑆),𝑓²𝑑𝜇=(𝑓𝑑𝜇)², (3.10)so for each 𝑓,𝑔∈𝐶₀(𝑆),(𝑓𝑔𝑑𝜇)²=[(𝑓𝑑𝜇)(𝑔𝑑𝜇)]²=(𝑓𝑑𝜇)²(𝑔𝑑𝜇)²=(𝑓²𝑑𝜇)(𝑔²𝑑𝜇), (3.11)and by Holder's inequality, there exist real numbers 𝛼 and 𝛽, not being zero, such that𝛼𝑓²=𝛽𝑔²a.e.(𝜇). (3.12)
Now, if 𝐴 and 𝐵 are the disjoint compact subsets of 𝑆 with 𝜇(𝐴)>0 and 𝜇(𝐵)>0, by the Urysohn's lemma, there exist 𝑓 and 𝑔 in 𝐶₀₀(𝑆) such that𝑓(𝐴)=0=𝑔(𝐵),𝑓(𝐵)=1=𝑔(𝐴). (3.13)But from (3.12) and 𝜇(𝐴)>0, there is 𝑥₀∈𝐴 such that𝛼𝑓(𝑥₀)²=𝛽𝑔(𝑥₀)². (3.14)So, 0=𝛽𝑔(𝑥₀)²=𝛽. Also, 𝜇(𝐵)>0 follows that there is 𝑦₀∈𝐵, such that𝛼𝑓(𝑦₀)²=𝛽𝑔(𝑦₀)², (3.15)and therefore 0=𝛼𝑓(𝑦₀)²=𝛼. This contradicts the fact that 𝛼 and 𝛽 are not both zero. Hence, if 𝐴 is a compact subset of 𝑆 with 𝜇(𝐴)>0 and 𝐵 is another compact subset of 𝑆 disjointed from 𝐴, then we must have 𝜇(𝐵)=0. Therefore, 𝜇(𝐴^𝑐)=0, that is, 𝜇(𝐴)=𝜇(𝑆). This proves that if 𝐴 is a compact subset 𝑆, then either 𝜇=0 or 𝜇(𝐴)=𝜇(𝑆).
Now, the regularity of 𝜇 follows that for each Borel subset 𝐵 of 𝑆, 𝜇(𝐵)=0 or 𝜇(𝐵)=𝜇(𝑆). Hence, either 𝜇=0 or the measure 𝜇/𝜇(𝑆) is a Dirac measure, say 𝜇/𝜇(𝑆)=𝜀_𝑎 [2].
Now, put 𝜇=𝜇(𝑆)𝜀_𝑎, we have𝜇(𝑆)=𝑑𝜇=1²𝑑𝜇=(1𝑑𝜇)²=(𝑑𝜇)²=𝜇(𝑆)², (3.16)and so 𝜇(𝑆)=1. Thus, 𝜇=𝜀_𝑎.  

Remark 3.2. If 𝑆 is a discrete semigroup, then 𝐶₀(𝑆)^∗=ℓ₁(𝑆), and so 𝑀₀(𝑆)^∗=𝐶₀(𝑆)^∗∗=ℓ₁(𝑆)^∗=𝑚(𝑆) [1]. In this case, the multiplication on 𝑀(𝑆)^∗ is just the point-wise multiplication as in 𝑚(𝑆). Let 𝑒 be the right identity of 𝑆, then(𝜇⊙(𝐹×𝐺))(𝜀_𝑒)=(𝐹×𝐺)(𝜇∗𝜀_𝑒)=(𝐹×𝐺)(𝜇)=𝜇(𝐹×𝐺),((𝜇⊙𝐹)×(𝜇⊙𝐺))(𝜀_𝑒)=(𝜇⊙𝐹)(𝜀_𝑒)(𝜇⊙𝐺)(𝜀_𝑒)=𝜇(𝐹)𝜇(𝐺), (3.17)since both 𝜇⊙𝐹 and 𝜇⊙𝐺 are in 𝑀(𝑆)^∗=𝑚(𝑆). Hence, 𝜇∈𝑀(𝑆) is multiplicative, if the condition of Theorem 3.1 is satisfied. Therefore, 𝜇 must be either 0 or a Dirac measure. But, when 𝑆 is a topological semigroup, the multiplication in 𝑀(𝑆)^∗ defined by (1.3), (1.4), and (1.5) is just a generic Arens product in the second dual of a Banach algebra, which is different from the point-wise multiplication.

It is known that 𝑀₀(𝑆) is weak∗ dense in the set of all means on 𝑀(𝑆)^∗. We give some characterizations theorems for the extreme amenability of locally compact semigroup.

Lemma 3.3. Let 𝑀(𝑆)^∗ be TLA. The following statements are equivalent:(i)𝑀(𝑆)^∗ is ETLA,(ii) for every 𝐹∈𝑀(𝑆)^∗ and 𝜇∈𝑀₀(𝑆), there exists a mean 𝑀 on 𝑀(𝑆)^∗ such that 𝑀(𝐹×𝐹)=𝑀(𝜇⊙𝐹)²,(iii) for every 𝐹∈𝑀(𝑆)^∗ and 𝜇∈𝑀₀(𝑆), there exists a mean 𝑀 on 𝑀(𝑆)^∗ such that 𝑀(𝐹×𝐹)=𝑀(𝐹)𝑀(𝜇⊙𝐹),(iv) for every 𝐹∈𝑀(𝑆)^∗ and 𝜇∈𝑀₀(𝑆), there exists a mean 𝑀 on 𝑀(𝑆)^∗ such that 𝑀(𝜇⊙(𝐹×𝐹))=𝑀(𝐹)².

Proof. (i)⇒(ii)⇒(iii)⇒(iv) are obvious.
(iv)⇒(i). Suppose that 𝐹,𝐺∈𝑀(𝑆)^∗ and 𝜇∈𝑀₀(𝑆). For 𝐹+𝐺 by (iv), there exists a mean 𝑀 on 𝑀(𝑆)^∗ such that𝑀(𝜇⊙((𝐹+𝐺)×(𝐹+𝐺))=𝑀(𝐹+𝐺)²)=𝑀(𝐹)²+2𝑀(𝐹)𝑀(𝐺)+𝑀(𝐺)², (3.18)and to expand the right-hand side, we get𝑀(𝜇⊙(𝐹×𝐺))=𝑀(𝐹)𝑀(𝐺). (3.19)Since 𝑀(𝑆)^∗ is topological left invariant, hence, 𝑀(𝐹×𝐺)=𝑀(𝐹)𝑀(𝐺). Therefore, 𝑀(𝑆)^∗ is ETLA.  

Theorem 3.4. Let 𝑀 be a topological left invariant mean on 𝑀(𝑆)^∗. The following statements are equivalent:(i)𝑀 is a multiplicative,(ii) there exists a net {𝜇_𝛼} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐺 in 𝑀(𝑆)^∗,𝑤^∗−lim_𝛼(𝐺×(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝜇_𝛼)=0. (3.20)

Proof. (i)⇒(ii). Let 𝑀 be a multiplicative topological left invariant mean on 𝑀(𝑆)^∗. By Lemma 3.3, for any 𝐹∈𝑀(𝑆)^∗ and 𝜇∈𝑀₀(𝑆),𝑀(𝜇⊙(𝐹×𝐹))=𝑀(𝐹)². (3.21) Let 𝐹,𝐺∈𝑀(𝑆)^∗, then𝑀(𝜇⊙((𝐹+𝐺)×(𝐹+𝐺)))=𝑀(𝐹+𝐺)². (3.22)We have𝑀(𝜇⊙(𝐹×𝐹+2𝐹×𝐺+𝐺×𝐺))=(𝑀(𝐹)+𝑀(𝐺))². (3.23)So,𝑀(𝜇⊙(𝐹×𝐹))+2𝑀(𝜇⊙(𝐹×𝐺))+𝑀(𝜇(𝐺×𝐺))=𝑀(𝐹)²+𝑀(𝐺)²+2𝑀(𝐹)𝑀(𝐺). (3.24)Thus,𝑀(𝜇⊙(𝐹×𝐺))=𝑀(𝐹)𝑀(𝐺). (3.25)
Note that we apply the commutativity of × in 𝑀(𝑆)^∗. Since 𝑀 is a mean on 𝑀(𝑆)^∗ and 𝑀₀(𝑆) is weak∗ dense in the set of all means on 𝑀(𝑆)^∗, hence, there exists a net {𝜇_𝛼} in 𝑀₀(𝑆) such that 𝑀=𝑤^∗−lim_𝛼𝜇_𝛼 in 𝑀(𝑆)^∗∗. Now for 𝐹∈𝑀(𝑆)^∗,𝑤^∗−lim_𝛼𝐹(𝐺×(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝜇_𝛼)=𝑤^∗−lim_𝛼(𝜇_𝛼(𝜇⊙(𝐹×𝐺))−𝑀(𝐺)𝜇_𝛼(𝐹))=𝑀(𝜇⊙(𝐹×𝐺))−𝑀(𝐹)𝑀(𝐺)=0. (3.26)Thus,𝐹(𝑤^∗−lim_𝛼(𝐺×(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝜇_𝛼))=0, (3.27)that is,𝑤^∗−lim_𝛼(𝐺×(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝜇_𝛼)=0. (3.28)
(ii)⇒(i). Since 𝑀 is a topological left invariant mean on 𝑀(𝑆)^∗, there exists a net {𝜇_𝛼} in 𝑀₀(𝑆) such that 𝑀=𝑤^∗−lim_𝛼𝜇_𝛼 in 𝑀(𝑆)^∗∗. If 𝜇∈𝑀₀(𝑆),𝑀(𝐹×𝐺)−𝑀(𝐹)𝑀(𝐺)=𝑀(𝜇⊙(𝐹×𝐺))−𝑀(𝐹)𝑀(𝐺)=𝑤^∗−lim_𝛼𝜇_𝛼(𝜇⊙(𝐹×𝐺))−𝑀(𝐺)(𝑤^∗−lim_𝛼𝜇_𝛼(𝐹))=𝑤^∗−lim_𝛼((𝐹×𝐺)(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝐹(𝜇_𝛼))=𝑤^∗−lim_𝛼(𝐹(𝐺×(𝜇∗𝜇_𝛼))−𝐹(𝑀(𝐺)𝜇_𝛼))=𝑤^∗−lim_𝛼𝐹(𝐺×(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝜇_𝛼)=𝐹(𝑤^∗−lim_𝛼(𝐺×(𝜇∗𝜇_𝛼)−𝑀(𝐺)𝜇_𝛼))=𝐹(0)=0. (3.29)Therefore, 𝑀(𝐹×𝐺)=𝑀(𝐹)𝑀(𝐺), that is, 𝑀(𝑆)^∗ is extremely topological left amenable (ETLA).

Lemma 3.5. If 𝑀 is a multiplicative topological left invariant mean on 𝑀(𝑆)^∗, then there is a net {𝜇_𝛽} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐹 in 𝑀(𝑆)^∗,lim_𝛽‖𝐹×(𝜇∗𝜇_𝛽)−𝑀(𝐹)𝜇_𝛽‖=0. (3.30)

Proof. We consider 𝑀(𝑆) with the norm topology. Let 𝒫=𝑀(𝑆)^{𝑀(𝑆)∗×𝑀0(𝑆)} with the product of the norm topologies, where 𝑀(𝑆)^∗×𝑀₀(𝑆) is the set theoretic cartesian product. Then, 𝒫 is a locally convex topological vector space [10]. Now, by Theorem 3.4 corresponding to 𝑀, there exists a net {𝜇_𝛼} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐹 in 𝑀(𝑆)^∗,𝑤^∗−lim_𝛼(𝐹×(𝜇∗𝜇_𝛼)−𝑀(𝐹)𝜇_𝛼)=0. (3.31)We define a linear map 𝐏∶𝑀(𝑆)→𝒫 by𝐏(𝜈)(𝐹,𝜇)=𝐹×(𝜇∗𝜈)−𝑀(𝐹)𝜈, (3.32)for all (𝐹,𝜇)∈𝑀(𝑆)^∗×𝑀₀(𝑆). Hence,𝑤^∗−lim_𝛼𝐏(𝜇_𝛼)(𝐹,𝜇)=0, (3.33)that is, 𝐏(𝜇_𝛼)→0 in the product of weak topologies [10]. Therefore, 0 lies in the weak closure of the convex set 𝐏(𝑀₀(𝑆)), and so is in the closure of 𝐏(𝑀₀(𝑆)) in the original topology of 𝒫. So, there is a net {𝜇_𝛽} in 𝑀₀(𝑆) such that for all (𝐹,𝜇)∈𝑀(𝑆)^∗×𝑀₀(𝑆),lim_𝛽‖𝐏(𝜇_𝛽)(𝐹,𝜇)‖=0, (3.34)that is,lim_𝛽‖𝐹×(𝜇∗𝜇_𝛽)−𝑀(𝐹)𝜇_𝛽‖=0, (3.35)and the proof is complete.

Theorem 3.6. Let 𝑆 be a locally compact semigroup. The following statements are equivalent:(i)𝑀(𝑆)^∗ is extremely topological left amenable,(ii) there exists a net {𝜇_𝛽} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐹 in 𝑀(𝑆)^∗,lim_𝛽‖𝜇∗𝜇_𝛽−𝜇_𝛽‖=0,lim_𝛽‖𝐹×𝜇_𝛽−𝐹(𝜇_𝛽)𝜇_𝛽‖=0, (3.36)(iii) there exists a net {𝜇_𝛾} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐹 in 𝑀(𝑆)^∗,𝑤^∗−lim_𝛾(𝜇∗𝜇_𝛾−𝜇_𝛾𝑤)=0,_∗−lim_𝛾(𝐹×𝜇_𝛾−𝐹(𝜇_𝛾)𝜇_𝛾)=0. (3.37)(3.38)

Proof. (i)⇒(ii). Let 𝑀 be a multiplicative left invariant mean on 𝑀(𝑆)^∗. Theorem 3.4 implies that there exists a net {𝜇_𝛼} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐹 in 𝑀(𝑆)^∗,𝑤^∗−lim_𝛼(𝐹×(𝜇∗𝜇_𝛼)−𝑀(𝐹)𝜇_𝛼)=0. (3.39)By Lemma 3.5, there exists a net {𝜇_𝛽} in 𝑀₀(𝑆) such that for any 𝜇 in 𝑀₀(𝑆) and 𝐹 in 𝑀(𝑆)^∗,lim_𝛽‖𝐹×(𝜇∗𝜇_𝛽)−𝑀(𝐹)𝜇_𝛽‖=0. (3.40)Without the loss of generality, we may assume that 𝜇_𝛽→𝑀₁𝜎(𝑀(𝑆)^∗∗,𝑀(𝑆)^∗) for some mean 𝑀₁ in 𝑀(𝑆)^∗∗. Therefore, for any 𝐹,𝐺 in 𝑀(𝑆)^∗ and 𝜇 in 𝑀₀(𝑆), we have𝑀₁(𝜇⊙(𝐺×𝐹))−𝑀(𝐹)𝑀₁(𝐺)=lim_𝛽{(𝜇⊙(𝐺×𝐹))(𝜇_𝛽)−𝑀(𝐹)𝑀₁(𝐺)}=lim_𝛽[𝐺(𝐹×(𝜇∗𝜇_𝛽))−𝐺(𝑀(𝐹)𝜇_𝛽)]=lim_𝛽𝐺(𝐹×(𝜇∗𝜇_𝛽)−𝑀(𝐹)𝜇_𝛽)=0. (3.41)In (3.41), we put 𝐹=1=𝐺, then𝑀₁(𝜇⊙(1×1))=𝑀(1)𝑀₁𝑀(1),₁(𝜇⊙1)=𝑀(1)𝑀₁(1),𝑀(1)=1. (3.42) Also, for 𝐹=1 and 𝐺 in 𝑀(𝑆)^∗,𝑀₁(𝜇⊙𝐺)=𝑀₁(𝐺), (3.43)and for 𝐺=1 and 𝐹 in 𝑀(𝑆)^∗,𝑀₁(𝜇⊙𝐹)=𝑀₁(𝐹). (3.44)Therefore, for any 𝐹 in 𝑀(𝑆)^∗, we have𝑀(𝐹)=𝑀₁(𝜇⊙𝐹)=𝑀₁(𝐹). (3.45)Now, from (3.30) for 𝐹=1, we getlim_𝛽‖𝜇∗𝜇_𝛽−𝜇_𝛽‖=0. (3.46)Also, let 𝐹 in 𝑀(𝑆)^∗ and 𝜀>0 be given. Since𝑀(𝐹)=𝑀₁(𝐹),𝐹(𝜇_𝛽)→𝑀(𝐹),‖𝜇_𝛽‖≤1, (3.47)it follows from (3.30) that for any 𝜇 in 𝑀₀(𝑆),lim_𝛽‖𝐹×(𝜇∗𝜇_𝛽)−𝑀₁(𝐹)𝜇_𝛽‖=0. (3.48)Now fix an arbitrary 𝜇∈𝑀₀(𝑆). This together with (3.46) implies that there exists a 𝛽₀ such that‖𝜇∗𝜇_𝛽−𝜇_𝛽𝜀‖<3(‖𝐹‖+1),∀𝛽⪰𝛽₀,‖𝐹×(𝜇∗𝜇_𝛽)−𝑀₁(𝐹)𝜇_𝛽𝜀‖<3. (3.49) Also, we may assume that|𝐹(𝜇_𝛽)−𝑀₁𝜀(𝐹)|<3. (3.50)Consequently,‖𝐹×𝜇_𝛽−𝐹(𝜇_𝛽)𝜇_𝛽‖≤‖𝐹×𝜇_𝛽−𝐹×(𝜇∗𝜇_𝛽)‖+‖𝐹×(𝜇∗𝜇_𝛽)−𝑀₁(𝐹)𝜇_𝛽‖+‖𝑀₁(𝐹)𝜇_𝛽−𝐹(𝜇_𝛽)𝜇_𝛽‖≤‖𝐹‖‖𝜇∗𝜇_𝛽−𝜇_𝛽‖+‖𝐹×(𝜇∗𝜇_𝛽)−𝑀₁(𝐹)𝜇_𝛽‖‖𝜇_𝛽‖|𝑀₁(𝐹)−𝐹(𝜇_𝛽<𝜀)|3+𝜀3+𝜀3=𝜀,∀𝛽⪰𝛽₀. (3.51) Obviously, (ii)⇒(iii).
(iii)⇒(i). Since 𝑀₀(𝑆) is weak∗ dense in the set of all means on 𝑀(𝑆)^∗, by passing to a subnet if necessary, we may assume that 𝜇_𝛾→𝑀 weakly∗ in 𝑀(𝑆)^∗∗ for some mean 𝑀. Thus, the assertion of (3.37) implies that 𝑀 is a topological left invariant mean. Also, (3.38) implies that 𝑀 is multiplicative because for any 𝐹,𝐺 in 𝑀(𝑆)^∗ and 𝜇 in 𝑀₀(𝑆),𝑀(𝐺×𝐹)−𝑀(𝐺)𝑀(𝐹)=𝑤^∗−lim_𝛾{(𝐺×𝐹)(𝜇_𝛾)−𝐺(𝜇_𝛾)𝐹(𝜇_𝛾)}=𝑤^∗−lim_𝛾{𝐺(𝐹×𝜇_𝛾)−𝐺(𝐹(𝜇_𝛾)𝜇_𝛾)}=𝑤^∗−lim_𝛾𝐺(𝐹×𝜇_𝛾−𝐹(𝜇_𝛾)𝜇_𝛾)=𝐺(0)=0. (3.52)Therefore, 𝑀(𝐺×𝐹)=𝑀(𝐺)𝑀(𝐹), that is, 𝑀(𝑆)^∗ is extremely left amenable.

Remark 3.7. The conclusions of Theorem 3.6 are different from the classical characterizations of extremely left amenable discrete semigroups [4, Theorem 2]. This difference in the two situations is that any multiplicative mean on 𝑚(𝑆) is the weak∗ limit of evaluation functionals, while taking weak∗ limits of all convergent nets of Dirac measures in 𝑀(𝑆)^∗ does not exhaust all multiplicative means on 𝑀(𝑆)^∗ [2, Theorem 2.7].

Theorem 3.8. Let 𝑆 be a locally compact semigroup. Define a function 𝐓∶𝑀(𝑆)^∗→𝑚(𝑆) by(𝐓(𝐹))(𝑎)=𝐹⊙𝜀_𝑎,∀𝑎∈𝑆. (3.53)Then,(i)𝐓 is bounded and linear,(ii)𝐓(1)=1,(iii)𝐓(𝐹)≥0 whenever 𝐹≥0,(iv)𝐓(𝐹×𝐺)=𝐓(𝐹)𝐓(𝐺) for all 𝐹 and 𝐺 in 𝑀(𝑆)^∗,(v)𝑙_𝑏(𝐓(𝐹))=𝐓(𝜀_𝑏⊙𝐹) for all 𝑏∈𝑆 and 𝐹∈𝑀(𝑆)^∗. 

Proof. (i),(ii), and (iii) are obvious.
(iv) For any 𝑎∈𝑆, we have(𝐓(𝐹×𝐺))(𝑎)=(𝐹×𝐺)⊙𝜀_𝑎=(𝐹⊙𝜀_𝑎)×(𝐺⊙𝜀_𝑎)(byLemma2.2(i))=𝐓(𝐹)(𝑎)×𝐓(𝐺)(𝑎)=(𝐓(𝐹)×𝐓(𝐺))(𝑎)=(𝐓(𝐹)𝐓(𝐺))(𝑎). (3.54)In the final equality, we use the fact that multiplication in 𝑚(𝑆) is a point-wise multiplication, see Remark 3.2 of Theorem 3.1.
(v) Let 𝑎∈𝑆 and 𝐹∈𝑀(𝑆)^∗, then𝑙_𝑏(𝐓(𝐹))(𝑎)=(𝐓(𝐹))(𝑏𝑎)=𝐹⊙𝜀_𝑏𝑎=𝐹(𝜀_𝑏∗𝜀_𝑎)=(𝜀_𝑏⊙𝐹)(𝜀_𝑎)=(𝐓(𝜀_𝑏⊙𝐹))(𝑎). (3.55)So, 𝑙_𝑏(𝐓(𝐹))=𝐓(𝜀_𝑏⊙𝐹).  

Remark 3.9. From Theorem 3.8, it follows that the map 𝐓^∗∶𝑚(𝑆)^∗→𝑀(𝑆)^∗∗ carries means to means, multiplicative means to multiplicative means, left invariant means to left invariant means, and multiplicative left invariant means to multiplicative left invariant means. But 𝐓^∗ does not carry a type of means in 𝑚(𝑆)^∗ onto the same type of means in 𝑀(𝑆)^∗∗. Indeed, if 𝑀 is a multiplicative topological left invariant mean which is not weak∗ limit of all convergent nets of Dirac measures in 𝑀(𝑆)^∗, then 𝑀 does not belong to 𝐓^∗(ℳ), where ℳ is the set of all means on 𝑚(𝑆).