THICKNESS IN TOPOLOGICAL TRANSFORMATION SEMIGROUPS

This article deals with thickness in topological transformation semigroups (x-semigroups). Thickness is used to establish conditions guaranteeing an invariant mean on a function space defined on a x-semigroup if there exists an invariant mean on its functions restricted to a sub-x-semigroup of the original x-semigroup. We sketch earlier results, then give many equivalent conditions for thickness on :-semigroups, and finally present theorems giving conditions tbr an invariant mean to exist on a function space.

Junghenn generalizes Mitchell's theorem thus: Thammt If T is a left-thick subsemigroup of S, then F has a ieft-invariant mean F[ T has a left-invariant mean.
Thickness can be defined in the more general setting of a transformation semigroup.This section defines such scmigroups and other necessary terms.
The abbreviated notion (S,X) will denote a transformation scmigroup whenever the mcaning of x is clear or whcncvcr x is generic.CI',Y) is a subtmnsformation semigroup of (S,X) T is a subsemigroup of S. YX. and TYY.
Dcf'mition 2.2, Let scmigroup S and set X both be endowed with Hausdorff topologies.
Transformation semigroup (S,X,g) is a topological transformation semigroup, or r-semigroup x is separately continuous in the variables and x.Again, a x-semigroup will be denoted briefly by (S,X).
Let C(X) denote the set of continuous and bounded complex-or real-valued functions on X.
In the preening definition F* may be replaced by C(X)* since every functional in F* can be extended to a functional in C(X)*.Also it can be shown that F* can be replaced by M(F), the set of all means on F. Definition 2_5.Let F be G-introverted, pF*, and keG*.The evolution product of J, and denoted .p, is def'med by J-If (Tsf) (['F).Note that leF* and that if G is norm-closed, conjugate-closed, and contains the constant functions, then .M(G) and IeM(F) imply ,IM(F).A mean on FC(X) is defined in the same way as a mean on B(S) was defined in section 1.If F is an algebra under pointwisc multiplication, then mean I is multiplicative Is(fg) p(f)P(g) (,ge').
Let M(F) set of all means on F, and MM(F) set of all multiplicative means on F. M(F) and MM(F) are both w*-compact, being closed subsets of the unit ball in F*.
An evaluation at xeX is defined by e(x)f f(x) (l'eF): clearly an evaluation is a mean.A finite mean on F is a convex combination of evaluations.
A mean is multiplicative if and only if it is the w*-Iimit of evaluations.
A special case of transformation semigroup is furnished by letting X S and g ,(.) where a.s: S--S is defined for any fixed sES by .s(t)st (S).If GgC(S) is a linear space, then Lsg(t g(st) (Vs,tES.gEG);also, X,IM(G) klaeM(G).If FgC(X) is a linear space then LsT TuT (VsES.IEM(F)).Mean IEM(G) is left-invariant p(Lsg I(g) (VgG).J unghenn's generalization of F-left thickness carries over in a straightforward way to transformation semigroups.The corresponding concept is defined in Definition 3.1, and a plethora of alternative characterizations is given by Theorem 3.3.Assumptions: (S,X) is a transformation semigroup; G=C(S) is a subalgebra; F=C(X) is an algebra which is norm-closed, S-invariant, G-introverted, and contains the constant functions; YX.
Remark 3.2.If X S and the action is left multiplication, then the definition is identical to Junghenn's.
To show that u 0 is invariant it suffices to prove that fcl e(y)Te(t)dft fOe(y)dO (VtE T).
rl Therefore, lz I (EloE2o...oEn) [ I (Ej) n fl (E,).Since this holds for arbitrary n, (T,Y) be a sub r-semigroup of (S,X); FeB(X) be a translation invariant, norm-closed, G-introverted subalgebra which contains the constant functions.
1.If F has an invariant mean with respect to if,Y) and T is G-thick in S, then F has an invariant mean with respect to (S,X).
2. lf G has a left-invariant mean and Y is F,S-thick in X, then Fly has an invariant mean with respect to (T,Y).
2. Because Y is F,S-thick in X, then by Theorem 3.3.f::ItM(F) such that vl(f) (''vM(G),fF(Y)).Let v be an invariant mean of G. Then vl is an invariant mean of F such that vt(f) (VftF(Y)).By Theorem 3.6 Fly has an invariant mean with respect to if,y).QED In the preceding theorem the thickness condition on T in (1) implies the thickness condition on Y in (2) according to the following lemma: Lemma 3.8.Let (S,X) be a -semigroup; if,Y) be a sub x-semigroup of (S,X); F=B(X) be a translation-invariant, norm-closed, G-introverted subaigebra which contains the constant functions.
If T is G-thick in S, then Y is F,S-thick in X.
PROOF: Let fF(Y): 0fl, fl on Y. Then Te(y)fF(T (yY).By Theorem 3.3.eapplied to L(S,G) ::IIM(G such that g(LsTe(y)f) (Te(y)Tsf) Ie(y)Tsf and lxTe(y)f te(y)f.Then lae(y)eM(F) has the properties required by Theorem 3.3.efor Y to be F,S-thick.. Multiplicative Mean.and Several results connect multiplicative means with thickness.F is assumed to be an S- invariant, norm-closed algebra =C(X) which contains the constant functions.