COMPACTNESS PROPERTIES OF A LOCALLY COMPACT GROUP AND ANALYTIC SEMIGROUPS IN THE GROUP ALGEBRA

Let G be a locally compact group with left Haar measure M, and let L1 (G) be the convolution Banach algebra of integrable functions on G with respect to p . In this paper we are concerned with the investigation of the structure of G in terms of analytic semigroups in L1 (G) . In this context, a well known result is the theorem of A . Sinclair which says that G is metrizable if and only if there is an analytic semigroup (az)Rez>0 in L'(G) such that [al *L 1 (G)] = [L'(G) * al]= L1 (G) ([26, p .41]). Other results relating the behavior of analytic semigroups on half-discs to certain linear properties of Ll (G) have been obtained in [6] . The following problem was raised by J . Esterle in [3, p.460] .


Introduction
Let G be a locally compact group with left Haar measure M, and let L1 (G) be the convolution Banach algebra of integrable functions on G with respect to p .In this paper we are concerned with the investigation of the structure of G in terms of analytic semigroups in L1 (G) .In this context, a well known result is the theorem of A. Sinclair which says that G is metrizable if and only if there is an analytic semigroup (az)Rez>0 in L'(G) such that [al *L 1 (G)] -= [L'(G) * al]-= L1 (G) ([26, p .41]).Other results relating the behavior of analytic semigroups on half-discs to certain linear properties of Ll (G) have been obtained in [6].The following problem was raised by J. Esterle in [3, p.460] .
Question E. If L'(G) has a non-zero analytic semigroup (a z) R, z>0 which is bounded on the line {Re z = 1} , does it follow that G is compact ?What about the converse ?Also, A. Sinclair asked in [26] for the relationships between the rate of growth on vertical lines of analytic semigroups in L'(G), and compactness properties of G. Recall that G is said to be of polynomial growth if for every compact neighborhood K of G there is a nonnegative integer m such that p(I1 n) = O(n'n) as n -> oo ([23, p.280]) .If the minimal m for which (*) is satisfied is the same for all compacts K then this number is called the degree of growth in G.Such a number exists if, for example, G is connected or G is compact (m = 0 in this *This research was started, and a part of the results were obtained, during the attendence of the author at "Semester on Automatic Continuity and Banach Algebras" in the University of Leeds, June 1987, supported by the U .K .Science and Engineering Research Council .The research has also been partially supported by the Spanish DGICYT, Proyecto PS87-0059, and the Caja de Ahorros de la Inmaculada, Zaragoza, Ayuda CB 1188 last case).Then, does G have polynomial growth if and only if there is an analytic semigroup (az)R, z>0 in L1(G) and a nonnegative integer N such that llal+iy11 = O(1 yj') as ly1 --> oo, y E IR? (See [26, p.81]) .Recently, T. Pytlik has shown that if an element b of a Banach algebra A is such that ¡le¡,,¡¡ = O(1tlw), as Iti --, oo (t E R) for some nonnegative integer N, then there exists an analytic semigroup (az )Rez>0 in A such that ~jal+`y11 = O(1yj'+1) as jy1 -> oo (y E R) ( [25]) .J .Dixmier had proved in [8, p.17] that, for a group G of polynomial growth, such an element b exists in L'(G) and, furthermore, the number N associated to b can be chosen as N = m + 1, if m satisfies condition (*) .Thus Sinclair's question has an affirmative answer in one way, but as far as we know nothing else is known about the converse.Notice that the first part of Question E can be viewed as a particular case of this converse .
Here, we present a first stage in the study of Question E. We prove that, if G is a central group, then L1 (G) has a non-zero analytic semigroup (az)Rez>0 such that {al+ 'y : y E R} is relatively weakly compact if and only if G contains a compact, open subgroup.If, furthermore, G is also connected then G must be compact .We obtain this theorem as a consequence of the following one, which we prove in a more abstract setting : if (az)R, z>0 is an analytic semigroup in a Banach algebra A such that {al+ 'y : y E R} is relatively weakly compact, then the spectrum a(al) is at most countable .The proof of this last result which we give here is based upon elementary properties of the weakly almost periodic functions on the real line (see [9], [10]) .Other essential ingredients in our general argument are the structural properties of central groups ( [17], [18]) .
We also show that for every locally compact group G containing a compact and open subgroup, L1 (G) has an analytic semigroup which is norm compact on {Re z = 1} .Then, if G is central (in this case G is of polynomial growth) its structure allows us to improve the number N appearing in Sinclair's question, with respect to the one we would get from the theorems given by Dixmier and Pytlik in [8] and [25], respectively.
The paper is divided into three sections.In the first one, we collect definitions and basic properties of weakly almost periodic functions, analytic semigroups in Banach algebras, and central groups, that we shall need in the two other sections.Section 2 is devoted to prove the result mentioned above for analytic semigroups in Banach algebras .In section 3 we discuss Question E, and we give there the precedings results about group algebras .
Acknowledgements .We wish to thank B .Aupetit, O. Blasco, J. C. Candeal, B .Cuartero, J. Esterle, N. Gronbwk, T. Ransford for helpful conversations or comments about subjects of this paper, and A .Hulanicki and T. Pytlik for reprints.

Preliminaries
Let IR be the real line, and denote by Cb(IR) the Banach algebra of bounded and continuous functions on R .If f E C6^a nd t E R we define ft E Cb(R) by ft(s) = f(s -}-t) (s E IR).A function f E Cb(R) is said to be weakly almost periodic if {ft : t E IR} is relatively compact in the weak topology of Cb(IR) .This notion, which was introduced by W. Eberlein in [9], is a generalization of the well known almost periodic functions of H. Bohr ([4]) .Denote by W(R) the class of weakly almost periodic functions on R .Every continuous function on R which is null at infinity, and Every positive definite function on R belong to W(R) .The class W(R) enjoys interesting properties.We shall only need the following mean ergodic theorem ([9, Theorem 15.2]) Let H be the open right-hand half-plane of C, and let A be a Banach algebra .An analytic semigroup in A is an analytic function z --> a', H -> A such that ax+w = azaw(z, w E H) .We identify an analytic semigroup with its range, denoted by (az )Rez>o .Throughout, whenever we consider analytic semigroups, we shall write a instead of al .We are interested in a non-zero analytic semigroup (az)R,Z>o with sup ..R j1a l +' v11 < oo .In this case, it is well known that if A is commutative, and ~¿A is the character space of A, then for each cp E q)A there is a E R such that W(az) = e"(z E H) (see [26, p.85]) .Also, the Banach algebra generated by the semigroup equals the Banach algebra polynomially generated by a ( [14, p.379]) .
Let G be a locally compact group, and let Z(G) be its centre, that is, Z(G) _ {t E G : ts = st for all s E G} .The group G is said to be central if the quotient group G/Z(G) is compact .Clearly, all compact groups and all locally compact abelian groups are central."In fact, many of the features common to there two classes appear in their natural setting only when viewed as being characteristic of central groups.In addition there are strong indications that the class of central groups marks the utmost degree of generality in which all there features are still present" ([18, p.361]) .We shall use the following results in section 3 Theorem 1 .2. Leí G be a central group .Then there exisis n >_ 0 such that G -Rn x Go where Go is a locally compact group containing a compact, open, normal, subgroup K such that Go/K is abelian.
A proof of this is in [17, p.331] .The class of all connected, central groups admits several interesting, and equivalent, descriptions ([21, p.14,15]).We shall only need the next one Theorem 1 .3.If G is a connected central group then G = Rn x K, where n > 0 and K is a compact (connected) group.
The corresponding results on the structure of locally compact abelian groups can be seen in [20 I] .
We finish this section with a simple observation Proof. .The centre Z(G) is of polynomial growth because it is commutative ([12, Theorem 7.8]).Besides that, G/Z(G) is compact by definition .Then the result follows from [22, p. 167,168] .

Analytic semigroups which are relatively weakly compact on vertical lines
The next proposition is a crucial step in our reasoning Proposition 2.1.Let A be a Banach algebra, and let A* be its dual Banach space.Suppose that there is an analytic semigroup (a z)Rex>0 in A such that {al+ 4y : y E R} is relatively weakly compact.Then Pro-f-(i) Put f(y) = cp(a2 +'y) (y E R) and T(b)(y) = cp(a1+`yb) (b E A, y E R) .Then T is a continuous linear function from A to Cb(R), which is also continuous with respect to the corresponding weak topologies, and we have that f, = T(al+iv) (v E R) .It follows that f E W(R) .
(ii) For each A E 9I, the function y -> a2 +'ye -'ay, R --> A is continuous and then it is Bochner integrable over each interval [0, r] (r E R), and its range Ao is separable .Put r b,,\ = (1/r) f a2+t ye -iay dy (A r E R) 0 By (i), there exists ba E A** such that (ba, cp) = lim r_ .cp(br a) .But, for each r E R, ba belongs to the (weakly) closed convex hull Ca of the set {a2 +'ye -t ay y E R} C Ao .Then Ca is a weakly compact subset of Ao, by the Krein's Theorem ( [11, p.553]) .Since ba belongs to CA , we obtain that ba E Ao .
(iii) Since Ao is separable we can choose a sequence (cPn)n>i in A* such that ¡lb¡¡ = supe>1 Icp n (b)j for every b E Ao .Let X be a countable subset of R such that cp,,(ba) = 0 if A q X, Set X = Un>1 X.Then, if A 1 X, ¡lba¡¡ = supra>, 1Wn(ba)j = 0.
The arguments which we have considered to prove parts (i), (ii), arad (iii) of the above proposition have been taken from [16, p.82,83], [16, p .84], and [1, p .43], respectively.They have been written in its present form in order to give a more elementary proof of such properties, in this context .
The following theorem is the key point of this paper.
Theorem 2.2.Le¡ A be a Banach algebra which contains ara analytic semigroup (a z)R, z>o such that {al+`y : y E R} is relatively weakly compact.Then the spectrum v(a) of a is at most countable.

J0
If cp is a character of A we have that cp(a =) = eaZ (z E H), for some a E R.Then, Thus, u(a) is at most countable .o A more formal proof of this fact is also available by using some theorems ora compact topological semigroups (see [15] for this topic) .Such a proof relies heavily ora ideas coming from the basic theory of regular quasimultipliers of Banach algebras, which can be seen in [13] .The proof, and more information about commutative Banach algebras generated by semigroups as in Theorem 2.2, are given in [27] .
Corollary 2.3.Let A be a non-unital, commutative, semisimple, Banach algebra.Suppose that its character space <DA is connected in the Gelfand topology.If (a z)Rez>o is ara analytic semigroup in A such that {a 1 +'y : y E R} is relatively weakly compact then a = 0.
Proof .By the hypothesis ora A, IA is non compact ([24, p.154]) .Also, the function ep -4 cp(a), OPA -+ C is continuous and null at infinity, whence 0 E {cp E OPA} -.Moreover, for each cp E O¿A there is a E 6B with W(a) = ea .Thus the spectrum u(a) of a is a connected subset of 1B .But u(a) is also countable (Theorem 2.2), and therefore a(a) = 0.
Analogous arguments to those of the proof of this corollary will be used to prove Theorem 3.3 below.
Remark .The condition on the semigroup in the above results cannot be replaced by the weaker assumption that sup yeR ¡lal+'yll < oo : let A = Co([0,1]), the Banach algebra of continuous functions on [0,1] which are null at t = 0. Set az(t) = ez lo g t, where log t is the branch of the logarithm which is real-valued on the positive real numbers.Then (a')R, z>0 is an analytic semigroup in A such that sup..R llal+'y11 < oo, but a(a) = [0,1].

Group algebras
We begin by giving a result concerning the second part of Question E. Let G be a locally compact group, and let Go be an open subgroup of G.It is straightforward to verify that the mapping 0 : given by O(f)(t) = f(t) (t E Go), and O(f)(t) = 0 (t E G -Go) is a continuous algebra homomorphism .On the other hand, if G is a compact group then there are sequences (cpn)n>1 of functions on G such that (cpn)n>1 C L1 (G), Proof. .Let us assume that G is compact and take a sequence (IPn)n>1 as before the proposition .Recall that H is the open right half plane .Put az(t) = En>1 e-nzWn (t) (t E G, z E H) .Take a > 0 and z E H such that Re z >_ a.
Then ¡le -nzWnil <_ e-n Re z < e-,n .Hence the function z --> az,H -> Ll(G) is analytic, and it is also clear that az * a'°= a z +' (z, w E H) .Furthermore, since a l +' y = al+i(y+21r) ( y E R) the continuity of the mapping y -> a l+4y , R -> L1 (G) implies the compactness of {al+'y : y E [0, 27r]} Consider now the general case and let Go be a compact, open, subgroup of G. Let (a z)Re z>0 be the semigroup in L 1 (Go) given before .Put V = O(az) (z E H), where 0 : L'(G0) -> L1(G) is as before .Clearly, (V)R,,>0 satisfies the required conditions.As Proposition 3.1 shows, there exist non compact locally compact groups G such that Ll (G) has non-zero analytic semigroups which are not only bounded, but compact, on {Re z = 1} .
We can give a partial converse to the above proposition.But, before doing this, let us say at this point some words about the question suggested by A .Sinclair (see Introduction) .As already said, if G is a group of polynomial growth, and if m is a nonnegative integer such that p(K") = O(nm) as n -> oa for some compact neighborhood K in G, then the results given by Dixmier and Pytlik in [8] and [25] imply that there is a non-zero analytic semigroup (az )Rez>0 in L'(G) such that j1a l+'yl1 = O(1yj') as ¡y¡ -> oo (y E H), with N = m + 2 .If G is central, we can improve substantially the relationship between m and the growth of j1al +' y11 at the infinity, by using the structure of G and the same ideas as above .Proposition 3.2.Le¡ G be a central group and leí m be the minimal nonnegative integer for which ¡he condition (*) holds.Then there is a non-zero analytic semigroup (az )Rez>0 in L'(G) such ¡ha¡ j1al +' Y11 = O((logly1)') as jy1 ->oo(yER),ifm>1 .
Next, we are going to show our main results concerning Question E. Let A be a Banach algebra .If X is a Banach space we denote by £(X) the space of continuous endomorphisms on X endowed with the uniform norm.Recall that a representation cp of A on X is a continuous algebra homomorphism cp : a --> cp(a),A --~C(X).A closed subspace Xl of X is said to be invarianí under cp if W(a)(X1 ) C Xl for all a E A. If (0) and X are the only closed subspaces of X that are invariant under cp, and cp is non null, then cp is said to be irreducible.If X is finite-dimensional, then ep is called finite-dimensional ( [5]) .We denote by F the set of finite-dimensional irreducible representations of A. We say that A is P-semisimple if W(b) = 0 for every cp E P and some b E A implies that b = 0.
Let us consider the convolution Banach algebra L' (R" ; A) of A-valued Bochner integrable functions on IRn .), which is an at most countable subset of R, by Theorem 2.2.Now, since ,C(X) is finite-dimensional, the spectrum function is continuous on ,C(X) ([2, p.8]) which implies that the spectral radius p is also continuous on ,C(X).It follows that {p(r 0 cp)(a)) = maxaE,,(r) 1 A 1: r E R n} is a connected subset of fl X 1 : \ E o(a)} .Moreover, limr,-,,(r ® cp)(a) = 0, whence we deduce that P((r ® cp)(a» = 0 for every r E Rn .So, for each r E Rn , the Banach subalgebra of ,C(X) generated by the semigroup ((r ® cp)(az»RQ z>o is radical.But this is not possible unless (r ® cp)(a) = 0 ([26, Proof of Theorem 5 .6,p.80]) .Then we have obtained that W(f. e -'r .sa(s) ds) = 0 for all r E Rn and cp E F. Since A is F-semisimple we obtain that fue .e -'r .sa(s)ds = 0 for every r E R, whence, as usually, by composition with continuous linear functionals on A, we deduce that a = 0.
The above proposition applies to A = Ll (G), if G is a (so-called) maaimally almost periodic group (see [19, p.428]), and this fact gives rise to the following result .
Theorem 3.4.Let G be a central group such ¡ha¡ V(G) has a non-zero analytic semigroup (az )R, z>o with {al +'y : y E R} relatively weakly compact in Ll (G) .Then G contains a compact, open, normal, subgroup K such that GIK is abelian.
Proof.By Theorem 1 .2,G = Rn x Go with n and Go as there.We have to prove that n = 0 .Suppose, if possible, that n > 1.Since Go is (isomorphic to) a closed subgroup of G, L'(G o) is semisimple for the set of its finite-dimensional irreducible representations ([21, p.15], [19]).Also, because of Ll (G) = LI(Rn x Go ) = Ll(Rn) ®,, Ll(Go) = Ll(Rn;Ll(Go) ([2, p.132]), it is enough to take A = L'(G o) in Theorem 3.3 to obtain that a = 0, which is a contradiction .
We write now the converse result announced after Proposition 3.1.Corollary 3.5.(i) Let G be a central group .Then G contains a compact, open, subgroup if and only if V(G) has a non-zero analytic semigroup (a')Rez>o such that {a`]-'Y : y E R} is norm compact (or relatively weakly compact) in Ll (G) .
(ii) Let G be a central and connected group .Then G is compact if and only if L 1 (G) has a non-zero analytic semigroup (az )Rez>o such that {al +'y : y E R} is norm compact (or relatively weakly compact) in L'(G) .This corollary is a consequence of Proposition 3.1 and Theorem 3 .4.For part (ii) we need also Theorem 1 .3.
As we have seen, the study of Question E for central groups depends on the study of the existence in L'(R) of analytic semigroups, bounded on {Re z = 1} .Tliis leads us to pose the following questions .
Suppose that Also, for any connected, locally compact group G, we may ask this other question Question 3 .Suppose that there exists a non-zero element f in L I (G) such that ¡he spectrum a(f) is at most countable.Does it follow ¡ha¡ G must be compact?

( i )
For every 9 E A*, the function y --> cp(a2+'y), R -+ C is weakly almost periodicfor each A E R, and so it defines an element ba of A. (iii) The set {.\ E R : ba :~0} is at most countable.