APPENDIX A PARACOMPACT TOPOLOGICAL SPACES We have collected here the topological material used to study the homological properties of commutative Banach algebras and their ideals, IV.§2 and IV.§3, V.§2; the basic theorems here, from the point of view of the relevant applications, are Theorems A6 and A12. DEFINITION A1. Let U be a system of subsets of a topological space Q (for example, an open cover). This system is said to be locally finite if every point in Q has a neighbourhood intersecting only a finite number of sets in U. DEFINITION A2. The system of sets U, is said to be a refinement of the system of sets U2 if every set from U, is contained in some set from U2 • DEFINITION A3 (Dieudonne). A Hausdorff topological space is said to be paracompact if every open covering has an open, locally finite refinement that is also a cover. The most important class of paracompact spaces is given by the following. ASSERTION A4 (Stone [58, Theorem 5.1.3]). Every metrizable space is paracompact. ASSERTION A5 [58, Theorem 5. 1. 5]. Let Q be a paracompact space. Then it is normal; as a consequence (of this fact and Urysohn's lemma) gi ven any closed disjoint subsets F, G c Q there exists a cont inuous function f: Q ~ [0, 1] that is zero on F and equal to one on G. THEOREM A6 (cf . [ 196, Lemma 2. 1] ). Let Q be a locally compact, paracompact space. Then it can be represented as a disjoint union of subspaces Q,,; v E Q, each of which is the union of an increasing family of compact subsets K~; n= 1,2" ... , such that K~ lies in the interior of K~+'. DEFINITION A7. A family {hI'; Il E A'} of cont inuous non-negative functions on the topological space Q is called a partition of unity (an n-partition of unity, respectively) on Q if, for each sEQ, the value h,.(s) is nonzero for no more than a countable family of indices (than n indices, respectively) and I: ,h,.(s) = 1. ,.EA 268 PARACOMPACT TOPOLOGICAL SPACES 269 PROPOSITION A8. Let !1 be a locally compact, paracompact topological space. Then it has a 2-partition of unity {hI': ilEA'} such that, for any compact Ke!1, there is a finite collection n Il" ... , Iln E A' such that I: h == 1 on K. k=l fLk ~ This is deduced from A5 and A6 .• PROPOSITION A9. Let !1 be a locally compact, paracompact, non-compact space. Then it has a sequence of points converging to 00. ~ This follows from Theorem A6 .• EXAMPLE A10. The space of all countable ordinals with the order topology (cf. [58, Example 3.1.27]) is not paracompact. EXAMPLE All. We consider ~~ - the spectrum of the algebra Cb (see O. §2. 1). The only convergent sequences in ~~ are those which are eventually constant [58, Corollary 3.6.15]. Therefore the complement of anyone-point subset of ~~\~ (= the spectrum of any maximal ideal in Cb) is not paracompact. THEOREM A12 (cf. [196, Lemmas 3.1 - 3.3]). Let!1 be a locally compact topological space such that there exists a continuous function G: QxQ--7[O, 1] with the following properties: ( I ) for any compact K e !1, G( s, t) --70 as s --700, uniformly for t E K and G( s, t) --70 as t --700, uniformly for s E K; (II) G(s, s) == 1. Then !1 is paracompact. DEFINITION A13. The topological space Q is said to be hereditarily paracompact if every subset, with the induced topology, is paracompact. ASSERTION Al4 (Dieudonne [103]). Q is hereditarily paracompact ~ every open subset is paracompact. ASSERTION A15. The following properties of a topological space !1 are equivalent: (I) Q x Q x Q is hereditarily paracompact; (I I) Q x !1 x!1 is heredi tari ly normal; (III)Q is metrizable. In this assertion the equivalence of (II) and (III) is a well-known result due to Katetov [110], the deduction of (II) from (I) is provided by Assertion A5 and the deduction of (1) from (II) is provided by Assertion A4. APPENDIX B INVARIANT MEANS ON LOCALLY COMPACT GROUPS We indicate here some of the concepts and facts in harmonic analysis that are necessary for characterizing amenable group algebras (VI I. §2). For a more detai led account of much of this material see Greenleaf [22] and Hewitt and Ross [52]. Let G be a group and E a set. Given f: G ~E and reG, the map rf: S Hf(rs) (respectively, r: s Hf(sr) is called the left (right) shift of f by r. We also put f-(s) = f(S-l). Now let G be a topological group and E a Banach space. DEFINITION B1. The vector function u: G ~E is said to be right (respectively, left) uniformly continuous on G if, for any c > 0, there exists a neighbourhood of the identity U in G such that, for any s e G and re U, The function u is said to be two-sidedly uniformly continuous if it is simultaneously right and left uniformly continuous. From now on, G is a locally compact group wi th a left-invariant Haar measure ds. As usual, LCO( G) denotes the Banach space (of equivalence classes) of essentially bounded measurable (with respect to the measure ds) complex-valued functions with norm IIrpll=esssupqrp(t)/: teG}=inf{sup{/rp(t)/: teG\N, N is a locally null set in G}. We recall that LCO(G) is isometrically isomorphic to Ll(G)· and moreover the function rp e LCO(G) corresponds to the functional f HI f(s)rp(s)ds; feL 1(G). G Obviously rp e LClO(G) is right (left, respectively) uniformly continuous _ the map G~LCO(G) : r Hrrp (r Hrpr) is continuous. The closed subspace in LClO(G) consisting of the bounded two-sidedly uniformly continuous functions is denoted by euc(G). Besides the case when fh he Ll(G) (cf. o. §2. 2) the convolution A set NeG is said to be locally null if the measure of the intersection of N with any compact subset of G is zero. 270 INVARIANT MEANS 271 fl*h(s) = J Ht)h(t-1S )dt G of two functions on G can still exist in two important situations. We select the following simple proposition. PROPOSITION B2. Let feL1(G) and 1peLOO (G). Then f*rp (respectively, rp* f~, f*1p* f~) is defined for all s e G and is a right (respectively, left, two-sidedly) uniformly continuous function on G. ~. DEFINITION B3. Let H be a linear functional on LOO(G) or Cuc(G)· Then 11 is said to be left-invariant if H(rrp) = H(rp) for all rp and r, normalized if H(1) = 1 (1 is the constant function with value one), and positive if H(1p)~O for all rp~O. It is said that H is a left-invariant mean (or a mean) if H has all three properties (or just the last two properties). PROPOSITION B4. If H is positive, then for all rp, l1(qi) = l1(rp) (here the bar indicates the complex conjugate). If H is a mean, then for all real-valued rp ess inf{1p(t): teG} ~11(1p) ~esssup{1p(t) teG}. 2 h COROLLARY B5. Any mean is a continuous functional. PROPOSITION B6. Suppose that there is a linear functional on one of the spaces of the pair (LOO(G) , Cuc(G)) which has some combination of the following properties: left-invariance, normality, positiveness, continuity. Then there is a linear functional on the other space of this pair with the same properties. ~ When the functional is given on LOO(G) it is sufficient to take its restriction on Cuc(G). Conversely, given a functional m: Cuc(G) ~IC, take any compact neighbourhood K of the identity in G such that {s S-1 e K} = K. Let XK e LOO(G) be the characteristic function of K. Then, for any rp e LOO(G) XK*rp*XK e Cuc(G) (Proposition B2), and the functional H: LOO(G) ~IC given by rp ~m(XK*rp*XK)' obviously has the same properties as m. _ COROLLARY B7. LOO(G) has a left-invariant mean if and only if Cuc(G) has one. We now come to the basic concept of this appendix. DEFINITION B8. A locally compact group G is said to be amenable if there is a left-invariant mean on either LOO(G) or (= and; see above) on Cuc(G). REMARK B9. Apart from LOO(G) and Cuc(G) there is a whole lot of other function spaces connected with G (for example Cb(G)) on which the existence of the analogue of a left-invariant mean is equivalent to the group G being amenable [22]. There are other equivalent definitions of 2 When rp E Cuc(G) "ess" is of course superfluous. 272 APPENDIX 8 the amenability of groups that enable them to be looked at from quite a different point of view [22, 79, 88]. REMARK 810. It can be shown that the existence of a left-invariant mean on LCO(G) or Cuc(G) implies the existence of right and two-sided invariant means defined in an obvious way [22]. REMARK 811. In applications (Theorem VII.2.33 is an example) the specific form of a left-invariant mean is, as a rule, of no particular value and only the fact that it exists is important.