Amenability, Weak Amenability and Approximate Amenability of ^(S)
by
Filofteia Gheorghe
A thesis submitted to
the Faculty of Graduate Studies
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
Department of Mathematics
University of Manitoba
Winnipeg, Manitoba
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Amenability, Weak Amenability and Approximate Amenability of £*(S)
BY
Filofteia Gheorghe
A Thesis/Practicum submitted to the Faculty of Graduate Studies of The University of
Manitoba in partial fulfillment of the requirement of the degree
Of
MASTER OF SCIENCE
Filofteia Gheorghe © 2008
Permission has been granted to the University of Manitoba Libraries to lend a copy of this thesis/practicum, to Library and Archives Canada (LAC) to lend a copy of this thesis/practicum, and to LAC's agent (UMI/ProQuest) to microfilm, sell copies and to publish an abstract of this thesis/practicum.
This reproduction or copy of this thesis has been made available by authority of the copyright owner solely for the purpose of private study and research, and may only be reproduced and copied as permitted by copyright laws or with express written authorization from the copyright owner. Abstract
We are doing a survey on amenability, weak amenability and generalized notions of amenability of semigroup algebras ^(5). Based on the characterizations of amenability of a Banach algebra, F. Ghahramani, R.J. Loy and Y. Zhang have introduced approximate amenability and pseudo-amenability for Banach algebras. Since they have been introduced, many results concerning them have been obtained by many researchers. We focus on this topic regarding semigroup algebras. We give some new characterizations for a Banach alge bra to be approximately amenable. For a semigroup 5 with a generating set E, we also give necessary and sufficient conditions so that ^(5) is amenable, weakly amenable or bound- edly approximately amenable.
It is known that if the semigroup algebra ^:(5) is approximately amenable then 5 must be a regular amenable semigroup. We prove that the converse is not true by examining the bicyclical semigroup Si which is an important semigroup and has been studied by many researchers from various aspects. Precisely, we show that, although Si is a regular amenable semigroup, I1 (Si) is not approximately amenable. In the appendix we also give a direct proof to the fact that l1^) is not approximately amenable, where 52 is the partially bicyclic semigroup denned by 52 =< 1, a, b, c \ ab — ac = 1 >.
l Acknowledgements
I would like to express my gratitude to my advisor, Dr. Yong Zhang whose expertise in the field, constructive comments and high expectations motivated me to achieve my goals.
Because he is a hard worker and actively involved in research, he was an excellent role model for me.
I also wish to thank Dr. F.Ghahramani for his clear explanations during my course work.
His approachability and feedback to my thesis helped to make my graduate experience a positive one.
I would also like to thank Dr. Liqun Wang for taking time to serve as my external advisor and Dr. Y.Choi whose help during the final stages of completion was most appreciated.
I would like to acknowledge the Departament of Mathematics at the University of Manitoba for their generous financial support.
I'm grateful to my friends in Winnipeg and family without whose help and encouragement this thesis would not have been possible.
ii Contents
Abstract i
Acknowledgements ii
1 Preliminaries 1
1.1 i1- semigroup algebra 1
1.2 More about Banach algebras 5
2 Amenability of ll{S) 10
2.1 Amenability of semigroup algebras 10
2.2 Amenability of f(S,uj) 15
2.3 Amenability of ll{QS) 19
3 Weak amenability of I1 (S) 21
3.1 Weak amenability of discrete semigroup algebras 21
3.2 Weak amenability of ^(S, ui) where S is commutative 22
3.3 H1 (l1 (S), e°°(S)) and H1 (tx (5), l1 (S)) for some classes of semigroups . . 25
4 Approximate amenability of I1 (S) 28
iii 1 Approximate amenability of a Banach algebra 28
2 Amenability of bicyclic and partially bicyclic semigroups 34
3 A characterization of approximate amenability of a Banach algebra 35
4 Approximate amenability of i1 (Si), where S\ is the bicyclic semigroup . . 40 Chapter 1
Preliminaries
1.1 i1- semigroup algebra
Terms and concepts of. basic real and functional analysis which we have not defined or discussed can be found in [8] and [49]. In this section we establish some notations and definitions.
Definition 1.1.1. A complex algebra is a vector space A over the complex field C in which a multiplication is defined, called an algebra product, A x A -+ A; (a,b) -+ ab, that satisfies:
a(bc) = (ab)c, (a + b)c = ac + be, a(b + c) = ab + ac (a, 6, c G A)
(aa)b — a(ab) = a(ab) (a € C, a, b e A).
We say that A is commutative if ab = ba for all a, b £ A. If in addition, A is a Banach space with respect to a norm that satisfies the submultiplicative inequality
||a6||<||o||||6|| (a,beA) (1.1)
1 2 then A is called a Banach algebra. If A contains a unit element e such that ae = ea'— a
(a € A) then A is called a Banach algebra with identity. If || e ||= 1 then A is a unital
Banach algebra.
Suppose that the algebra A does not have a unit. Then we define the unitization of A to be A* = C 0 A. A* is a unital algebra, with unit (1,0), for the product
(a, a)(j3, b) = (a/3, ab + /3a + ab) (a, (3 € C, a, b 6 A).
If A is a Banach algebra, then so is A* for the norm ||(a,a)|| = \a\ + \\a\\.
A subalgebra of an algebra A is a linear subspace B of A such that ab € 5 for all a, 6 € 5.
A left ideal in an algebra A is a subalgebra I QA such that, if a £ A and 6 € /, then ab € /.
Similarly, we can define a right ideal and a two-sided ideal for A.
The radical of an algebra A is defined to be the intersection of the maximal left ideals of
A#; it is denoted by rad A. The algebra A is semisimple if rad{A} = {0} (see [9]).
We recall some definitions. For further details see [38].
Definition 1.1.2. A groupoid (S,n) is defined as a non-empty set S on which a binary operation /i: S x S —> S is defined. We say that (5, /x) is a semigroup if the operation -/x is associative, that is to say, if, for all x,y and z in 5,
fi(n(x,y),z)) = n(x,n(y,z)). (1.2)
We shall follow the usual algebraic practice of writing the binary operation as multiplication.
Thus n{x,y) becomes xy, and formula (1.2) takes the simple form
(xy)z = x(yz), the familiar associative law of elementary algebra. A non-empty subset T of S is a subsemi- group if T is a semigroup for the product in S. For s 6 S, we set
(s) = {sn :n€N}, the semigroup generated by s; the subsemigroup of S generated by a subset T is denoted by{T).
If the semigroup 5 has the property that, for all x, y in S,
xy = yx, we shall say that S is a commutative semigroup. (The term abelian is also used, by analogy with the group theoretic term). If a semigroup S contains an element 1 with the property that, for all x in S,
xl — lx, we say that 1 is an identity element (or just an identity) of S, and that 5 is a semigroup with identity or (more usually) a monoid. We now define
S if S has an identity element { 5U{1} otherwise. We refer to 51 as the monoid obtained from S by adjoining an identity if necessary.
If a semigroup S with at least two elements contains an element 0 such that, for all x in S,
Ox = xO = 0, we say that 0 is a zero element (or just a zero) of S, and that S is a semigroup with zero.
By analogy with the case of S1, we define
S if 5 has a zero element
5 U {0} otherwise. 4 and refer to 5° as the semigroup obtained from S by adjoining a zero if necessary.
If a is an element of a semigroup S without identity then Sa need not contain a. The following notations will be standard:
S1a = SaU {a},aS1 =aSU {a},S1aS1 = SaSUSaliaSU {a}.
Let S be a semigroup. For subsets A and B of S, we set A • B = { st : s G A, t € B }; we write SPl for S • S.
S is called simple if it contains no proper (two-sided) ideal.
Definition 1.1.3. Let S be a semigroup, and consider the Banach space
eHS) = {f:S^ClY,\f(s)\ l We write Ss for the characteristic function of { s } for s £ S, so / € i (S) has the form : s€S where II / Hi= Y, I as |< oo. Definition 1.1.4. Let S be a semigroup, and let / — ^ar<5r and g = YlPs8s belong to ^(S). Set / * 9 = (^arSr)* (J2&5s) =][](£ arPs)St teS rs=t where Ylrs=tarPs = 0 when there are no elements r and s in S with rs = i. Let S be a semigroup, and set A = (^(S),*, || • ||i). yl with the usual pointwise addition, scalar multiplication, the product (convolution) * and with the norm || / ||i is a Banach algebra called the discrete semigroup algebra of S. Moreover if S — G is a group then i1 (S) is the discrete group algebra I1 (G). 5 Definition 1.1.5. Let (.°°(S) be the space of all bounded complex-valued functions on S. The dual space of £1(S) is t°°(S) with the duality s£S Given a function f on S the left (right) translation of f by x £ S is a function on S denoted by lxf such that lxf(s) = f(xs) (resp. rxf(s) = f{sx)). A discrete semigroup S is left amenable if the space £°°(S) admits a functional m, called left invariant mean, such that m(l) = 1 =|| m || and m(lxf) = m(f), x € S, f € ^°°(5). Similarly one can defines right amenable. If S is both left and right amenable, it is amenable. 1.2 More about Banach algebras Definition 1.2.1. Let A be an algebra. A left A-module is a linear space E over C and a map (a, x) i-+ a • x : A x E —> £, such that : a • (ax + /?y) = aa- x + /3a-y, (aa + /?&) • £ = era • x + /#> • x a-(b-x) — ab-x (a,[3 €C,a,b £ A,x,y € E). A right A-module is defined similarly. An A-bimodule is a space E which is both a left A-module and a right A-module and which is such that: a • (x • b) — (a • x) • b (a,b £ A,x € E). 6 A Banach space E is said to be a Banach left A-module if it is a left A-module and there exists k > 0 such that || ax \\< k \\ a ||j| x ||, a G A, x G E. Banach right A-modules are defined similarly. E is said to be a Banach A-bimodule if it is both a Banach left A-module and a Banach right A-module and the module multiplications are related by a(xb) — (ax)b, a,b G A, x G E. For any Banach algebra A, A itself is a Banach A-bimodule with the product of A giving the module multiplications. If E is a Banach left(right) A-module, then E*, the conjugate space of E, is a Banach right (resp.left) A-module with the natural module multiplications defined by (x,fa) = (ax,f) (jKsp.(x,af) = (xa,f)), (1.3) for / G E*,a G A and x G E. We call this module the dual right (resp. left) module of E. Here, for x G E and / G E*, {x, f) denotes the value f(x). If E is a Banach A-bimodule, then multiplications given by Equation (1.3) make E* into a Banach A-bimodule, called the dual module of E. Since the only bimodules we are concerned with in the following are Banach bimodules we will refer to them simply as A-bimodules. Suppose that X and Y are Banach spaces. We denote the projective tensor product of X and Y by X§>Y, and denote the elementary tensor of x G X and y G Y by x ® y; we refer to [51] for the details about this kind of product space. Suppose that A is a Banach algebra. Then A®A is a Banach A-bimodule with the multi plications specified by a • (b ® c) = ab We use 7r to denote the linear map from A® A into A specified by n{a®b) = ab (a,b&A). A directed set is a partially ordered set A (admitting Reflexivity, Antisymmetry and Transitivity) such that, given Ai, A2 G A, there exist A G A with A > Afc (k = 1,2). Let X be a topological spaced A net in X is a mapping from a directed set A into X. m ls sa A net {ZA}A€A X id to converge to x G X, denoted by lima: A = x, if for every AeA neighborhood U of x, there exist Ao G A such that x\ G U for all A > Ao- Amenability of Banach algebras has been one of the major themes in the homology theory of Banach algebras [34]. The definition of amenability was introduced by B.E. Johnson in 1972. Amenable Banach algebras have since proved themselves to be widely applicable in modern analysis (for example see [10] and [50]). Definition 1.2.2. Suppose that A is a Banach algebra and let E be a Banach A-bimodule. A (continuous) derivation from A into E is a (continuous) linear mapping D : A —» E which satisfies D(ab) - a • D{b) + D{a) -b {a,beA). For any x G E, the mapping adx : A —> E given by adx(a) = a- x — x • a (a €. A), is a continuous derivation, called an inner derivation, x is called the implementing element 1 for adx. Denote by Z (A,E) the space of all continuous derivations from A into E and by ^(A, E) the space of all inner derivations from A into E. Then iV1^, E) is a subspace of 8 Z1(A,E). The quotient space TIM A F\ - Zl{AE) is called the first cohomology group of A with coefficients in E. For the general theory of the Banach cohomology group Hn(A,E), where n G N, see [40], [35] and [10]. A Banach algebra A is said to be contractible if H1 (A, E) = 0 for all Banach A-bimodules E, amenable if ^(AjE*) = 0 for all Banach .A-bimodules E, and weakly amenable if HX(A,A*) = 0, where A* denotes the dual space of A with natural .A-bimodule action. Throughout, unless otherwise stated, by a derivation we mean a continuous derivation. A Banach algebra A is amenable if for every A-bimodule E every derivation D : A —> E* is inner (i.e. 3 x € E* such that D{a) =a-x-x-aVa£A). Trivially, an amenable Banach algebra is weakly amenable; however the class of weakly amenable Banach algebras is considerably larger. See Example 4.1.1 below. There are many alternative formulations of the notion of amenability, of which we note the following, for further details see [2, 6, 10, 35, 50]. The Banach algebra A is amenable if and only if any, and hence all, of the following hold, where ir: A (i) (Johnson [41]) A has a bounded approximate diagonal, that is, a bounded net {mi) C A®A such that for each x G A, mi • x - x • m.i —>• 0, 7r(m;) • x —> x; (ii) (Johnson [41]) A has a virtual diagonal, that is, an element M £ (A® A)** such that for each x G A, x • 1 = M • x, (ir**M) • x = x; (iii) (Gourdeau [24]) any derivation of A into any Banach A-bimodule is the strong limit of a net of inner derivations which have a bounded net of implementing elements. 9 Definition 1.2.3. Let A be a normed algebra. A left approximate identity for A is a net {e\}\eA m -"4- such that for all x € A, lim eAX = x. (1.4) AeA i?i^/ii approximate identities are similarly defined by replacing e\x with zeA in Equation (1.4). A two-sided approximate identity is a net that is both a left and a right approximate identity. If {e\} is norm bounded, then we have a bounded (left/right) approximate identity. B.E. Johnson [40] proved the following general implication: Theorem 1.2.4. // a Banach algebra A is amenable then A has a bounded approximate identity. Chapter 2 Amenability of ^(S) 2.1 Amenability of semigroup algebras The notion of amenability for Banach algebras is well-known as a general principle. The problem of determining which Banach algebras in certain classes are amenable is often a sub stantial problem; there are some major theorems. For example, the amenable C*-algebras, the amenable group algebras, and the amenable measure algebras have been determined in famous theorems. Let S be a semigroup, and let £l(S) be the corresponding semigroup algebra. We classify the semigroups S for which i1 (S) is amenable. Let us first recall the known theory of the amenability of Banach algebras on locally compact groups G. This result combines two famous theorems of B. E. Johnson [40, 43, 15]. Theorem 2.1.1.' Let G be a locally compact group. Then: (i) L1 (G) is an amenable Banach algebra if and only if G is an amenable group; (ii) LX{G) is weakly amenable. 10 11 In particular, Ll(G) is amenable for each locally compact abelian (LCA) group G. Given a semigroup S the problem of the amenability and weak amenability of tx (S) as a Ba- nach algebra is rather more complicated. We will present some results regarding amenability of €1(5) as a Banach algebra and shall determine exactly when £x(5) is amenable as a Ba- nach algebra. This is due to various authors [16, 12, 32, 31, 44]. We recall some further standard notions from semigroup theory. Definition 2.1.2. An element n 6 S is idernpotent ifuu — u and denote with E the set of idempotents. If S is commutative and satisfies the condition that each u in 5 is idernpotent we call S a semilattice. On E there is a usual order: e,/ G E, e < f if ef = fe = e. An element p G E is minimal if q = p whenever q G E with q Definition 2.1.3. A semigroup S is called left cancellative if for all r,s,t G S, rs = rt implies s = t. Similarly, we can define right cancellative. Definition 2.1.4. A semigroup 5 is a regular semigroup if for each s G S there exists s* G S with ss*s = s and s*ss* — s*. If s* is unique for each s € S, we say that S is an inverse semigroup. A group G is a regular semigroup with E — {e<3}, the bicyclic semigroup (defined in 3.3 below) is easily seen to be an elementary inverse semigroup. Theorem 2.1.5 ([16], Theorem 8). Let S be an inverse semigroup with E finite. Then £l(S) is amenable if and only if each maximal subgroup of S is amenable. Definition 2.1.6. Let 5 be a semigroup and suppose that there is a semilattice E and disjoint subsemigroups Ss (s G E) of S such that S = Uses <5« and SaSp C Sap (a, p G E). 12 Then 5 is called a semilattice of the subsemigroups Sa (a £ E). If E is a finite set, we say 5 is a finite semilattice of subsemigroups. Definition 2.1.7. (a) A semigroup S is left reversible if for all x,y € S, xSDyS ^ 0. (b) H C S is a left ideal group if H is a left ideal in S, as well as being a group under the semigroup operation. (c) The minimum ideal K(S) is called the kernel of S in [38, §3.1]. There are several known partial results, which we summarize in the following theorem and which determine exactly when il (S) is amenable as a Banach algebra. Theorem 2.1.8. Let S be a semigroup. (i) Suppose that S is abelian and E = S. Then ^(S) is weakly amenable [12, Proposition 10.5]; (ii) Suppose that t1 (S) is an amenable Banach algebra. Then: S is an amenable semigroup [16, Lemma 3]; S is (left and right) reversible [25], [52, Lemma 1]; S has only finitely many idempotents and each ideal I in S is regular and, in particular, I = 7'2] [17, Theorem 2], S has a minimal idempotent; ^(S) has an identity and K(S) exists and is an amenable group [12, Corollary 10.6]; ll{S) is a semisimple algebra (this follows from [18, Theorem 5.11]); S contains exactly one left ideal group So, which is also the only right ideal group; further more So is amenable [44> Theorem 4-4]- (Hi) Suppose that S is unital and left or right cancellative. Then i1 (S) is amenable if and only if S is an amenable group [31, Theorem 2.3]. 13 (iv) Suppose that S is abelian. Then£1(S) is amenable if and only if S is a finite semillatice of amenable groups [32, Theorem 2. If. The force of these results seems to be that I1 (S) is amenable if and only if S " is built up from amenable groups ". It can be shown that (}(S) is " left-amenable " if and only if 5 is a left-amenable semigroup [46]. In these results it is apparent that the condition of amenability imposes strong algebraic constraints on the semigroup. In fact a characterization is given in [12, Theorem 10.12]. We proceed to describe a kind of semigroup which is the utmost importance in the algebraic theory of semigroups (see [5, §3.1]). Definition 2.1.9. Let G be a group, I and A be arbitrary non-empty sets, and G° = Gu{0} be a group with zero adjoined (see Definition 1.1.2). A sandwich matrix P — (p\i) is a A x I matrix with entries being elements of G° such that each row and column of P has at least one non-zero entry. The set 5 = G x I x A with the composition (a,i,j)o(b,l,k) = (aPjlb,i,k), (a,i,j),(b,k,l)eS is a semigroup that we denote by M (G; I, A; P). Similarly if P is a A x / matrix over G°, then 5 = Gx/xAU {0} is a semigroup under the following composition operation: (aPjib,i,k) UPji^O {a,i,j)o(b,l,k) = < 0 ifPji = Q (a,i,j)o0 = 0o(a,i,j) = 0oQ = 0. 14 This semigroup which is denoted by M°( G; I,A;P) also can be described in the following way. An 7 x A matrix A over G° that has at most one nonzero entry a — A(i,j) is called a Rees I x A matrix over G° and is denoted by (a)y. The set of all Rees I x A matrices over G° form a semigroup under the binary operation A • B = APB, which is called the Rees 7 x A matrix semigroup over G° with the sandwich matrix P and is isomorphic to M°(G;I,A;P) [37, pp. 61-63]. Definition 2.1.10. A principal series of ideals for S is a chain S = h 2 h 2 - 2 Im-l ?Im = K(S) where h, 72,...,7m are ideals in S and there is no ideal of S strictly between Ij and Ij+{ for each j 6 Nm_i. Theorem 2.1.11 ([12], Theorem 10.12). Let S be a semigroup. Then the Banach algebra il{S) is amenable if and only if the minimum ideal K{S) exists, K(S) is an amenable group, and S has a principal series S = h 2 h 2 ••• 2 Im-\ 2 Im = K(S) such that each quotient Ij/Ij+\ is a regular Rees matrix semigroup of the form M°(G, P, n), where n € N, G is an 1 amenable group, and the sandwich matrix P is invertible in Mn(£ (G)). Definition 2.1.12. The Brandt semigroup S over a group G with index set I is the semi group consisting of all canonical I x I matrix units over G U {0} and a zero matrix ©. Writing S = {(g)ij : g € G,i,j £ 7} U {Q}, where (g)ij is the matrix with (k, Z)-entry equal to g if (k, I) = (i,j) and 0 if (k, I) ^ (i, j) we get . (9h)a iij = k {g)ij • {h)kl 9 j?k 15 Theorem 2.1.13 ([16], Theorem 7). Let S be the Brandt semigroup over a group G with finite index set I. Then t>}{S) is amenable if and only if G is amenable. For the situation in which the index set is infinite the result is false. Theorem 2.1.14 ([16], Theorem 12). Let S be a Brandt semigroup with an infinite index set over an arbitrary group. Then ^(S) is not amenable. 2.2 Amenability of £l(S, u) Definition 2.2.1. A weight (function) to on a semigroup S is a function from S to the positive reals, satisfying ui(st) < u(s)u(t) V t, s 6 S. Then ^1(5,o;) is the Banach space of functions / from S to C for which Yj \f(s)\uj(s) < oo, ses this sum being the norm of /, which we denote by ||/lim its dual can be identified with ^(SjW-1), the Banach space of functions C for which sup{-!—T-T-U < oo, the norm of (f> being this supremum. ses ^(s) We define the convolution of two functions f,g € ^(SjU) by f *g(s) — \] f(u)g(v). uv=s With multiplication taken to be convolution, 0} (S, u) becomes a Banach algebra. In [31] N. Gronbaek gives a complete description of amenability of £1(G,w), where G is an infinite group. fi{G,u)) is often called a Beurling algebra. Put Q(g) = L0(g)L0(g-1) (geG). Theorem 2.2.2 ([31], Theorem 3.2). Let G be a discrete group. Then the following are equivalent: 16 (i) 0}{G,u)) is amenable; (ii) I1 (G, Cl) is amenable; (Hi) there is a bounded net (TJ Hgfig)iei in fi{G,Sl) satisfying: 9 f^ lim(X>p-> 1; g CfejlimH^-zi'lln-O (h€G). (iv) there is a positive left-invariant mean on £°°(G, H-1); (v) G is amenable and sup{ £l(g) | g G G } < oo. Also N. Gronbaek obtained the following generalization of [1, Theorem 2.1]. Corollary 2.2.3 ([31], Corollary 3.3). If G is an abelian group then tl(G,u)) is amenable if and only if sup{w(£)cj(£-1) | g G G) < oo. In Theorem 2.2.2, on the basis of the amenability criterion obtained by A. Ya. Helemskii (see [34]) it was proved that an algebra £l(G,u>) is amenable if and only if the group G is amenable and the associated weight fl is bounded above. As a consequence of this assertion R.I. Grigorchuck obtained the following statement. Theorem 2.2.4 ([27], Theorem 2). The algebra I1 (G,u) is amenable if and only if the group G is amenable and the weight u> is equivalent to some multiplicative character x '• G —* R+. Thus, up to equivalence, the only amenable Beurling algebras are those of the form d}(G,x)> where G is an amenable group and x '• G —> K+ is a multiplicative character. 17 Similar remarks apply for weighted convolution semigroup algebras. In fact, most of the previous theorems are consequences of the following results due to J. Duncan and A.L.T. Paterson [17]. We first introduce some notation. [mz-1] = {x 6 S : xu = u}, [u_1it] = {x 6 S : ux = u}, X(u)=uSf)[uv,-1]. Theorem 2.2.5. Let S be a semigroup that contains an infinite pairwise disjoint sequence x of sets X{un). Then £ {S,uS) is not amenable for any weight function u. Corollary 2.2.6. Let S be an inverse semigroup with E infinite. Then ^(S, uS) is not amenable for any weight u. Corollary 2.2.7. Let S be a left cancellative semigroup with identity with£1(S,oj) amenable for some weight u>. Then S is a group. Corollary 2.2.8. Let S be an abelian semigroup with ^{S,^) amenable for some weight OJ. Then S is a finite semilattice of abelian groups. Theorem 2.2.9. Let S be a semigroup with ^(S,u>) amenable for some weight u>. Then S is a regular semigroup with E finite. G.H. Esslamzadeh introduced in [18] the Z^Munn algebras, defined as follows. Definition 2.2.10. Let A be a unital Banach algebra, I and J be arbitary index sets, .and P be a J x / nonzero matrix over A such that || P ||oo= sup{ \\ Pji ||: j € J, i € / } < 1. Let CM (A, P) be the vector space of all I x J matrices A over A such that Mlli= E !l^ll<°°- 18 Then £M(A,P) with the product AoB = APB, A,B e CM{A,P), and the ^-norm is a Banach algebra that is called the il-Munn I x J matrix algebra over A with sandwich matrix P or, briefly, the I1 -Munn algebra. G.H. Esslamzadeh proved the following results for weighted semigroup algebras but for simplicity he considered just the unweighted case. Lemma 2.2.11 ([18], Lemma 5.2). If S is a regular semigroup with E finite, then S has a principal series S = Si D S2 D S3 D ... D Sm D Sm+i — 0. Moreover for every k = 1, ...,m — 1 there are natural numbers rikJk, a group Gk and a regular Ik x nfc matrix Pk on G°k such that Sk/Sk+i = M°(Gk,Pk)- Also Sm = M(Gm,Pm) for some lm x nm matrix Pm over a group Gm. Theorem 2.2.12 ([18], Theorem 5.9). With the notations of Lemma 2.2.10 the following conditions are equivalent: (i) •£1(£) is amenable. (ii) CM(H1(Gk),Pk) has an identity and£1(Gk) is amenable, k = l,...,m. l 1 Definition 2.2.13. For a in a semigroup S, J(a) is the principal ideal of S aS and Ja is the set of elements b € J(a) such that J(b) — J (a). The inclusion among the principal ideals induces the following order among the equivalence classes Ja: Ja < J\, if J (a) C J (b) (Ja < Jf, if J(a) £ J{b)). Let 1(a) denote the ideal {b € J(a) : J(, < JG}, i.e., 1(a) = J(a)\Ja. The factors J(a)/I(a), a € S are called the principal factors of 5. Proposition 2.2.14. For a semigroup S, ^(S) is amenable if and only if S has a principal l series S = Si D S2 D S3 D ... D Sm D Sm+i = 0 and l (T) is amenable for every principal factor T of S. 19 G.H. Esslamzadeh also gave a generalization of [16, Theorem 8]. Theorem 2.2.15. Let S be a regular semigroup with a finite number of idempotents. The following conditions are equivalent: (i) ll{S) is amenable. (ii) every maximal subgroup of S is amenable and (}{T) is semisimple for every principal factor TofS. In particular if tl(S) is amenable, then it is semisimple. 2.3 Amenability of i1 ((3S) In this section we consider the two products • and 0 on the Stone-Cech compactification (3S of S such that ((3S, •) and ((3S, 0) are semigroups. They are the topics of the monograph [36]. Measure algebra M(G): Let G be a locally compact group. The measure algebra M(G) is the unital Banach algebra of all finite complex regular Borel measures on G, with the convolution product defined by (/, fi * u) = jG(JG f(gh)dn(g))dv(h), n, v € M(G) and / e C0(G), where Co(G) is the space of all continuous functions on G vanishing at infinity. H. G. Dales, F. Ghahramani and A. Ya. Helemskii proved in [11] that a measure algebra is amenable if and only if G is a discrete and amenable group. Recently it was proved in [12] that amenability and weak amenability of €J(5) is related to the amenability and weak amenability of M(j3S). 20 Proposition 2.3.1 ([12], Lemma 11.6). Let S be a semigroup such that M(@S,D) is amenable. Then fi{S) is amenable, S is amenable, S has a finite group ideal, E is fi nite, and each ideal in S is regular. Further i1 (f3S, •) is amenable. Theorem 2.3.2 ([12], Theorem 11.9). Let S be a semigroup such that £l((3S,0) is an amenable Banach algebra. Then S is finite. Definition 2.3.3. A semigroup S is weakly cancellative if for any a,b £ S, the sets {x € S : xa — b}, {y £ S : ay = b} are finite. Proposition 2.3.4 ([12], Proposition 11.13). Let S be a weakly cancellative semigroup. Suppose that M(@S,0) is weakly amenable. Then fi(S) is weakly amenable. Chapter 3 Weak amenability of £l(S) 3.1 Weak amenability of discrete semigroup algebras The notion of weak amenability for commutative Banach algebras was introduced by W. G. Bade, P. C. Curtis, Jr., and H. G. Dales in [1], and in the general case in [43]. Recall that a Banach algebra A is weakly amenable if every derivation D : A —• A* is inner (i.e. 3 x € A* such that D(a) = a-x-x • aV a € A). The question whether bounded derivations are necessarily zero has also been considered in [28] and [29]. As the name of concept suggests, weak amenability is derived from the stronger concept of amenability and a principal aim of the paper [1] was to exhibit classes of weakly amenable Banach algebras which are not amenable. It is noted that a commutative Banach algebra is weakly amenable if and only if H1 (A, E) = 0 for each symmetric Banach A-module E. Prom here on, the term weakly amenable will be abbreviated to WA. It is known that I1 (G) is weakly amenable for all groups G (in fact L1 (G) is WA for all locally compact groups 21 22 G [43]). For the case tl{G), Johnson [42] gives an explicit construction for the implementing element of the inner derivation. Grigorchuk [26, Remark 1.16] provides motivation for the study of cohomology over semigroups. Throughout S denotes a (discrete) semigroup. Definition 3.1.1. A (generalized) inverse of s G S is an element t € S such that sts = s, tst = t. If s has an inverse it is called regular and if not singular. A completely regular element is one for which there is t € S, sts = s and ts = st (then tst is an inverse for s). A semigroup is called (completely) regular if each of its elements is (completely) regular. The completely regular elements of a semigroup are those which lie in a subgroup. Completely regular semigroups are those which can be regarded as the disjoint unions of their maximal subgroups. T.D. Blackmore studied in [4] weak amenability of discrete semigroups algebras where S is completely regular and commutative. Theorem 3.1.2 ([4], Theorem 3.6 and [33], Corollary 2.8). If S is completely regular or is a commutative union of groups then £1(S) is WA. 3.2 Weak amenability of £l(S, co) where S is commutative T.D. Blackmore proved that for S commutative, certain conditions ensure that (.1{S, Proposition 3.2.1. If S is commutative and ^(S,UJ) is WA for some weight u then ^(S) is WA. 23 We denote the set of singular elements by Ns- Theorem 3.2.2. Suppose that S is a commutative semigroup and satisfies one of the fol lowing: i) There is s € Ns such that for all t € Ns\ {s}, s £ Sxt; ii) S contains one singular element (only); Hi) There exists non-empty M C Ns such that for all s,t € M, s e S1t and for all s € M and ueNs\M, s £ S1u; iv) There exists non-empty, finite subset M of Ns, such that if s € M and u G Ns\M then s i Sxu; v) S has a homomorphic image T such that NT ^ 0 is finite. Then 11{S,UJ) is not WA for any weight u. Proposition 3.2.3. Let S be a commutative semigroup andT a homomorphic image of S. Then 0{T) is WA if tl{S) is WA. Proof. The homomorphic image of a commutative WA Banach algebra is WA, • Proposition 3.2.4. If S is a commutative, finite semigroup then ^(S) is amenable if (and only if) it is WA. Theorem 3.2.5. Let S be a semigroup, possibly with zero 0. If S satisfies the conditions CI to C3 given below then P(S) is WA. CI. Whenever u,v,w,z e 5 are such that uv = wz ^ 0 then there is an a € S with v = az and w = ua. C2. Ifu,v,w,z£S\ {0} are such that vu = 0 and uv = wz then zw = 0. " 24 C3. Ifu,v£S\ {0} are such that uv = vu = 0 then there are b and c in S with u = be and cv = vb = 0. These conditions are satisfied if S is a Rees matrix semigroup. Hence: Corollary 3.2.6. If S is a Rees matrix semigroup then£l(S) is WA. Also in [33] N. Gronbaek characterized the weak amenability of ^(5,w) for certain commutative semigroups, in terms of the non-existence of homomorphisms from certain subsets of these semigroups into the semigroup (C, +) that satisfy a boundedness condition depending on u. Definition 3.2.7. On a commutative semigroup S we define the preorder s < t by t G s+S. We define the following sets: V(t) = {s\s V V(t)* = { / € C ® \ f(si + 82) - /(-Si) + /(S2), SX + S2 G V(t) }. Proposition 3.2.8 ([33], Proposition 4.2). Suppose that there exist t e S and f 6 V"(i)*\(0) such that , 1/001 , sup{ . , , , s + u = t\ = a < co. U)(S)UJ(U) ThenP(S,u) is not WA. Theorem 3.2.9 ([33], Theorem 4.7). Suppose that S is a commutative semigroup and satisfies one of the following: (a) S is a cancellative semigroup; (b) Every element of S is divisible by some n € N, n > 2; 25 (c) Ifu€ S\(0) then (V(u)) = S, where (V(u)) is the subsemigroup generated by V(u). Then for every weight u> : S —» M+, (}{S,UJ) is WA if and only if {/ e V(t)* | supi^^rrr I s + u = t} < 00} = {0} for all t A consequence of these results is that if G is an abelian group then ^1(Gr,w) is WA if and only if there are no non-zero homomorphisms, suP{-7MfL:,€G} See [33, Corollary 4.8]. In [48] it is shown that if G is an abelian group and the weight u satisfies for every g EG, then ^(G,^) is WA. 3.3 H1(£1{S)J0O{S)) and F1(^1(5')^1(^)) for some classes of semigroups In this section we present the results obtained by S. Bowling and J. Duncan who investigated in [3] two notions of cohomological triviality for Banach algebras: weak amenability and cyclic amenability. Definition 3.3.1. Recall that 5 is a Clifford semigroup if it is an inverse semigroup with each idempotent central (i.e. ex = xe for every idempotent e and every x in S), or equiv alent!^ if it is a (strong) semilattice of groups (see [37, Chapter IV]). So we can write the 26 1 Clifford semigroup as S = U{ Ge : e € E } where E is the semilattice of idempotents and each Ge is a group. B. E. Johnson and J. R. Ringrose proved in [39] that if G is a group then H1 (£l (G), i1 (G)) = {0}. An example given in [3] shows that this conclusion fails for Clifford semigroups, in general. We recall that a derivation D : ^(S) -> £°°(S) is cyclic if Ds[t] + Dt[s] = 0 for all s,t € ^(S). Every inner derivation is cyclic. We write Hl(£1(S),£°°(S)) for the bounded cyclic derivations modulo' the inner derivations and we say that £l (S) is cyclically amenable ifH{(£1(S),£°°(S)) = {0}. Theorem 3.3.2 ([3]). tf1^1^),*00^)) = {0} and Hl(£1(S),£0O(S)) = {0} for any Clifford semigroup S and for any Rees semigroup S. Definition 3.3.3. If A is a non-empty set, let us denote by FA the set of all non-empty finite words aia2.~am in the "alphabet" A. A binary operation is defined on FA by juxtaposition: (ai02...am)(&i&2-&m) = aia2...am&i&2-&m With respect to this operation FA is a semigroup, called the free semigroup on A. The set A is called the generating set of A. Example 3.3.1. The bicyclic semigroup is the semigroup Si = (e,p,q \ pq = e), Si = {qmpn\m>0,n>0}. Then ^(Si) is not weakly amenable [3]. It is proved that F1(^1(5i),^O0(5i)) ^ ^°°(N) and H{(£1(Si),£°°(Si)) = 0. 27 Let FC2 denote the free commutative semigroup on generators p, q and let F2 denote the corresponding free semigroup. It is also proved that H1(£1{FC2),£0O(FC2)), H{(£1(FC2),£0o(FC2)) ~ f°(N). The result clearly extends to any finite generators. In fact, for a semigroup S of a countable number of generators, the H1(£1(S),£°°(S)) is always isomorphic to £°°(N). The situation is similar for the non-commutative case but the argument is more difficult. Theorem 3.3.4. H1(£1(F2),£°°(F2)) ~ £°°(N), Hl(£1(F2),£0o{F2)) = 0 [30]. Theorem 3.3.5. i) Let S = l){Ge • e € E} be a Clifford semigroup with identity 1 and 1 1 1 suppose that eGi = Ge for every e€E. Then H (e (S),£ (S)) = 0. 1 1 l ii) LetS = U{Ge:eeE} be a Clifford semigroup with E finite. Then H (e (S),£ (S)) = 0. S. Bowling and J. Duncan showed that H1 (i1 (S), I1 (S)) = 0 for the bicyclic semigroup, the free commutative semigroup on two generators and the free semigroup on two generators, i.e. Theorem 3.3.6. ^(^(Si),*1^)) = H1(e1(FC2),£l(FC2)) = H1(£1(F2),f1(F2)) = 0. Chapter 4 Approximate amenability of £ (S) 4.1 Approximate amenability of a Banach algebra Based on the characterizations of amenability of a Banach algebra, F. Ghahramani, R.J. Loy and Y. Zhang have introduced several new notions of amenability. In particular, by dropping the requirement that aforementioned nets are bounded, definitions of approximate and pseudo-amenability were given [20, 22, 21]. The corresponding class of Banach algebras is larger than that of the amenable algebras. Definition 4.1.1. Let A be a Banach algebra and let E be a Banach yl-bimodule. A continuous derivation D : A -+ E is approximately inner if there is a net £a in E such that D(a) = limad^a(a) (a £ A), where the limit is taken with respect to the norm topology on E. When E is a dual module, D is weak*-approximately inner if the net converges with respect to the weak*-topology. Definition 4.1.2. Let A be a Banach algebra. 28 29 (i) A is approximately amenable if, for each Banach A-bimodule E, every derivation D : A —» E* is approximately inner. (ii) A is approximately contractible if, for each Banach A-bimodule E, every derivation D : A—> E is approximately inner. The qualifier sequential prefixed to the above definitions specifies that there is a sequence of inner derivations approximating each given derivation. Similarly, the qualifier weak* prefixed to the definition of approximate amenability specifies that the convergence is in the weak* topology over E*. Moreover, if the implementing net ad^a can be chosen to be bounded in an approximately amenable (contractible) Banach algebra, we call it boundedly approximately amenable (contractible). Of course, each amenable Banach algebra is approximately amenable. Some approximately amenable Banach algebras which are not amenable are constructed in [20] and [23]. Definition 4.1.3, Let A be a Banach algebra. A is weakly approximately amenable if every derivation D : A —* A* is approximately inner, where A* denotes the dual space of A with natural A-bimodule action. Definition 4.1.4. A Banach algebra A is pseudo-amenable if there is a net (ua) € A ® A called an approximate diagonal for A, such that limfa • ua — uaa) — 0 and lim7r(wQ)a"= a a a for each a €. A. All the notions of approximate amenability concern with the question of whether every derivation D : A —» E* is approximately inner. We know that when G is a locally compact group, amenability, approximate amenability, and pseudo-amenability coincide for the group algebra L1 (G) [20, 22]. 30 The Beurling algebra: Recall that a weight wona locally compact group G is a continuous function u : G —• (0, oo) satisfying uj{xy) < u(x)u>(y), (x,y € G). For a locally compact group G and a weight u> on G, L1^) = Ll{G, 1 w(g)=W(fl- ) (g£G). For any weight u, its symmetrization is the weight defined by • no^wG/Ms-1) (g£G). The following is essentially in [31]. Theorem 4.1.5. Let G be a locally compact group, u a weight on G with uj(e) = 1. The following are equivalent: (i) Lx(u) is amenable; (ii) L1(Q) is amenable; (Hi) G is amenable and 0 is bounded. Recently F. Ghahramani, R.J. Loy and Y. Zhang proved in [21] that the Theorem 4.1.5 is still true without condition o>(e) = 1. They also gave a direct proof of (iii) =*• (i) by constructing a diagonal for L1 (w). We can say more in the special case G = Z. Let u be a weight on Z such that w(0) = 1. Let .00 tl(u) = { a = (a(n) : n € 1) : ^ | a(n) | u(n) < oo }. —oo . 31 Then ll{u>) is a commutative Banach algebra with respect to convolution multiplication oo (a*b)(n) = ^a{n- k)b(k) (n e Z), —oo and the norm 00 || a ||= ^ I a(n) I w(n) (oG^(w)). —oo An algebra of this type is a Beurling algebra on Z. Theorem 4.1.6 ([1], Theorem 2.2). Let u be a weight sequence on Z such that ui{n)ui(—n) 0 n as n —*• oo. Then the Beurling algebra ll{u)) is weakly amenable. Theorem 4.1.7 ([1], Theorem 2.3). Let w be a weight sequence on Z such that , oj(m + n) . 1+ | n | . _ , sup{ ; , . ;(-—j—!—!—r):m,n€Z} is finite. Then the Beurling algebra C1 (w) is not weakly amenable. a Theorem 4.1.8 ([1]). Let ua(n) = (1+ | n \) (n e Z), a > 0. x (i) If a = 0, then t (ua) is amenable. l (ii) If a > 0, then i {oJa) is not amenable. (Hi) IfO x (iv) If a> 1/2, then t {(jja) is not weakly amenable. Theorem 4.1.9 ([21], Corollary 8.5). The Beurling algebras e1(Z,ui), oj{n) = (1+ | n |)Q with a > 0, are not sequentially approximately amenable. Proposition 4.1.10 ([21], Proposition 8.1). Suppose the weight u is bounded away from 0, and that L1 (G) is approximately amenable. Then G is amenable. 32 It is conjectured in [21] that L1(LJ) will fail to be approximately amenable whenever Q —> oo and a weaker result is proved. Suppose that G is a locally compact group, u a continuous weight on G. Define ib{x) = lim inf ^^ (x e G): r-»oo (j(r) Theorem 4.1.11 ([21], Theorem 8.4). Let u be a weight function on G. 1 , (1) Suppose that there is a net (ra) C G such that limra = oo and (w(r~ )u;(r a)) is bounded. a Then Ll{uj) is boundedly approximately contractible if and only if it is amenable; (2) Suppose that lim ui(x~l)ui{x) = oo. Then Lx{u) is not boundedly approximately x—+oo amenable. M. Lashkarizadeh Bami and H. Samea studied in [47] the approximate amenability of the discrete semigroup algebras &l(S) for left cancellative semigroups S. It is shown that a left cancellative semigroup S ( not necessarily with identity ) is left amenable whenever the Banach algebra il(S) is approximately amenable. The converse is not true. As a conse quence it is proved that if S is a finite semigroup and il{S) is approximately amenable, then S is amenable. Also for finite-dimensional Banach algebras A, approximate amenability and amenability are equivalent. Therefore, for a finite semigroup S, approximate amenability and amenability of £l(S) are equivalent. Corollary 4.1.12 ([47], Corollary 1.11). Let S be a Brandt semigroup over an amenable group G with infinite index set. Then ^(S) is approximately amenable but not amenable. The Corollary 4.1.12 shows that in general the approximate amenability of a semigroup 33 algebra is not equivalent to its amenability. In [7] I1 -convolution algebras of totally ordered sets are studied. Let A be a non-empty, totally ordered set, and regard it as a semigroup by defining the product of two elements to be their maximum. The resulting semigroup which is denoted by Av, is a semilattice. For every t £ Av denote the point mass concentrated at t by et. r a s an The definition of multiplication in^(Av) ensures that eset = emax(s,t) f° ll d *• Theorem 4.1.13 ([7], Theorem 6.1). Let T be any totally ordered set. Then tl{%) is boundedly approximately contractible. Theorem 4.1.14 ([7], Theorem 6.4). Let A be an uncountable well-ordered set. Then X £ (AV) is not sequentially approximately amenable. While sequential approximate amenability implies bounded approximate amenability, the converse is false from the previous two theorems. Other examples of semigroup algebras of the form ^(S) that are approximately amenable but not amenable are given in [12]. Example 4.1.1. Let 5 be the semigroup N with product mn — min{rn,n} and take AA = £l(S) with convolution product. Because A/\ is abelian and E(S) — S according to Theorem 2.1.8 (i), Ah is weakly amenable but not amenable [17, Theorem 2]. It is sequentially approximately amenable from Theorem 4.1.14. See also [21, Example 4.6]. 34 4.2 Amenability of tricyclic and partially bicyclic semigroups It is known that if i1 (5) is approximately amenable, then 5 must be a regular amenable semigroup [21]. In section 4.4 we will show that the converse is not true by examining the bicyclical semigroup Si. We prove that £1(5i) is not approximately amenable. In this section, we reveal the class of partially bicyclic semigroups. We have already defined the bicyclic semigroup in Section 3.3. It is the semigroup generated by a unit e and two more elements p and q subject to the relation pq = e. We denote it by Si = (e,p,q\pq = e). Many of its properties can be found in [5, §2.7]. The semigroup generated by a unit e and three more elements a,b and c subject to the relations ab = ac = e is denoted by S2 = {e) ai b, c I ab = e, ac = e); and the semigroup generated by a unit e and four elements a, b, c, d subject to the relations ac = bd = e is denoted by Si,i = (e,a,b,c,d\ ac = e,bd = e). 52 and 5i,i will be called partially bicyclic semigroups. J. Duncan and I. Namioka showed in [16] that Si is an amenable semigroup by studying the maximal group homomorphic image of Si. In [45] A.T.-M. Lau and Y. Zhang showed the same result directly by constructing a left and right invariant mean on ^°°(5i) and also proved that the partially bicyclic semigroups 52 and 5i,i are not left amenable and 52 is right amenable. 35 Because of the recent interest in approximate amenability, we shall give attention to this case. Theorem 4.2.1 ([21], Theorem 9.2). Let S be a semigroup such that fi(S) is approximately amenable. Then: i) S is regular; ii) S is amenable. Since the partially bicyclic semigroups 52 and 5i,i are not amenable, the Banach algebras 1 r l ^ (S 2) and £ (Siti) are not approximately amenable according to the previous theorem. In this chapter we will also give a direct proof of the fact that ^(S^) is not approximately amenable. 4.3 A characterization of approximate amenability of a Ba nach algebra The following theorem contains a list of conditions relating approximate amenability, ap proximate contractibility, pseudo-amenability, and the existence of certain diagonal-type nets. Theorem 4.3.1. (1) For a Banach algebra A the following statements are equivalent: (i) A is approximately amenable; (ii) A is approximately contractible; (Hi) A is weak* -approximately amenable; 36 (iv) the unitization A# of A is pseudo-amenable; (v) the unitization A# of A is approximately amenable; (vi) there are nets (m\) in A® A and (f\), (g\) in A such that for each a £ A, (a) a • m\ - m\ • a + f\ (b) af\ —> a, g\a —» a, and (c) •n{m\) -fx-g\-*0. (2) If A has a bounded approximate identity, then A is approximately amenable if and only if A is pseudo-amenable. In part (1), the equivalence of statements (i), (ii) and (iii) is [21, Theorem 2.1], the equivalence of statements (i) and (v) is [20, Proposition 2.4], while the equivalence of con ditions (ii), (iv), and (vi).is [20, Proposition 2.6]. Part (2) of Theorem 4.3.1 is [22, Proposition 3.2]. Also there is another characterization for approximate amenability of a Banach algebra: Proposition 4.3.2 ([13], Proposition 2.1). Let A be a Banach algebra. Then A is approx imately amenable if and only if, for each s > 0 and each finite subset S of A, there exist F £ A ® A and u,v £ A such that ir(F) = u4- v and, for each a £ S: (i) || a • F - F • a + u ® a - a ® v ||< e; (ii) || a — au ||< e and \\ a - va ||< e. We give a characterization for a the Banach algebras A to be approximately amenable. 37 Proposition 4.3.3. A Banach algebra A is approximately amenable if and only if the mapping D : A —> A#§>A# defined by D(a) — a Proof. One implication is straightforward since ker(7r) is an A-bimodule, therefore D is approx imately inner. Conversely, suppose that the derivation D : A —• ker(7r) is approximately inner. Then 3 £a G ker(7r) such that D(a) = lim(a • £Q — £a • a). Hence a — lima • (e <8> e — £a) (e <8> e — £a) • a = 0. Let ua — e We have that 7r(«a) = e. We write uQ = ^P a^ ® 6^ where ^ ||a„|| ||6^|| < oo. i=\ i=l Let us consider any derivation A : A# —*• X and the mapping ip : A#®A# -* X where X is a unital A#-bimodule and tp(a®b) = aA(b). We prove that A is approximately inner. For: OO 00 l a • i!>{ua) = aiPiJ^ < ® b a) = a £ <£A(6J,) t=l i=l oo oo 1=1 1=1 = ip(a • ua) = ip(a • ua — ua • a) + ^(ua • a). 00 00 i){ua • a) = TKE ai ® 6*,a) = ]£ oj,A(6j,a) j=i »=i oo = Y^( = A(a) + V'Cua) • a. 38 Combining these identities, we obtain: a • ip(ua) = ijj{a -ua-ua- a) + A (a) + ip(ua) • a and A (a) = lim(a • ip(ua) — i/)(ua) • a). Ot Thus A is approximately inner, therefore A* is approximately contractible and so A is approximately amenable using the previous theorem. • We can also give a shorter proof using Theorem 4.3.1. If D is approximately inner then 3 £a C ker(-7r) such that D(a) = lim (a • £a - £a • a), a £ A. Then a—>oo lim a(e <8> e — £Q) - (e Take tta := e ® e — £a. Then lim(a • ua - ua • a) = 0 and 7r(ua) = e. Therefore -A# is a pseudo-amenable and by Theorem 4.3.1 the Banach algebra A is approximately amenable. Proposition 4.3.4. Let S be a semigroup with generating set E. Then £l(S) is amenable (respectively weakly amenable) if and only if for every continu ous derivation D : f-(S) —> X* (respectively D : ^(5) —> i°°(S)) there exists £ m X* (respectively £°°(S)) such that D(6S) = 6S • £ - £ • <5S /or even/ s €. E. Proof. " =4>" is trivial. " <$=": 5 = (E). For each s G 5 let Z(s)=min{n: s is a product of n elements of E}. We prove D(5S) = 6S • £ — £ • S$ for all s G S by induction on n. 39 Suppose that V s G S with l(s) ^ n, D(ds) — Ss • £ — £ • Ss. Let t € S, l(t) = n + 1. We can write t = sx where l(s) < n, x € E. Then jD(<5t) - D(SSSX) = £>(<5S) • 5X + Ss • D(6X) = Ss • Sx • £ - £ • Ss • 5X = St • £ - £ • St. By linearity and continuity of D we then have D(f) = f-£ — £-fVf€£1(S), which means that I1 (S) is amenable. • Proposition 4.3.5. Let S be a semigroup with generating set E. Then ll{S) is boundedly approximately amenable if and only if for every continuous derivation D : C1 (S) —> X* there is a net (£j) in X* such that ad^ is bounded and D(SS) = lim(S3 • & — & • S3) for every i SEE. Proof. "=*>" is clear, so we only need to prove "<="'. Let A = ad^. Prom hypothesis || Dj || is bounded by a constant M > 0. For / G il(S) and e > 0, there exists a finite sum E™=i cn$s„ such that m 6 || / - Y, Cn sn ||< 1 + ||£,||+M- m TO Di(Y,Cn5Sn) -> DQTCAJ- n=l , n=l Hence there is «o such that for alH > io, m m £ II A-(E CnSsJ - D(£ *A») ||< 1+|Lp ||+M n=l n=l " " 40 So for i > io, II A(/) - D(f) || =|| A(£CAJ + Di(f - E^J ~ D(f) || n=l n=\ m m <|| A(E^n) - D(f) || + || A mi / - X>5Sn || n=l n=l mm m <|| A(E^) - ^(E^n) II + II ^(E^») - ^ n=l n=l n=l + M- 1 + ||D||+Af < , . „L . „ + II ^ II , . n ",, . ,, + M l + IPH + M 1 + ||£>|| + M 1 + ||I>||+M e. Therefore ]imDi(f) = D(f) (feiHS))- • 4=4 Approximate amenability of ^(Si), where 5i is the tri cyclic semigroup We consider the problem of approximate amenability for ^(S) where S is the bicyclic semigroup. We have the following result: Theorem 4.4.1. The Banach algebra ^(Sx) is not approximately amenable. Proof. In the proof we will use S to denote Si. Let TT : tx(S x 5) —> tx{S) be the product mapping. Consider the derivation D : tx(S) —>• ker(7r) defined by D(f) = f®5e-6e®f (feeHs)). 41 lf£1(S) is approximately amenable, then by Proposition 4.3.3 there exists a net (&) C ker(7r) C such that D(f) = lim(/ • £,; - & •/),/€ ^(S). Let & = E ™," If u £ S, u ^ e, the above implies up E cU - E cu,k = i uk=u ku—e c _1 lip E 4,« - E U = (*) ufc=e fcu=u Cfc n c lim m.fe t E 2L/ - JL* ^ (m,n)f^(u,e),(e,u) ufc=ra fcu=n Taking u = p we have : lipCe.e + cW^1 (4.1) nl _C li 4,P e,e = -l (4.2) Taking u = gwe have : Iip4,e-4,P = 1 (4.3) imc c = 1 (4.4) l e,e + e,gp We prove that lim ci e = 1. In the relation (*) take u = g and let i ' r = { (m,n) e S x 5 : m = «f+\n=/,r>l,rSN}. Then T C {(m,n) e S x S : (m,n) ^ (q,e),(e,q)}. 42 So li™ Y I Y2 cln - Y ckk \= °- (m,n)er qk=m kq=n If qk = gr+1 then k = qr, and if fcg = pr then k = pr+1. Hence oo ^Y I CV,Pr ~ cgr+1,Pr+1 l= °' and therefore YlfY^cV,PT ~ cV+1,Pr+1) = °" r=l c But /J(Cgr)Pr — cqr+itPr+i) = Cqp ( since the series 2J I m,n I converges which implies r=l m,n l lim c r+i „r+i =0). 7—too y >P ' Therefore, limcgp = 0. So, using relation (4.2) we have that liniCee = 1. i ' In the relation (*) take u — q and let T = {(m,n) £ S x S : m = qi+1pr ,n = qrpl : r > 1,1 > l,r,l &N}. Then TC{(s,t)eSxS:(s,t)^(p,e),(e,p)} and UP Y I 2 4,n - 2 4,fe 1= 0- (m,n)er qk=m kq=n If gfc = ql+1pr then A; = g'pr> and if kp — qrpl then k = 5rp/+1. Hence ^YY \Cq'pr,qrpl ~ Cg'+1Pr>9rP'+1 l= °' and so l>l r>l r>l Therefore, liJnII4p>-,O = 0- r>l In the relation (*) take u — qp and let r = { (m, n) € 5 x S : m = qpr+1, n = gr, r > 1, r G N }. Then r C { (s,t) € 5 x 5, (s,t) jk (qp,e), (e,qp) }. If qpk = qpr+1 then k = pr or k — qpr+1, r > 1. Also fegp = gr does not have any solutions k for r > 1. Therefore, r>l c r r an But since ^J c^^ = 0, c\e + ^2 p ,q ~ 0 d so lim^J Cprgr = — 1. So mn=e r>l r>l In the relation (*) take u=-qp and let r = { (m, n) € 5 x 5 : m = pr, n = qr+1p, r > 1, r Then r C { (a,i) € 5 x 5, (M) ^ (gp.e), (e.gp) }. The equation qpk = pr, r > 1 does not have any solutions. If kqp = qr+1p then k = qr OT k = qr+1p. Hence lim^|4Mr+4r(rflp|=0. r>l So arguing as before, we have r>l l Also we have that V] c mn = 0, so we get mn=qp cqp,e ~f" ce,qp + cg,p + 2—t CQPr+1,qr "*" A^ CP'\9r+1P "*" ^_/ Cqpr,qrp ~ ®' r>l r>\ r>\ We have also that limc^,e = limCe^, = limCqp = 0 by (4.1), (4.2) and (4.4). So using relations (4.6) and (4.7), r>\ which is a contradiction with relation (4.5). Therefore, C1 (S) is not approximately amenable. Since we couldn't answer to the problem if ^(Si) is or not approximately amenable we wander this fact. 45 Appendix We know from Theorem 4.2.1 that £1{S2) is not approximately amenable since 52 is not amenable, but the proof is not strightforward. In this Appendix we will give a direct proof to this result. Let 52 be the partially bicyclic semigroup. 52 = (e,p,vo,v\ \ pvo = pv\ = e) and consider the free semigroup of three generators F = (e,vo,vi). An element of 52 has the r form xp , x G F, r € N. Consider the set Ap — {VQ,V\}. We make the convention that every element to the power zero is the identity e. Denote l(x) the length of the element x £ F (i.e. example x — vfy^vi has the length 6 and we write l(x) = 6, 1(e) = 0). The following three claims are true for partially bicyclic semigroup 52- Claim 1 xpr ^ e, V r > 1, x G F. Proof. Suppose by contradiction that xpr — e. Multiplying to the right with vr we get r x = v V v € Ap. This implies that VQ = v\. If r = 1 we have VQ = v\ which is a contradiction. If r > 2 in the relation VQ — v\ multiply to the left with pr_1 we get vo = v\ which is a contradiction. Therefore, xpr ^ e, Vr > 1, x G F. • Claim 2 Let irreducible a, f3 € 52, ap = e. Then a = pr for some r G N and (3 € F. 46 Proof, a,(3 € S2 => a = xpr,(3 = ypl for some r,f€N and x,y £ F a(3 — e =*• xprypt = e Suppose t > 1. Then xprypt — zps for some z £ F,s> 1. Therefore, zps = e which is a contradiction with Claim 1. Hence t = 0 (3 = y£F a(3 = xpry = e • If Z(y) =r=4>a; = e=^a = pr. • If /($/) < r =*> :rpry = xpT~1^ = e which is a contradiction with Claim 1 since r — /(#) > 1. • If l(y) > r in the relation xpry = e =*• 2 = e for some z £ F, l(z) > 1. In the relation z = ewe multiply to the left with p'^) and we get pl^z> — e which is a contradiction with Claim 1. • Claim 3 Let x,y € F such that there exists v £ Ap, xv = yv. Then x = y. Proof. For suppose that x ^ y. If l(x) ^ l(y) then for sure we have contradiction with Claim 1 by multiplying with some pmaxWx)Mv)}+^ to the left getting pl'(z)_%)l = e. Therefore, l(x) = l{y). Then there exist 0 < n < l(x), n € N such that multiplying xv = yv with pn to the left we 47 get VQZ = V\Z for some z € F. • But V0S2 H ^1^2 = 0. Hence we have a contradiction. • The following six results are true for 52 = {e,p,VQ,V\ \ pvo = pv\ = e). F = {e, vo, v\), Ap = {v0, VI}, V a, (3 G 52, a(3 = e and V v € Ap. Lemma 1 { k \ vopk = vopa } C { a } U { vpa | v € Ap }. Proof. For, suppose that vopk = vopa has the solution fc = xp* for some 16F and t 6 N. From Claim 2, a = pr for some r £ N. vopk = vopa 43- vopxp* = vopr+1 43- pxp* = pr+1 (1) i+1 r+1 • If l(x) = 0 thenp = p =4- i = r (otherwise multiplying to the right with vmin{r+l,t+l} we get contradiction with Claim 1). So, t — r =$• k = pr = a =$• k = a. • Suppose l(x) = 1. Then p* = pr+1. Therefore by the same argument t = r + 1 which implies that k = vpr+l = vpa • Suppose l(x) > 2. Then in the relation (1), ypt = pr+1,l(y) > l,y € F,y = px. In the relation ypl = pr+1 multiply to the left with p1^ and we get p* = pl(v)+r+1. By the same argument t = l(y) + r + 1. But from (1) pxp* = pr+1 =$> pccp^i''+r+1 = pr+1 =*• pxp1^ — e. Therefore j/p'M = e which is a contradiction by Claim 1 since l(y) > 1. D Lemma 2 For each k € 52, /rap 7^ /? and wpA; ^ a Vv G. Ap. Proof. For suppose, that there exists k € S2 such that fe = xpr (x € F and r € N) then kvp — fj =*• xprup = /?. We showed in Claim 2 that j3 £ F. . 48 Multiplying to the left with p1^ we get p"^xprvp = e-& zp = e for some z € F. But this is a contradiction with Claim 1. Therefore, kvp ^ /? V v G Ap. For the second equation suppose that vpk = a =» vpk = /, r £ N since a = pr by Claim 2. Let consider the solution k = xpf, x € F, t € N. vpzp* = pr (2) • If l(x) = 0 then the relation (2) <^> upt+1 = pT. Multiplying to the left with p we get pt+i _ pr+i _^ ^ _ r gQ ^e reiation (2) «=>• upp* = j?' =*• vp — e which is a contradiction with Claim 1. • If l(x) = 1 then the relation (2) becomes vp1 = pr =$> p* = pr+1 =4> t = r + 1 (otherwise mm t r 1 we have contradiction with Claim 1 since if we multiply to the right with v i > + } we get pa = e for some s > 1). So the relation (2) =*> vpr+1 = pr =$• vp = e which is a contradiction with Claim 1. • If l(x) > 2 then the relation (2) =*> vyp1 = pr,l(y)>l. Multiplying to the left with pl+l(v) we get that vyp1 = pr 44- pl = pT+l(v)+l •& t — r + 1 + Z(y) (otherwise we have contradiction by the same argument). So, vyp1 — pr <$ vypr+1+l^ = pr •& vyp1+l<-^ = e which is a contradiction with Claim-1. Therefore vpk ^ a Vv € Ap. D Lemma 3 { k | fcvp = /?vp } = { ft, (3vp } 49 Proof. For suppose, that k = xpr for some x € F and r € N is a solution of the equation kvp = (3vp. xprvp = (3vp (3) If we multiply relation (3) with some v € Ap to the right we get xprv — j3v (4). • If r = 0 then the relation (4) becomes xv = (3v =• x = f3 by Claim 3. • If r > 1 then (4) 4*- arpr-1 = /3v. Multiplying to the right with vr~l we get x = f3vr Hence xpT = (3vTpT. So from (3) (3vTpTvp = (3vp. Multiplying with pW> to the left deduce that vrpr = vp <3> vr~1pr~1 — e. If r > 2 we have contradiction with Claim 1. So r = 1. Therefore, k = /3vp. We proved that { k \ kvp = j3vp } = {/?, f3vp }. D Lemma 4 For each r e N*, { k \ kv = (3pr } = {/3pr+1}. Proof. For suppose k — xpl for some x £ F and i € N is a solution of kv — ppr, xptv — fipr (5) • If l(x) = /(/?) then in the relation (5) multiplying to the left with p1^ we get pfv = pr. lit — 0 then v = pr <$• pr+1 = e (multiply withp to the left). This is a contradiction with Claim 1. Hence t > 1. p'v == pr •$=>• pt_1 = pr «> i — 1 — r<^i = r + l. Therefore relation (5) becomes • If /(a;) > Z(/3) then we multiply relation (5) to the left with p1^ and we get yptv = pr (6) 50 for some y €E F, l(y) > 1. If t = 0 then yptv = pr •& yv — pr •&• yvr+l = e ( the last equality is obtained multiplying the previous one to the right with vr). So, yvr+x = e which is a contradiction. ( we get pl{y)+r+i _ e which is a contradiction with Claim 1). Hence t > 1. Therefore, yptv — pr ^ ypl~l = pT. Multiplying with p1^ to the left we get pl~x = pr+l(y) & t - 1 = r + l(y) •» t = 1 + r + l(y). So the relation (6) becomes ypi+r+i(y)v = pr ^. ypr+i(y) _ pT ^ ypi{y) _ e which is a contradiction with Claim 1 since Kv) > i. • If l(x) < /(/?) then we multiply relation (5) with p1^ to the left and we get plv = ypr (7) for some y € F, l(y) > 1. Multiplying the relation (7) with p1^ to the left we get plM+tv = pr •&• p^1^'1 = pr <& t + l(y) - 1 = r =*» l(y) = r + 1 -1. But l(y) > 1. Therefore r+l-t>1 In the relation (7) we have that ptv = ypr. If t = 0 from (7) we have that v = ypr. Multiplying with p to the left we get e = zpr for some z 6 F. But r > 1 and so we have a contradiction with Claim 1. If t > 1 the relation (7) imply that p*_1 = ypr. Multiplying with w*-1 to the right we get e = ypr~t+x. This is a contradiction with Claim 1 since we proved that r + 1 — £ > 1. • Lemma 5 i) {k\pk = p} C {e} \J{vp \ v £ Ap}; ii) {k | kv — v} = {e} U{'UP}- 51 Proof, i) Suppose k = xpf, (x € F, t € N) is a solution of equation pk = p. Then • If l(x) = 0 then t — 0 and A; = e • If Z(x) = 1 then t = 1 and k — vp • If Z(x) > 2 then yp* = p, y € F, l(y) > 1. Then ypt_1 = e which is a contradiction with Claim 1. (If i = 0 we have y — p =» pl^+l = e which is a contradiction with Claim 1 and if t = 1 we have y = e =^ p'^' = e which contradict Claim 1. ii) we want to prove that {k \ kv = v} = {e} \J{vp}. Suppose k = xp1, (x G F,t G N) is a solution of equation kv = v. Then xptv = v • If t = 0 and l(x) = 0 then k = e. If l(x) = 1 then multiplying with p to the left we get v = e which is a contradiction with Claim i. If l(x) > 2 then multiplying to the left with pl(x)+1 we get p1^ = e which is a contradiction with Claim 1. • If t = 1 then x = v =» k = vp • If t > 2 then xpt-1 = v. Multiplying with p to the left we get ypt~1=e for some y € F which is a contradiction with Claim 1 since t - 1 > 1 . • Lemma 6 {(m,n) G 52 x 52 | ran = vp} C {(vpa, /?), (va, j3p), (a, (3vp)} Proof. We want to show that the only decomposition of vp is vp = vpe — evp = vep 52 For, suppose that vp = xpl (8) x£F,t€N If l(x) =0 and t = 0 we have vp = e which is a contradiction with Claim 1. If l(x) = 0 and t > 1 then vp = p*. Multiplying with p to the left we have p = pt+1 => pf = e which is a contradiction with Claim 1. If l(x) > 1 and t = 0 we have up = z. Multiplying with p1^ to the left we get p1^ = e which is a contradiction with Claim 1. If l(x) > 1 and t > 1 , multiplying relation (8) with v to the right we get v^xp*'1 (9). If t = 1 we have u = If t > 2 multiply the relation (9) with ut_1 to the right and we get v* = x. In the relation (8) from the beginning we have vp = vtpt, t > 2. This imply vt~1pt~1 — e which is a contr. with Claim 1 since t — 1 > 1. • We summarize the previous six lemmas in the following Proposition: Proposition Let 52 be the partially bicyclic semigroup. Denote Ap = {VQ, v\}, \AP\ = 2. The following six conditions are fulfilled for every a, {3 € S2, a(3 = e, V v € Ap : (i) { k I vopk = vopcn} C {a} U { vpa \v £ Ap}; (ii) For each k G 5, fcup ^ /? and upfc ^ a; (iii) { A; | kvp = pvp } = {/3, j3vp }; (iv) For each r e N*, { k | feu = /3pr } = { /?pr+1 }; (v) {fc|pfc = p}C{e}U{up|uGAp}; {fc|fc?; = w} = {e}U{vp} ; 53 For each k £ S, kp ^ e and vk =£ e; (vi) { (m,n) | ran = vp} C { (vpa,/?), (va,/3p), (a,/3up) \ a(3 = e}. then the semigroup algebra ^(S^) is not approximately amenable. Based on the proof of non-approximate amenability of fi(S) where S is the bicyclic semigroup we will try to prove similarly that Z1^) is not approximately amenable. Theorem The Banach algebra Z1^) is not approximately amenable. Proof. Note: In the proof when we are using conditions (i)-(vi) are in fact conditions from the above proposition. Let 7r: ^(S x S) —> £1(5) be the product mapping, consider the derivation D : £x(5) -> ker(7r) defined by D{f) = f®5e-5e®f (f£l\S)). If £1(5) is approximately amenable, then there exists a net (&) C ker(7r) such that D(f) = Let & = Y, cm,n^"i,ii where cj^^ satisfy ^ ^ cln,n = 0 ^or each £ € 5, and ^ ^ | cj^^ |< oo. m,n mn=t m,n Then for every u £ 5, we have : linl C c <*u,e - Se,u = lim]TcJn,n( The convergence is in the norm topology of i1 (S x S) 54 If u € S, u 7^ e, the above implies 1 c = l TY 4,e- X! U- ufc=u feu=e lim c c = _1 X^ l« ~ Y lk (*) uk=e ku=u (m,n)^(u,e),(e,u) uk=m ku=n Take u = uo, we have limcee - cj, p = 1. We will prove that lirnCge = 1 Take U = VQ and let r +1 r T = {(m,n) e S x S : m = v 0 ,n = p ,r> l.reN}. where N denotes the set of all positive integers. Then PC {(m,n) £ S x S : (m,n) ^ (v0, e), (e, v0) } lifi Y i Y cU - Y c™* i= o (m,n)er vok—m kvo=n r The sequence (vo,p), (VQ,P2), ..., (vo,p ) is infinite since p and i>o have infinite order: Suppose, by way of contradiction, that ph+k'" = ph for some positive integers h and k. Mul tiplying on the right by vft, we obtain pk = e. Then vo = ei>o = pkvo = pk~1e = pfc_1 and «oP = pk — e, which is a contradiction with Claim 1. the equation vok = VQ+1 =*• k = VQ and kvo = pr => k = pr+1 by (iv). 00 OO Precisely lim ]T I ^^-^+1^+1 1= 0 and this implies that limX](45,p'--c^+i,pr+i) = 0. r=l r=l oo But ^(C^ - Cvr+\pr+l) = <0,P l Note (the series ^ | c^ | converges then lim c qr+i^r+i = 0), therefore limc^op = 0. So we have that limce e = 1. " 55 In the relation (*) take u = vop , VQ is fixed and let r = {(m, n) £ S x S : m = vopa, n = j3 : (a, (3) ^ (e, e) } Then r C {(s, t)£SxS:{s,t)^ (vop, e), (e, v0p) } and (m,n)er uopfc=m kvop=n The function (a, /?) -+ (m, n) is injective. For suppose vopa = vopai and a/3 = aifS = e. If ai T^Q then by (i) ai = vpa for some i; € Ap. Therefore -up = vpa(3 — ai/3 = e which is a contradiction. So a\ = a. the equation vopk = vopa have solutions k = a and A; G {vpa | u G Ap} due to (i) and kvop = j3 do not have solutions due to (ii). Therefore, "f1 £ «/J+£cW> = 0 (4.8) a/3=e veAp (a,/3)?4(e,e) In the relation (*) take tt = vp , v G Ap is fixed and let I" = { (m, n)eSxS:m = Q,n = 0vp : (a, 0) ^ (e, e) } Then T' C {(*,t) e S x S : (s,t) J= (vp,e), (e,vp)} and "f1 £ i £ cU - 2 ckk i= ° (m,n)sr vpk=m kvp=n 56 the equation vpk = a do not have solution due (ii) and kvp = (3vp have two solutions k — (3 and k = flvp due to (Hi). The function (a,0) —• (m,ri) is injective. For suppose (3vp = /?iup and af3 = a/?i = e. If /?i 7^ /? then by (iii) /?i = 0vp. Therefore vp = aflvp — a(3\ = e which is a contradiction. So A = /?. Therefore, "f1 E (CM + CU) = ° (V«G^,) (4.9) a,3=e (a,/3)#(e,e) and we have \Ap\ relations. So, if we are adding relations (4.8) and (4.9) and use the last condition (vi) : c C = 0 lim[(|,4p| + l)( £ <*)+£( £ U/3+ E W>] cfc/3=e i;Gylp aft=e a/3=e (.a,/3)^(e,e) (a,/3)^(e,e) (a,/3)^(e,e) C C C C c lim[(|.4p| + 1)( E kn ~ 4,e) + E ( E kn ~ E va,0P ~ vp,e ~ Up)} = 0 ran—e veAp mn=vp a/3=e C Hm[(\Ap\ + 1)( E 4,n - Taking in the relation (*) u = v we have: r" = { {m,n) £SxS:m = vr+1a,n = (3pr : a(3 = e,r € N*} Then T" C {(m, n) G 5 x 5 : (m, n) ^ (v, e), (e,«)} since urpr ^ e V r 6 N. lT E I E 4,n - E Cm,fc 1= 0 (m,n)er vk=m kv=n The equation vfc = ur+1a has the solution k = i/a and the equation kv = /3pr has the solution k = /3pr+1 due to (iv). The function (a, (3) —• (m, n) is also injective and the r r sequence (v a,(3p )r is infinite. 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