FIXED POINT SET FOR SEMIGROUP OF MAPPINGS

ON BANACH SPACES RELATED TO HARMONIC ANALYSIS

Anthony To-Ming LauUniversity of Alberta

Outline of Talk

1. The general problem

2. Amenable semigroups

3. Locally compact groups

4. Fixed point set in the group von Neumann algebra V N (G)

5. Metric semigroup

6. Fixed point set of power bounded elements in the Fourier-Stieltjes algebra B(G)

7. Weak∗-fixed point properties on B(G)

2

1. The general problem

Let K be a non-empty compact convex subset of a separated locally convex

space and S = {Ts : s ∈ S} be a semigroup of continuous mappings from K into

K.

Problem 1. When is the fixed point set of S

F (S) = {x ∈ K; Tsx = x for all x ∈ S}

non-empty?

Problem 2. Suppose we know that F (S) is non-empty, what can we say about

elements in F (S)? or span F (S)?

3

• Schauder’s Fixed Point Theorem: If S is the free commutative semigroup

on one generator, then F (S) 6= ∅.

• Markov-Kakutani Fixed Point Theorem: If S is commutative and affine,

then F (S) 6= ∅.

• W.M. Boyce (1969, TAMS). There are two continuous functions f and

g : [0, 1] → [0, 1] which commute under composition, and do not have a common

fixed point.

4

2. Left amenable semigroups

A semigroup S is left amenable if `∞(S) = space of bounded complex-valued

of S has a left invariant mean m i.e. m ∈ `∞(S)∗, ‖m‖ = m(1) = 1 and

m(`af) = m(f) for all a ∈ S, and f ∈ `∞(S).

For a convex subset C of a vector space, a map Λ : C → C is said to be affine

if

Λ(tx+ (1 − t)y

)= tΛ(x) + (1 − t)Λ(y)(x, y ∈ C, t ∈ [0, 1]).

Theorem (Day, 1961). A semigroup S is left amenable if and only if S has the

following fixed point property:

(∗) : Whenever S = {Ts; s ∈ S} is a representation of S as continuous affine

mappings on a non-empty compact convex subset K of a separated locally convex

space, then there is a x0 ∈ K such that Ts(x0) = x0 for all s ∈ S.

5

Corollary. Any commutative semigroup is left amenable.

A (discrete) semigroup group S is left reversible if aS∩bS 6= ∅ for any a, b,∈ S.

Examples:

• S commutative or left amenable =⇒ S is left reversible.

• Any group is left reversible.

• A finite semigroup is left amenable ⇐⇒ S is left reversible.

• The solvable group: G = all 2× 2 matrices

[x y

0 1

], x, y ∈ IR, x 6= 0 with

matrix multiplication (called “ ax+ b ” group), is left amenable.

• Free semigroup, or free group on 2 generators is not left amenable.

6

K = {m ∈ `∞(S)∗ : ‖m‖ = m(1) = 1}, the set of means on `∞(S).

S = {`∗s; s ∈ S}, 〈`∗sm, f〉 = 〈m, `sf〉, s ∈ S, m ∈ K.

F (S) = set of left invariant means on `∞(S)

Then S is left amenable ⇐⇒ F (S) 6= ∅.

Theorem (E. Granirer, Illinois J. Math., 1963, M. Klawe, TAMS, 1977). Let S be a

left amenable semigroup. Then dim 〈F (S)〉 = n⇐⇒ S contains exactly n finite left

ideal group

In particular, if S is a group, then dim 〈F (S)〉 < ∞ ⇐⇒ S is a finite group.

In this case, `∞(S) has exactly one left invariant mean.

7

Theorem 1 (Lau, 1976, PAMS). Let E be a dual Banach space with a fixed predual

E∗ i.e. (E∗)∗ = E. Let S be a semigroup of linear operators from E into E

satisfying:

(i) ‖sx‖ ≤ ‖x‖ for all x ∈ E, s ∈ S.

(ii) Each s ∈ S is a continuous linear map from (E, weak∗) into (E,weak∗).

Let G ⊆ co W ∗OT (S) be a closed subsemigroup of co W ∗OT (S) in the W ∗OT

(weak∗ operator topology). Let X be a G -invariant subset of E. Let K be a

weak∗-closed subset of F (S) = {x ∈ E; s(x) = x for all s ∈ S}. If G(x) ∩K 6= φ

for each x ∈ X, then there exist P ∈ G such that for each x ∈ X, P (x) ∈ K.

8

A translation invariant subspace X ⊆ `∞(S) is called left introverted

(M.M. Day 1957) if for each f ∈ X, φ ∈ `∞(S)∗, the function φ`(f)(s) = 〈φ, `sf〉,

s ∈ S, is in X.

Example. Let S be a semitopological semigroup i.e. S is a semigroup with a

topology such that for each a ∈ S, the mappings s → a · s, and s → s · a are

continuous. Let LUC(S) = {f ∈ CB(S); s → `sf is continuous when CB(S) has

the ‖ · ‖∞ -topology }. Then LUC(S) is left introverted.

Corollary (Granirer-Lau, 1971, Illinois J. Math.). Let X be a translation invariant

left introverted subspace of `∞(S). Then X has a left invariant mean ⇐⇒

(∗) for each f ∈ X, there exist m ∈ X∗, (depending on f), ‖m‖ = m(1) = 1,

and 〈m, `af〉 = m(f) for all a ∈ S.

Open problem 1: Does condition (∗) imply X has a left invariant mean when X

is not left introverted?

Example. The space CB(S) is in general not left introverted in general.

9

3. Locally compact groups

A topological group (G, T ) is a group G with a Hausdorff topology T such

that

(i) G×G→ G

(x, y) → x · y

(ii) G→ G

x→ x−1

are continuous.

G = topological group then

LUC(G) = all f ∈ CB(G)

such that for any ε > 0, there exists a neighbourhood U of the identity e such that

|f(x) − f(y)| < ε if yx−1 ∈ V, x, y ∈ G

(i.e. f is uniformly continuous with respect to the right uniformity of G).

10

A topological group G is (left) amenable if LUC(G) has a left invariant

mean.

• (M. Gromov and V. Milman, American J. Math. 1983).

G = group of unitary operators on a Hilbert space with strong operator topology.

Then G is extremely amenable i.e. LUC(G) has a multiplication left

invariant mean.

In the case G has the following fixed point property (T. Mitchell, Trans. A.M.S.

1970): Whenever G acts on a compact T2-space X, then X contains a common

fixed point for G.

(See Lau-Ludwig: “Fourier Stieltjes algebra on topological groups”, Advances of Math.

2012 for more examples.)

11

G is locally compact if the topology T is locally compact i.e. there is a basis

for the neighbourhood system of the identity consisting of compact sets.

Ex: Gd, IRn, (E,+), T, Q, GL(2, IR), E = Banach space, T = {λ ∈ C; |λ| = 1}

Note that: (Q,+) is not locally compact.

(E,+) is locally compact if and only if E is finite dimensional.

Examples: Compact groups, abelian groups are amenable.

Let G be a locally compact group.

• There is a left Haar measure λ on G i.e. λ(E) = λ(gE) for each g ∈ G and

Borel subset E ⊆ G.

• “λ” is unique up to multiplication by positive constant.

Examples:

• If G = (IR,+), λ = Lebesgue measure.

• If G = discrete group, λ = counting measure

i.e. λ(E) = |E| if E is finite, λ(E) = ∞ if E is infinite.

12

Theorem 2 (Graniner and Lau 1971). Let G be a locally compact group. If G is

extremely amenable, then G = {e}.

A map S : X → X is affine-linear if there exist a (complex) linear map

S1(x) : X → X as well as an element xS ∈ X such that S(x) = S1(x)+xS (x ∈ X).

The following theorem can be regarded as a variant of Day’s fixed point theorem

of amenable groups concerning affine-linear actions on a dual Banach space E∗ rather

than affine actions on weak∗-compact convex subsets sets of E∗.

Theorem 3 (Chen-Lau-Ng, Studia Math. 2015). G is amenable if and only if any

weak∗-continuous affine-linear action α of G on any dual Banach space E∗ with one

norm-bounded orbit (and equivalently, with all orbits being bounded) has a fixed point.

It should be noted that “affine actions” in this result cannot be replaced by “linear

actions” since any linear action always has a common fixed point, namely “0”.

13

Let G be a locally compact group.

L1(G) = group algebra of G i.e. f : G 7→ C measurable such that

∫|f(x)|dλ(x) <∞

(f ∗ g)(x) =

∫f(y)g(y−1x)dλ(y)

‖f‖1 =

∫|f(x)|dλ(x)

(L1(G), ∗

)is a Banach algebra i.e. ‖f ∗ g‖ ≤ ‖f‖ ‖g‖ for all f, g ∈ L1(G)

L∞(G) = essentially bounded measurable functions on G.

‖f‖∞ = ess - sup norm. = inf{M : {x ∈ G; |f(x)| > M is a locally null set}

}

L∞(G) is a commutative C∗-algebra containing CB(G)

L1(G)∗ =L∞(G) : 〈f, h〉 =

∫f(x)h(x)dλ(x)

14

G = locally compact abelian group. Then L1(G) is a commutative Banach alge-

bra.

A complex function γ on G is called a character if γ is a homomorphism of G

into (T, ·).

G = all continuous characters on G

⊆ L∞(G) = L1(G)∗.

Example

G = IR G = IR

G = T G = Z

G = Z G = T.

Equip G with the weak∗-topology from L1(G)∗ (or the topology of uniform

convergence on compact sets). Then

G with product: (γ1 + γ2)(x) = γ1(x)γ2(x)

is a locally compact abelian group called the “dual group” of G.

15

Let G be a locally compact group.

L2(G) = all measurable f : G→ C∫

|f(x)|2dλ(x) <∞

〈f, g〉 =

∫f(x) g(x)dλ(x)

L2(G) is a Hilbert space.

Left regular representation:

{ρ, L2(G)},

ρ : G 7→ B(L2(G)

),

ρ(x)h(y) = h(x−1y), x ∈ G, h ∈ L2(G).

16

Let G be a topological group.

A continuous function φ : G → C is positive definite if ∀ λ1 . . . λn ∈ C ,

x1 . . . xn ∈ G

n∑

i,j=1

λiλjφ(xix−1

j ) ≥ 0

⇐⇒ ∃ a continuous unitary representation {π,Hπ} of G, ξ ∈ Hπ, such that

φ(x) = 〈π(x)ξ, ξ〉 ∀ x ∈ G

⇐⇒ φ is of positive type if G is locally compact

i.e. 〈φ, f∗ ∗ f〉 ≥ 0 ∀ f ∈ L1(G), f∗ =1

∆(x)f(x−1)

∆ = modular function of G; ∆(x) =λ(Ex)

λ(E), x ∈ G,

E a Borel subset of G, 0 < λ(E) <∞.

17

P (G) = all continuous positive

definite functions on G

⊆ CB(G)

(P (G), ·

)is a commutative semigroup with pointwise multiplication.

Let B(G) = 〈P (G)〉 . Then B(G) is a translation invariant subalgebra of CB(G).

B(G) is called the Fourier Stieltjes algebra of G.

18

We now assume that G is locally compact.

C∗(G) = completion of(L1(G), ‖| · ‖|

)

(Group C∗-algebra of G)

‖|f‖| = sup {‖π(f)‖ : π continuous unitary representation of G} ≤ ‖f‖1

〈π(f)ξ, n〉 =

∫〈π(x)ξ, n〉f(x)dλ(x) ξ, n ∈ Hπ ,

f ∈ L1(G).

If G is amenable, then

‖|f |‖ = ‖ρ(f)‖, where ρ(x)h(y) = h(x−1y), h ∈ L2(G).

19

For φ ∈ B(G) = C∗(G)∗

‖φ‖ = sup {|〈φ, f〉| : f ∈ L1(G), ‖|f‖| ≤ 1}

≥ ‖φ‖∞ (sup norm)

(B(G), ‖ · ‖

)is a commutative Banach algebra

A(G) = B(G) ∩ Cc(G)‖·‖

B(G)⊆ B(G)

(Fourier algebra of G) where Cc(G) are continuous functions f : G → C with

compact support.

A(G) is called the Fourier algebra of G.

A(G) = {All φ : G → C of the form φ(x) = 〈ρ(x)h, k〉 , h, k ∈ L2(G), x ∈ G} ⊆

C0(G).

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– A(G) is a closed ideal in B(G)– A(G)∗ = V N (G)

= 〈 ρ(x) : x ∈ G 〉WOT

⊆ B(L2(G)

)

When G is abelian, then A(G) ∼= L1(G) V N (G) ∼= L∞(G)

B(G) ∼= M (G) C∗(G) ∼= C0(G)

Example. G = T = {λ ∈ C : |λ| = 1}, G = (Z,+).

A(T) ∼= `1(Z), V N (T) = `∞(Z)

B(T) ∼= M (Z) = `1(Z), C∗(T) ∼= c0(Z).

References. (1) P. Eymard, Bull. Soc. Math. France 92 1964.

(2) E. Kaniuth and A.T.-M. Lau, Fourier and Fourier-Stieltjes Algebras on Locally

Compact Groups, A.M.S. Survey and Monograph 2018.

21

4. Fixed point set in the group von Neumann algebra V N (G)

For σ ∈ B(G), T ∈ V N (G), define σ · T ∈ V N (G)

〈σ · T, ψ〉 = 〈T, σψ〉, ψ ∈ A(G).

Let Iσ = {σφ− φ : φ ∈ A(G)}‖·‖

⊆ A(G).

Then

(i) Iσ is a closed ideal in A(G)

(ii) I⊥σ = {T ∈ V N (G) : σ · T = T} = fixed point set of σ

(σ-harmonic functionals on A(G) : see Chu-Lau, Springer Verlag Lecture Notes

(2002), Math. Annalen (2006)) and Math. Zeitschrift (2011))

is a weak∗-closed subspace of V N (G)).

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(iii) If σ ∈ P 1(G) = {φ ∈ P (G); φ(e) = 1}, then H = {x ∈ G; σ(x) = 1} is a closed

subgroup of G, and

Fixed point set of σ = I⊥σ = V NH(G)

= 〈ρ(h) : h ∈ H〉W

∗

⊆ V N (G).

In particular I⊥σ is a W ∗-subalgebra of V N (G).

Theorem (Choquet-Deny 1960). Let G be a locally compact abelian group. µ be

a probability measure on G, i.e. µ ∈M (G), µ ≥ 0 and ‖µ‖ = 1. Then TFAE:

(i) support µ generates a dense subgroup in G

(ii) any µ-harmonic function h ∈ L∞(G) is constant i.e. µ∗h = h implies h = λ·1

for some λ ∈ C.

23

Let σ ∈ B(G),

I⊥σ = {T ∈ V N (G) : σ · T = T}.

Theorem (Choquet and Deny). Let G be a locally compact abelian group TFAE

for σ ∈ P1(G)

(i) For T ∈ V N (G), σ · T = T =⇒ T = λI

(ii) σ(g) 6= 1 if g 6= e.

Theorem (Granirer 86). Let σ ∈ B(G)

(a) dim (I⊥σ ) = n <∞ ⇐⇒ {g ∈ G : σ(g) = 1} is a finite set.

In this case, dim(I⊥σ ) = |{g ∈ G : σ(g) = 1}|.

(b) If G is amenable, I⊥σ is reflexive =⇒ I⊥σ is finite dimensional.

24

Theorem (Granirer 86). Let S ⊆ B(G) be a norm bounded semigroup and

F = {T ∈ V N (G); σ · T = T for all σ ∈ S}.

Then there exists a bounded linear projection P from V N (G) onto F such that

P (φ · T ) = φ · P (T ) for all φ ∈ A(G).

In particular if σ ∈ B(G), ‖σ‖ = 1, there exists a contractive projection

P : V N (G) onto I⊥σ satisfying

P (φ · T ) = φ · P (T ) for all φ ∈ A(G).

Proof. Apply Markov-Kakutani Fixed Point Theorem and Theorem 1.

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Let A be a commutative Banach algebra with a BAI.

For f ∈ A∗ and a ∈ A, by a · f we denote the functional on A defined by

〈a · f, b〉 = 〈f, ab〉 .

A projection P : A∗ → A∗ is said to be “invariant”(or A-invariant) if, for an

a ∈ A and f ∈ A∗, the equality P (a·f) = a·P (f) holds. Similarly, a closed subspace

X of A∗ is said to be “invariant” if, for each a ∈ A and f ∈ X, the functional a ·f

is in X (i.e. X is an A-module for the natural action (a, f) 7→ a · f). If there is

an invariant projection from A∗ onto a closed invariant subspace X of A∗ then X

is said to be “invariantly complemented in A∗”.

26

We say that a projection P : A∗ 7→ A∗ is “natural” if, for each γ ∈ ∆(A), either

P (γ) = γ or P (γ) = 0.

If X is a closed invariant subspace of A∗ and if there is natural projection P

from A∗ onto X we shall say that X is “naturally complemented” in A∗.

Remark: Every invariant projection P : A∗ → A∗ is natural.

Theorem 4 (Lau and Ulger, Trans. A.M.S. 2014). Let G be an amenable locally

compact group, and I be a closed ideal in A(G). Then X = I⊥ is invariantly

complemented in V N (G) ⇐⇒ X is naturally complemented. In particular, if σ ∈

B(G), ‖σ‖ = 1, there is an invariant projection onto I⊥σ = {T ∈ V N (G) : σ ·T = T}

(the fixed point set of σ) ⇐⇒ I⊥σ is naturally complemented.

27

5. Metric semigroup

A metric semigroup is a semitopological semigroup whose topology is determined

by a metric d. We consider the following fixed point property for a metric semigroup

S.

(FU ) : If S = {Ts : s ∈ S} is a separately continuous representation of S on a compact

subset K of a locally convex space (E,Q) and if the mapping s 7→ Ts(y) from

S into K is uniformly continuous for each y ∈ K, then K has a common

fixed point for S.

Note that the mapping s 7→ Ts(y) is uniformly continuous if for each τ ∈ Q

and each ε > 0 there is δ > 0 such that

τ(Ts(y) − Tt(y)

)≤ ε

whenever d(s, t) ≤ δ.

28

For example, suppose that S is a subset of a locally convex space L that acts

on E such that (a, y) 7→ ay : L×E → E is separately continuous; if a 7→ ay is linear

in a ∈ L for each y ∈ E, then the induced action of S on E, (s, y) 7→ sy(s ∈ S),

is uniformly continuous in s for each y ∈ E.

A Banach algebra A is a F -algebra (Lau [1983]) if it is the (unique) predual of

a W ∗-algebra M and the identity e of M is a multiplicative linear functional on

A.

Since A∗∗ = M∗, we denote by P1(A∗∗) the set of all normalized positive linear

functionals on M, that is,

P1(A∗∗) = {m ∈ A∗∗ : m ≥ 0, m(e) = 1}.

In this case P1(A∗∗) is a semigroup with the first (or second) Arens multiplication.

29

Examples of F -algebras include the predual algebras of a Hopf von Neumann

algebra (in particular, quantum group algebras), the group algebra L1(G) of a locally

compact group G, the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G)

of a topological group G. They also include the measure algebra M (S) of a locally

compact semigroup S.

A mean m ∈ P1(A∗∗) on A∗ = M is called a topological left invariant mean,

abbreviated as TLIM, if a ·m = m for all a ∈ P1(A) : in other words, m ∈ P1(A∗∗)

is a TLIM if m(x ·a) = m(x) for all a ∈ P1(A) and x ∈ A∗, i.e. A is left amenable.

Lemma A. The F -algebra A is left amenable if and only if the metric semigroup

P1(A) has the fixed point property (FU ).

Theorem 5 (Lau and Zhang, Trans. A.M.S. 2016). Let A be an F -algebra. Then

A is left amenable if and only if P1(A) has the fixed point property (FE).

30

(FE) : Every jointly continuous representation of S on a non-empty compact Hausdorff

space C has a common fixed point in C i.e. the semigroup P1(A) is extremely

left amenable.

Theorem 6 (A.T.-M. Lau, C.K. Ng and N.C. Wong, Oxford Quarterly Journal of

Math. 2018). Two locally compact groups G and H are

isomorphic as topological groups if and only if any one of the following holds

(1) L1(G)1+∼= L1(H)1+ as metric semigroups;

(2) M (G)1+∼= M (H)1+ as metric semigroups.

Open problem 2. Let G and H be locally compact groups, and Hop be the

opposite group of H. If there is a metric preserving semigroup isomorphism from

A(G)1+ (respectively, B(G)1+) onto A(H)1+ (respectively, B(H)1+), does either G =

H or G = Hop?

31

Theorem 7 (A.T.-M. Lau, C.K. Ng and N.C. Wong, 2019).

(a) Let A1 and A2 be two F -algebras with A∗1 being a semi-finite W ∗-algebra.

Any metric semi-group isomorphism Φ : P1(A∗∗1 ) → P1(A

∗∗2 ) extends to an

isometric isomorphism from A1 onto A2 .

(b) Let G1 and G2 be two locally compact groups such that G1 is unimodular.

If there is a metric semi-group isomorphism

Ψ : P1

(V N (G1)

∗)→ P1

(V N (G2)

∗)

or Ψ : P1

(B(G1)

∗∗)→ P1

(B(G2)

∗∗)

then there exists either a homeomorphis group isomorphism or a homeomorphic

group anti-isomorphism Θ : G2 → G1 such that Ψ(f) = f ◦ Θ.

32

6. Fixed point set of power bounded elements in B(G)

A (not necessarily linear) mapping T on a Banach space X is said to be non-

expansive if ‖T (x) − T (y)‖ ≤ ‖x− y‖ holds for all x, y ∈ X.

Theorem (Ishikawa 1976). Let X be a Banach space, let T : X → X be a nonex-

pansive mapping and let 0 < λ < 1. Consider Sλ = λI + (1 − λ)T. Then, for each

x ∈ X, we have

limn→∞

‖Sn+1

λ (x) − Snλ (x)‖ = 0.

33

Let T : X → X be a linear operator from a Banach X into X. Then T is

called power bounded if there exists M > 0 such that ‖Tn‖ ≤M for all

n = 1, 2, . . . .

Lemma B. Let X be a Banach space and let T : X → X a power bounded linear

map. Let 0 < λ < 1 and Sλ = λI + (1 − λ)T. Then, for each x ∈ X, we have

limn→∞

‖Sn+1

λ (x) − Snλ (x)‖ = 0.

Proof. Let c = sup{‖Tn‖ : n ∈ IN0} and define a new norm | · | on X by |x| =

sup {‖Tn(x)‖ : n ∈ IN0}, x ∈ X. Then ‖x‖ ≤ |x| ≤ c‖x‖, so the norms ‖ · ‖ and

| · | on X are equivalent. With respect to the norm | · |, the operator norm of T

is at most 1. Thus T : (X, | · |) → (X, | · |) is nonexpansive. By Ishikawa Theorem,

|Sn+1

λ (x)−Snλ(x)| → 0 and hence also ‖Sn+1

λ (x)−Snλ (x)‖ → 0 for every x ∈ X. �

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Let E be a dual Banach space with a (fixed) predual E∗ , i.e. E∗ is a Banach

space such that (E∗)∗ = E.

Let B(E)be the space of bounded linear operators from E into E. By the

weak∗-operators topology on B(E), denoted by W ∗OT, we shall mean the locally

convex topology determined by the family of seminorms {px,φ; x ∈ E and φ ∈ E∗ }

where px,φ(T ) = |φ(Tx)| T ∈ B(E). Then the unit ball of B(E) is compact in the

W ∗OT.

Lemma C. Let A be a Banach algebra. Let (Sα)α be a bounded net in B(A∗)

and let R,S ∈ B(A∗). Then we have the following:

(i) Sα → S in the W ∗OT if the and only if 〈Sα(f), a〉 → 〈S(f), a〉 for each

f ∈ A∗ and a ∈ A;

(ii) if Sα → S in the W ∗OT, then Sα ◦R→ S ◦R in the W ∗OT ;

(iii) if Sα → S in the W ∗OT, then T ∗ ◦ Sα → T ∗ ◦ S in the W ∗OT for any

bounded linear operator T : A→ A.

35

Let A be a commutative Banach algebra. A linear operator T : A→ A is called

a multiplier of A if T (ab) = aT (b) holds for all a, b ∈ A. When A is faithful i.e.

for any a ∈ A, the condition aA = {0} implies a = 0, then every multiplier of A

is bounded and the set M (A) of all multipliers of A is a unital commutative closed

subalgebra of B(A) of bounded linear operators on A.

• If A = L1(G), then M (A) ∼= M (G) (measure algebra)

• If A = A(G) and G is amenable, then M (A) ∼= B(G) (Fourier Stieltjes

algebra)

Theorem 7 (Kaniuth, Lau, Ulger, JLMS 2010). Given any power bounded multiplier

T of a commutative Banach algebra A, there exists an A-invariant projection P :

A∗ → A∗ such that

P ◦ T ∗ = T ∗ ◦ P = P and P (A∗) = ker (I − T ∗) = {φ ∈ A∗;T ∗φ = φ}.

36

Proof. Consider S = 1

2(I + T ). The sequence

((S∗)n

)n

is a bounded sequence in

B(A∗). Then there exists a subnet((S∗)nα

)nα

that converges in the W ∗OT -topology

to some P ∈ B(A∗). Since, for all a ∈ A, f ∈ A∗ and n ∈ IN, we have

|〈(S∗)n+1(f) − (S∗)n(f), a〉| = |〈f, Sn+1(a) − Sn(a)〉|

≤ ‖f‖ · ‖Sn+1(a) − Sn(a)‖,

we conclude from Lemmas B and C that S∗ ◦ P = P ◦ S∗ = P. This implies that

(S∗)n ◦ P = P for all n ∈ IN. Passing to the W ∗OT limit in B(A∗), we get

that P 2 = P. Since T = 2S − I, we also get T ∗ ◦ P = P ◦ T ∗ = P, and in

particular (I−T ∗)◦P = 0. This shows that P ∗(A∗) ⊆ ker (I−T ∗). To see the reverse

inclusion, let f ∈ ker (I − T ∗) be given. Then T ∗(f) = f and so S∗(f) = f. Then

(S∗)n(f) = f for all n and by the usual w∗-limit argument this gives P ∗(f) = f.

Thus ker (I − T ∗) ⊆ P ∗(A∗). �

37

For a discrete group D, let R(D) denote the Boolean ring of subsets of D

generated by all left cosets of subgroups of D.

Let Rc(G) = {E ∈ R(Gd) : E is closed in G}

Gd = denote G with the discrete topology.

Theorem (J. Gilbert, B. Schreiber, B. Forrest). E ∈ Rc(G) ⇐⇒

E =n⋃

i=1

(aiHi\

mi⋃j=1

bi,jKij

), where ai, bi,j ∈ G, Hi is a closed subgroup of G and

Kij is an open subgroup of Hi .

38

If u ∈ B(G), let

Eu = {x ∈ G; |u(x)| = 1} and

Fu = {x ∈ G; u(x) = 1}.

Remark: If u ∈ B(G) is power bounded, w = 1

2(1 + u), then Ew = Fw = Fu .

Theorem 8 (Kaniuth-Lau-Ulger 2010, JLMS). Let G be any locally compact group

and u ∈ B(G) be power bounded (i.e. sup{‖xn‖; n = 1, 2, . . . } <∞). Then

(a) The sets Eu and Fu are in Rc(G).

(b) The fixed point set of u in V N (G) = {T ∈ V N (G); u · T = T} is the range of

a projection P : V N (G) → V N (G) such that u · P (T ) = P (u · T ) for all T ∈

V N (G). If G is amenable, then {T ∈ V N (G); u·T = T} = 〈ρ(x); x ∈ Fu〉W

∗

=

〈ρ(x); x ∈ Ew〉W

∗

where w = 1

2(1 + u).

39

7. Weak∗-fixed point property for B(G)

• A dual Banach space E is said to have the weak∗-fixed point property if for

each weak∗-compact compact convex subset of E has the fixed point property

for non-expansive mappings.

• We say that a dual Banach space E has the weak∗ f.p.p. for left reversible

semigroup if whenever S is a left reversible semigroup and K is a weak∗

compact convex subset of E and S = {Ts : s ∈ S} is a representation of S as

non-expansive mappings from K into K, then K has a common fixed point

for S.

Theorem (T.C. Lim, Pacific J. Math. 1980). `1 = c∗0 has the weak∗-f.p.p. property.

Theorem (C. Lennard, PAMS 1990). Let H be a Hilbert space. Then B(H)∗ has

the weak∗-f.p.p.

40

Theorem (N. Randrianantoanina, JFA 2010). Let H be a Hilbert space. Then

B(H)∗ has the weak∗ f.p.p. for left reversible semigroups.

Let C be a non-empty subset of a Banach space X and {Dα|α ∈ Λ} be a

decreasing net of bounded non-empty subsets of X. For each x ∈ C, and α ∈ Λ, let

rα(x) = sup{‖x− y‖ | y ∈ Dα}

r(x) = limαrα(x) = inf

αrα(x)

r = inf{r(x)|x ∈ C}.

The set (possibly empty)

AC({Dα|α ∈ Λ}) = {x ∈ C|r(x) = r}

is called the asymptotic centre of {Dα|α ∈ Λ} with respect to C and r is called the

asymptotic radius of {Dα|α ∈ Λ} with respect to C.

41

Theorem 9 (Fendler-Lau-Leinert JFA 2013). Let G be a compact group. Let C be

a nonempty weak∗ closed convex subset of B(G) and {Dα : α ∈ Λ) be a decreasing

net of nonempty bounded subsets of C. Let r(x) be as defined above. Then for each

s ≥ 0, {x ∈ C : r(x) ≤ s} is weak∗ compact and convex, and the asymptotic center

of {Dα : α ∈ Λ} with respect to C is a nonempty norm compact convex subset

of C.

Corollary. Let G be a compact group, then B(G) = A(G) has the weak∗-f.p.p.

for left reversible semigroups.

• G-separable (Lau-Mah 2010 JFA)

• Corollary above can be derived from a result in [N. Randrainantoanina JFA 2010]

using non-commutative integration theory.

42

Theorem 10 (Fendler-Lau-Leinert JFA 2013). Let G be a locally compact group.

Then B(G) has the w∗-fixed point property if and only if G is compact. In this

case, B(G) has the weak∗-fixed point property for left reversible semigroups.

Open problem 3. Let G be a locally compact group. Let Bρ(G) denote the reduced

Fourier-Stieltjes algebra of B(G), i.e. Bρ(G) is the weak∗ closure of C00(G)∩B(G).

Then Bρ(G) = C∗ρ(G)∗ where C∗

ρ(G) = norm closure of {ρ(f) : f ∈ L1(G)} in

V N (G). Does the weak∗ fixed point property on Bρ(G) imply G is compact? This

is true when G is amenable by Theorem 10, since B(G) = Bρ(G) in this case.

Open problem 4. Let G be a locally compact group. Does the asymptotic centre

property on Bρ(G) imply that G is compact?

43

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