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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)
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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)?
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• 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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.)
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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.
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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”.
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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)
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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.
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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).
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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) <∞.
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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.
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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).
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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.
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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.
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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.
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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∗”.
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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.
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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.
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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.
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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).
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(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?
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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.
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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.
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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.
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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}.
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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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