Locally compact quantum groups and amenability
Ami Viselter
University of Alberta
Joint Mathematics Meetings
San Diego, January 11, 2013
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
1 / 20
Quantum groups basics
Motivation
No Pontryagin duality for locally compact (non-Abelian) groups.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
2 / 20
Quantum groups basics
Motivation
No Pontryagin duality for locally compact (non-Abelian) groups.
However, if G is a locally compact Abelian group, then
VN(G) L ∞ (Ĝ)
(implemented spatially by the Fourier transform U : L 2 (G) → L 2 (Ĝ)).
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
2 / 20
Quantum groups basics
Motivation
No Pontryagin duality for locally compact (non-Abelian) groups.
However, if G is a locally compact Abelian group, then
VN(G) L ∞ (Ĝ)
(implemented spatially by the Fourier transform U : L 2 (G) → L 2 (Ĝ)).
This suggests that—for non-abelian groups—VN(G) could play the
role of L ∞ of the “dual” of G!
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
2 / 20
Quantum groups basics
Motivation
No Pontryagin duality for locally compact (non-Abelian) groups.
However, if G is a locally compact Abelian group, then
VN(G) L ∞ (Ĝ)
(implemented spatially by the Fourier transform U : L 2 (G) → L 2 (Ĝ)).
This suggests that—for non-abelian groups—VN(G) could play the
role of L ∞ of the “dual” of G!
The goal
To find a category that will contain all locally compact groups, in which
every object has a dual, such that:
1
The dual of a LCAG is the usual one
2
We have a “Pontryagin duality theorem”.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
2 / 20
Locally compact quantum groups
Definition (Kustermans & Vaes, 1999)
A locally compact quantum group is a pair G = (M, ∆) such that:
1
M is a von Neumann algebra
2
∆ : M → M ⊗ M is a co-multiplication: a normal faithful
homomorphism which is co-associative:
(∆ ⊗ id)∆ = (id ⊗ ∆)∆
3
(both are M → M ⊗ M ⊗ M).
There are two (n.s.f.) weights ϕ, ψ over M (the Haar weights) with:
ϕ((ω ⊗ id)∆(x)) = ω(1)ϕ(x) when ω ∈ M∗+ , x ∈ M + and ϕ(x) < ∞.
ψ((id ⊗ ω)∆(x)) = ω(1)ϕ(x) when ω ∈ M∗+ , x ∈ M + with ψ(x) < ∞.
ϕ and ψ are called left and right invariant weights, respectively.
The novelty: existence of both left and right invariant weights is required!
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
3 / 20
Locally compact quantum groups
Definition (Kustermans & Vaes, 1999)
A locally compact quantum group is a pair G = (M, ∆) such that:
1
M is a von Neumann algebra
2
∆ : M → M ⊗ M is a co-multiplication: a normal faithful
homomorphism which is co-associative:
(∆ ⊗ id)∆ = (id ⊗ ∆)∆
3
(both are M → M ⊗ M ⊗ M).
There are two (n.s.f.) weights ϕ, ψ over M (the Haar weights) with:
ϕ((ω ⊗ id)∆(x)) = ω(1)ϕ(x) when ω ∈ M∗+ , x ∈ M + and ϕ(x) < ∞.
ψ((id ⊗ ω)∆(x)) = ω(1)ϕ(x) when ω ∈ M∗+ , x ∈ M + with ψ(x) < ∞.
ϕ and ψ are called left and right invariant weights, respectively.
The novelty: existence of both left and right invariant weights is required!
Notation: L ∞ (G) := M, L 1 (G) := M∗ .
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
3 / 20
Two classical examples
G – a locally compact group.
Example 1 (commutative LCQGs)
L ∞ (G) := L ∞ (G)
∆ : L ∞ (G) → L ∞ (G) ⊗ L ∞ (G):
using the identification L ∞ (G) ⊗ L ∞ (G) L ∞ (G × G), we let
∆(f ) (t, s) := f (ts)
for f ∈ L ∞ (G).
Take integration w.r.t. left and right Haar measures as the left and
right Haar weights, respectively:
Z
Z
ϕ(f ) :=
f (x) dx, ψ(f ) :=
f (x) dxr
for 0 ≤ f ∈ L ∞ (G).
G
Ami Viselter (University of Alberta)
G
LCQGs and Amenability
Joint Math Meetings 2013
4 / 20
Two classical examples
G – a locally compact group.
Example 2 (co-commutative LCQGs)
L ∞ (G) := VN(G)
∆ : VN(G) → VN(G) ⊗ VN(G) is defined by
∆(λg ) := λg ⊗ λg
ϕ = ψ = the “Plancherel weight”, defined roughly by
ϕ(λ(f )) := f (e)
Ami Viselter (University of Alberta)
for all 0≤f ∈ Cc (G).
LCQGs and Amenability
Joint Math Meetings 2013
5 / 20
Two classical examples
G – a locally compact group.
Example 2 (co-commutative LCQGs)
L ∞ (G) := VN(G)
∆ : VN(G) → VN(G) ⊗ VN(G) is defined by
∆(λg ) := λg ⊗ λg
ϕ = ψ = the “Plancherel weight”, defined roughly by
ϕ(λ(f )) := f (e)
for all 0≤f ∈ Cc (G).
If G is abelian, then this quantum group is the same as the
commutative (L ∞ (Ĝ), ∆).
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
5 / 20
Duality
LCQGs have a very rich structure
the (unbounded!) antipode plays a prominent role.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
6 / 20
Duality
LCQGs have a very rich structure
the (unbounded!) antipode plays a prominent role.
Every LCQG G admits a dual Ĝ such that:
1
2
if G is commutative: L ∞ (G) = L ∞ (G), then its dual is its
co-commutative counterpart: L ∞ (Ĝ) = VN(G)
Pontryagin duality theorem:
ˆ
Ĝ = G.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
6 / 20
Duality
LCQGs have a very rich structure
the (unbounded!) antipode plays a prominent role.
Every LCQG G admits a dual Ĝ such that:
1
2
if G is commutative: L ∞ (G) = L ∞ (G), then its dual is its
co-commutative counterpart: L ∞ (Ĝ) = VN(G)
Pontryagin duality theorem:
ˆ
Ĝ = G.
Moreover, both L ∞ (G) and L ∞ (Ĝ) are naturally represented on the
same (canonical) Hilbert space.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
6 / 20
Types of LCQGs
Proposition
If G is a LCQG, then L 1 (G) becomes a Banach algebra as follows:
for ω, ρ ∈ L 1 (G), define ω ∗ ρ ∈ L 1 (G) by
(ω ∗ ρ)(x) := (ω ⊗ ρ)∆(x), x ∈ L ∞ (G).
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
7 / 20
Types of LCQGs
Proposition
If G is a LCQG, then L 1 (G) becomes a Banach algebra as follows:
for ω, ρ ∈ L 1 (G), define ω ∗ ρ ∈ L 1 (G) by
(ω ∗ ρ)(x) := (ω ⊗ ρ)∆(x), x ∈ L ∞ (G).
The commutative case: L ∞ (G) = L ∞ (G), L 1 (G) = L 1 (G), ∗ = convolution.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
7 / 20
Types of LCQGs
Proposition
If G is a LCQG, then L 1 (G) becomes a Banach algebra as follows:
for ω, ρ ∈ L 1 (G), define ω ∗ ρ ∈ L 1 (G) by
(ω ∗ ρ)(x) := (ω ⊗ ρ)∆(x), x ∈ L ∞ (G).
The commutative case: L ∞ (G) = L ∞ (G), L 1 (G) = L 1 (G), ∗ = convolution.
Definition
Property
Discreteness
Compactness
Groups
is unital
The Haar measure(s) are finite
Ami Viselter (University of Alberta)
L 1 (G)
LCQGs and Amenability
LCQGs
is unital
ϕ(1) < ∞
L 1 (G)
Joint Math Meetings 2013
7 / 20
Types of LCQGs
Proposition
If G is a LCQG, then L 1 (G) becomes a Banach algebra as follows:
for ω, ρ ∈ L 1 (G), define ω ∗ ρ ∈ L 1 (G) by
(ω ∗ ρ)(x) := (ω ⊗ ρ)∆(x), x ∈ L ∞ (G).
The commutative case: L ∞ (G) = L ∞ (G), L 1 (G) = L 1 (G), ∗ = convolution.
Definition
Property
Discreteness
Compactness
Groups
is unital
The Haar measure(s) are finite
L 1 (G)
LCQGs
is unital
ϕ(1) < ∞
L 1 (G)
Theorem
G is discrete ⇐⇒ Ĝ is compact for every LCQG G.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
7 / 20
Amenable groups
G – a locally compact group.
Definition
A mean (= state) m of L ∞ (G) is left invariant if m(Lt x) = m(x) for all
x ∈ L ∞ (G) and t ∈ G.
G is amenable if it has a left invariant mean.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
8 / 20
Amenable groups
G – a locally compact group.
Definition
A mean (= state) m of L ∞ (G) is left invariant if m(Lt x) = m(x) for all
x ∈ L ∞ (G) and t ∈ G.
G is amenable if it has a left invariant mean.
Numerous equivalent characterizations
G is amenable ⇐⇒
topological amenability: there is a mean m ∈ L ∞ (G)∗ with
m(ω ∗ x) = ω(1)m(x)
for all x ∈ L ∞ (G), ω ∈ L 1 (G)
Leptin’s theorem: VN(G)∗ has a left bounded approximate identity
[But also: means on algebras other than L ∞ (G), Hulanicki’s theorem,
Følner’s condition, Reiter’s condition(s), Rickert’s theorem, etc...]
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
8 / 20
Amenability for LCQGs
Definition
G – a LCQG.
A left invariant mean on G is a state m ∈ L ∞ (G)∗ with
m((ω ⊗ id)∆(x)) = ω(1)m(x)
for all x ∈ L ∞ (G), ω ∈ L 1 (G).
G is amenable if it admits a left invariant mean.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
9 / 20
Amenability for LCQGs
Definition
G – a LCQG.
A left invariant mean on G is a state m ∈ L ∞ (G)∗ with
m((ω ⊗ id)∆(x)) = ω(1)m(x)
for all x ∈ L ∞ (G), ω ∈ L 1 (G).
G is amenable if it admits a left invariant mean.
G is co-amenable if L 1 (G) admits a left bounded approximate
identity.
How are these connected?
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
9 / 20
Amenability for LCQGs
Definition
G – a LCQG.
A left invariant mean on G is a state m ∈ L ∞ (G)∗ with
m((ω ⊗ id)∆(x)) = ω(1)m(x)
for all x ∈ L ∞ (G), ω ∈ L 1 (G).
G is amenable if it admits a left invariant mean.
G is co-amenable if L 1 (G) admits a left bounded approximate
identity.
How are these connected?
Theorem (one direction of Leptin’s)
Ĝ is co-amenable =⇒ G is amenable.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
9 / 20
Amenability for LCQGs
Examples
Every group is co-amenable (as a LCQG)
Every compact LCQG is amenable (use the left invariant weight!)
Every discrete LCQG is co-amenable
The bicrossed product machinery allows the construction of
non-Kac, non-discrete, non-compact amenable LCQGs
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
10 / 20
Amenability for LCQGs
Examples
Every group is co-amenable (as a LCQG)
Every compact LCQG is amenable (use the left invariant weight!)
Every discrete LCQG is co-amenable
The bicrossed product machinery allows the construction of
non-Kac, non-discrete, non-compact amenable LCQGs
Other direction of Leptin’s theorem
?
G is amenable =⇒ Ĝ is co-amenable
This is an important open problem.
Ruan (1996): true for discrete Kac algebras
Tomatsu (2006): true for all discrete LCQGs
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
10 / 20
Amenability and injectivity
Theorem
G – a locally compact group.
1
(Guichardet, 1969) G is amenable =⇒ VN(G) is injective.
2
(Lance, 1973) If G is discrete, the opposite also holds.
This is not true for a general G (Connes; example: SL (2, C)).
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
11 / 20
Amenability and injectivity
Theorem
G – a locally compact group.
1
(Guichardet, 1969) G is amenable =⇒ VN(G) is injective.
2
(Lance, 1973) If G is discrete, the opposite also holds.
This is not true for a general G (Connes; example: SL (2, C)).
Theorem (Enock & Schwartz, 1986; Bédos & Tuset, 2003)
G – a LCQG.
G is amenable =⇒ L ∞ (Ĝ) is injective. What about the other direction?
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
11 / 20
Amenability and injectivity
Theorem
G – a locally compact group.
1
(Guichardet, 1969) G is amenable =⇒ VN(G) is injective.
2
(Lance, 1973) If G is discrete, the opposite also holds.
This is not true for a general G (Connes; example: SL (2, C)).
Theorem (Enock & Schwartz, 1986; Bédos & Tuset, 2003)
G – a LCQG.
G is amenable =⇒ L ∞ (Ĝ) is injective. What about the other direction?
Theorem (Ruan, 1996)
It is true for discrete Kac algebras.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
11 / 20
Amenability and injectivity
Joint work with Piotr Sołtan.
Conjecture
If G is discrete, then L ∞ (Ĝ) is injective =⇒ G is amenable.
Unknown even when L ∞ (Ĝ) is of type I!
More generally?
This isn’t always true (see the groups case)
The injectivity of L ∞ (Ĝ) doesn’t take into account the quantum
structure of G (namely, ∆)
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
12 / 20
Amenability and injectivity
Instead of injectivity—“quantum injectivity”!
Theorem (P. Sołtan and A.V.)
The following are equivalent for a LCQG G:
G is amenable
there is a conditional expectation E : B(H) → L ∞ (Ĝ) with
E(L ∞ (G)) ⊆ Center(L ∞ (Ĝ))
there is a conditional expectation E : B(H) → L ∞ (Ĝ) with
E(L ∞ (G)) = C1.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
13 / 20
Amenability and injectivity
Instead of injectivity—“quantum injectivity”!
Theorem (P. Sołtan and A.V.)
The following are equivalent for a LCQG G:
G is amenable
there is a conditional expectation E : B(H) → L ∞ (Ĝ) with
E(L ∞ (G)) ⊆ Center(L ∞ (Ĝ))
there is a conditional expectation E : B(H) → L ∞ (Ĝ) with
E(L ∞ (G)) = C1.
The proof employs the Effros–Kishimoto Theorem of approximating a
(cb) bimodule map by “elementary operators”.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
13 / 20
Applications
1) This characterization is new even in the groups case:
Corollary
A locally compact group G is amenable ⇐⇒ there is a conditional
expectation E : B(L 2 (G)) → VN(G) such that
E(L ∞ (G)) ⊆ Center(VN(G)).
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
14 / 20
Applications
1) This characterization is new even in the groups case:
Corollary
A locally compact group G is amenable ⇐⇒ there is a conditional
expectation E : B(L 2 (G)) → VN(G) such that
E(L ∞ (G)) ⊆ Center(VN(G)).
2) New proof of Ruan’s theorem:
Ruan proved directly that if G is discrete and Kac, then L ∞ (Ĝ) is
injective =⇒ Ĝ is co-amenable
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
14 / 20
Applications
1) This characterization is new even in the groups case:
Corollary
A locally compact group G is amenable ⇐⇒ there is a conditional
expectation E : B(L 2 (G)) → VN(G) such that
E(L ∞ (G)) ⊆ Center(VN(G)).
2) New proof of Ruan’s theorem:
Ruan proved directly that if G is discrete and Kac, then L ∞ (Ĝ) is
injective =⇒ Ĝ is co-amenable
His approach uses the “biregular” representation, and relies on
the fine structure of compact Kac algebras
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
14 / 20
Applications
1) This characterization is new even in the groups case:
Corollary
A locally compact group G is amenable ⇐⇒ there is a conditional
expectation E : B(L 2 (G)) → VN(G) such that
E(L ∞ (G)) ⊆ Center(VN(G)).
2) New proof of Ruan’s theorem:
Ruan proved directly that if G is discrete and Kac, then L ∞ (Ĝ) is
injective =⇒ Ĝ is co-amenable
His approach uses the “biregular” representation, and relies on
the fine structure of compact Kac algebras
Our proof is very short, and is similar to the original proof for
discrete groups.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
14 / 20
Possible directions
Using the “quantum injectivity” theorem to prove the conjecture?
(G – discrete quantum group, L ∞ (Ĝ) is injective)
Modifying a conditional expectation
We have two conditional expectations:
E : B(H) → L ∞ (Ĝ) (by assumption), F : B(H) → L ∞ (G) (atomic)
F is normal, E usually isn’t.
Can F be used to “modify” E so that E(L ∞ (G)) ⊆ Center(L ∞ (Ĝ))?
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
15 / 20
Possible directions
Using the “quantum injectivity” theorem to prove the conjecture?
(G – discrete quantum group, L ∞ (Ĝ) is injective)
Modifying a conditional expectation
We have two conditional expectations:
E : B(H) → L ∞ (Ĝ) (by assumption), F : B(H) → L ∞ (G) (atomic)
F is normal, E usually isn’t.
Can F be used to “modify” E so that E(L ∞ (G)) ⊆ Center(L ∞ (Ĝ))?
Problem: L ∞ (Ĝ) is infinite (otherwise: Kac algebra!)
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
15 / 20
Possible directions
Using the “quantum injectivity” theorem to prove the conjecture?
(G – discrete quantum group, L ∞ (Ĝ) is injective)
Modifying a conditional expectation
We have two conditional expectations:
E : B(H) → L ∞ (Ĝ) (by assumption), F : B(H) → L ∞ (G) (atomic)
F is normal, E usually isn’t.
Can F be used to “modify” E so that E(L ∞ (G)) ⊆ Center(L ∞ (Ĝ))?
Problem: L ∞ (Ĝ) is infinite (otherwise: Kac algebra!)
The construction of E is usually done by a Hahn–Banach
argument
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
15 / 20
Possible directions
Using the “quantum injectivity” theorem to prove the conjecture?
(G – discrete quantum group, L ∞ (Ĝ) is injective)
Modifying a conditional expectation
We have two conditional expectations:
E : B(H) → L ∞ (Ĝ) (by assumption), F : B(H) → L ∞ (G) (atomic)
F is normal, E usually isn’t.
Can F be used to “modify” E so that E(L ∞ (G)) ⊆ Center(L ∞ (Ĝ))?
Problem: L ∞ (Ĝ) is infinite (otherwise: Kac algebra!)
The construction of E is usually done by a Hahn–Banach
argument
We have one sufficient condition for non-existence (communicated
by N. Ozawa)
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
15 / 20
Translations on LCQGs
Definition
A left translation of G is an automorphism Φ of L ∞ (G) that is left
covariant:
(Φ ⊗ id) ◦ ∆ = ∆ ◦ Φ.
One defines right translations similarly.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
16 / 20
Translations on LCQGs
Definition
A left translation of G is an automorphism Φ of L ∞ (G) that is left
covariant:
(Φ ⊗ id) ◦ ∆ = ∆ ◦ Φ.
One defines right translations similarly.
Example
For G = G, these are left and right translations of L ∞ (G)!
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
16 / 20
Translations on LCQGs
Definition
A left translation of G is an automorphism Φ of L ∞ (G) that is left
covariant:
(Φ ⊗ id) ◦ ∆ = ∆ ◦ Φ.
One defines right translations similarly.
Example
For G = G, these are left and right translations of L ∞ (G)!
Theorem (M. Kalantar and M. Neufang, 2010)
The group LT(G) of all left translations, endowed with the SOT, is
locally compact.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
16 / 20
Translations on LCQGs
Examples
1
For G = G we have LT(G) G (as locally compact groups)
2
When G = Ĝ, LT(G) is the group of continuous characters of G
(may be trivial!)
3
For −1 < q < 1 we have LT(SUq (2)) T
4
For 0 < µ < 1 we have LT(Eµ (2)) T
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
17 / 20
Translations on LCQGs
Examples
1
For G = G we have LT(G) G (as locally compact groups)
2
When G = Ĝ, LT(G) is the group of continuous characters of G
(may be trivial!)
3
For −1 < q < 1 we have LT(SUq (2)) T
4
For 0 < µ < 1 we have LT(Eµ (2)) T
The group LT(G) preserves compactness and discreteness, and
encodes some other properties of the quantum group G. Particularly:
Theorem (M. Kalantar and M. Neufang)
If G is discrete and amenable, then LT(G) is amenable.
Whether this is true for general a LCQG G was left open.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
17 / 20
Translations on LCQGs and amenability
Theorem (M. Kalantar, V. Runde and A.V.)
G is amenable =⇒ LT(G) is amenable.
Main ingredients of the proof:
1
2
Leptin’s Theorem for “translation amenability” in the quantum level
A result of Baaj and Vaes about when a subalgebra N ⊆ L ∞ (G)
with ∆(N) ⊆ N ⊗ N induces a locally compact quantum group
(N, ∆).
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
18 / 20
Translations on LCQGs and amenability
Consider all elements x ∈ L ∞ (G) which are left-translation invariant:
Φ(x) = x
for all Φ ∈ LT(G).
Similarly: right-translation invariant, two-sided translation invariant.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
19 / 20
Translations on LCQGs and amenability
Consider all elements x ∈ L ∞ (G) which are left-translation invariant:
Φ(x) = x
for all Φ ∈ LT(G).
Similarly: right-translation invariant, two-sided translation invariant.
Example
When G = Eµ (2), the vN algebra of two-sided translation invariant
elements is abelian, and is generated by one (normal) operator.
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
19 / 20
Translations on LCQGs and amenability
Consider all elements x ∈ L ∞ (G) which are left-translation invariant:
Φ(x) = x
for all Φ ∈ LT(G).
Similarly: right-translation invariant, two-sided translation invariant.
Example
When G = Eµ (2), the vN algebra of two-sided translation invariant
elements is abelian, and is generated by one (normal) operator.
Question
If the algebra of all such elements is “small”, perhaps
G is amenable =⇒ Ĝ is co-amenable ?
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
19 / 20
Thank you for listening!
Ami Viselter (University of Alberta)
LCQGs and Amenability
Joint Math Meetings 2013
20 / 20