Generating functionals on quantum groups Adam Skalski 1 Ami Viselter - - PowerPoint PPT Presentation

Generating functionals on quantum groups

Adam Skalski1 Ami Viselter ⋆,2

1IMPAN, Warsaw 2University of Haifa

Quantum Groups and their Analysis University of Oslo August 8, 2019

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 1 / 25

Convolution semigroups of measures

G – locally compact group

Convolution of measures

For positive Borel measures µ, ν on G, define their convolution µ ⋆ ν by (µ ⋆ ν)(A) :=

• G
• G

IA(gh) dµ(g)

• dν(h)

(∀measurable A). For convenience, write µ(f) :=

• G f dµ.

Definition

A convolution semigroup of probability measures on G is a family (µt)t≥0 of probability measures on G satisfying µ0 = δe and µs ⋆ µt = µs+t (∀s, t ≥ 0). It is w∗-continuous if µt(f) − − − − →

t→0+ µ0(f) = f(e) for all f ∈ C0(G).

It is symmetric if every µt is invariant under inversion.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25

Convolution semigroups of measures

G – locally compact group

Convolution of measures

For positive Borel measures µ, ν on G, define their convolution µ ⋆ ν by (µ ⋆ ν)(A) :=

• G
• G

IA(gh) dµ(g)

• dν(h)

(∀measurable A). For convenience, write µ(f) :=

• G f dµ.

Definition

A convolution semigroup of probability measures on G is a family (µt)t≥0 of probability measures on G satisfying µ0 = δe and µs ⋆ µt = µs+t (∀s, t ≥ 0). It is w∗-continuous if µt(f) − − − − →

t→0+ µ0(f) = f(e) for all f ∈ C0(G).

It is symmetric if every µt is invariant under inversion.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25

Convolution semigroups of measures

G – locally compact group

Convolution of measures

For positive Borel measures µ, ν on G, define their convolution µ ⋆ ν by (µ ⋆ ν)(A) :=

• G
• G

IA(gh) dµ(g)

• dν(h)

(∀measurable A). For convenience, write µ(f) :=

• G f dµ.

Definition

A convolution semigroup of probability measures on G is a family (µt)t≥0 of probability measures on G satisfying µ0 = δe and µs ⋆ µt = µs+t (∀s, t ≥ 0). It is w∗-continuous if µt(f) − − − − →

t→0+ µ0(f) = f(e) for all f ∈ C0(G).

It is symmetric if every µt is invariant under inversion.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25

Convolution semigroups of measures

G – locally compact group

Convolution of measures

For positive Borel measures µ, ν on G, define their convolution µ ⋆ ν by (µ ⋆ ν)(A) :=

• G
• G

IA(gh) dµ(g)

• dν(h)

(∀measurable A). For convenience, write µ(f) :=

• G f dµ.

Definition

A convolution semigroup of probability measures on G is a family (µt)t≥0 of probability measures on G satisfying µ0 = δe and µs ⋆ µt = µs+t (∀s, t ≥ 0). It is w∗-continuous if µt(f) − − − − →

t→0+ µ0(f) = f(e) for all f ∈ C0(G).

It is symmetric if every µt is invariant under inversion.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 2 / 25

Convolution semigroups of measures

Probabilistic importance

w∗-cont. conv. semigroups of

• prob. measures on G

←→ G-valued Lévy processes

Definition

A G-valued Lévy process is a family X = (Xt)t≥0 of random variables from a probability space to G such that:

1

X0 = 0;

2

X has independent and stationary increments;

3

X is continuous. Given a Lévy process X, define (µt)t≥0 to be its family of distributions: for t ≥ 0, µt is the probability measure on G defined by µt := P ◦ X−1

t

.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 3 / 25

Convolution semigroups of measures

Probabilistic importance

w∗-cont. conv. semigroups of

• prob. measures on G

←→ G-valued Lévy processes

Definition

A G-valued Lévy process is a family X = (Xt)t≥0 of random variables from a probability space to G such that:

1

X0 = 0;

2

X has independent and stationary increments;

3

X is continuous. Given a Lévy process X, define (µt)t≥0 to be its family of distributions: for t ≥ 0, µt is the probability measure on G defined by µt := P ◦ X−1

t

.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 3 / 25

Convolution semigroups of measures

Probabilistic importance

w∗-cont. conv. semigroups of

• prob. measures on G

←→ G-valued Lévy processes

Definition

A G-valued Lévy process is a family X = (Xt)t≥0 of random variables from a probability space to G such that:

1

X0 = 0;

2

X has independent and stationary increments;

3

X is continuous. Given a Lévy process X, define (µt)t≥0 to be its family of distributions: for t ≥ 0, µt is the probability measure on G defined by µt := P ◦ X−1

t

.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 3 / 25

Convolution semigroups of measures

(µt)t≥0 – w∗-cont. conv. semigroup of prob. measures on G.

Definition

The generating functional of (µt)t≥0 is the functional γ on C0(G) given by γ(f) := lim

t→0+

µt(f) − µ0(f) t with domain consisting of all f ∈ C0(G) s.t. the limit exists. What can we say about D(γ)?

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 4 / 25

Convolution semigroups of measures

(µt)t≥0 – w∗-cont. conv. semigroup of prob. measures on G.

Definition

The generating functional of (µt)t≥0 is the functional γ on C0(G) given by γ(f) := lim

t→0+

µt(f) − µ0(f) t with domain consisting of all f ∈ C0(G) s.t. the limit exists. What can we say about D(γ)?

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 4 / 25

Convolution semigroups of measures

Elementary C0-semigroup tools D(γ) is dense in C0(G).

Theorem (Hunt, Hazod–Siebert)

D(γ) contains a dense subalgebra of C0(G). G – Lie group: C2

0(G) ⊆ D(γ).

Generally: C∞

c (G) ⊆ D(γ).

The form of γ is completely understood – the Lévy–Khintchine formula: When G = Rn, there exist a positive matrix a ∈ Mn(R), b ∈ Rn and a positive measure on Rn\ {0} with

• Rn\{0} min(y2 , 1) dν(y) < ∞ s.t. for

all f ∈ C2

0(G),

γ(f) =

n

• i,j=1

aij ∂2f ∂xi∂xj (0) +

n

• k=1

bk ∂f ∂xk (0) +

• Rn\{0}

      f − f(0) − χB1(0)(y)

n

• ℓ=1

∂f ∂xℓ (0)yℓ        dν(y).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25

Convolution semigroups of measures

Elementary C0-semigroup tools D(γ) is dense in C0(G).

Theorem (Hunt, Hazod–Siebert)

D(γ) contains a dense subalgebra of C0(G). G – Lie group: C2

0(G) ⊆ D(γ).

Generally: C∞

c (G) ⊆ D(γ).

The form of γ is completely understood – the Lévy–Khintchine formula: When G = Rn, there exist a positive matrix a ∈ Mn(R), b ∈ Rn and a positive measure on Rn\ {0} with

• Rn\{0} min(y2 , 1) dν(y) < ∞ s.t. for

all f ∈ C2

0(G),

γ(f) =

n

• i,j=1

aij ∂2f ∂xi∂xj (0) +

n

• k=1

bk ∂f ∂xk (0) +

• Rn\{0}

      f − f(0) − χB1(0)(y)

n

• ℓ=1

∂f ∂xℓ (0)yℓ        dν(y).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25

Convolution semigroups of measures

Elementary C0-semigroup tools D(γ) is dense in C0(G).

Theorem (Hunt, Hazod–Siebert)

D(γ) contains a dense subalgebra of C0(G). G – Lie group: C2

0(G) ⊆ D(γ).

Generally: C∞

c (G) ⊆ D(γ).

The form of γ is completely understood – the Lévy–Khintchine formula: When G = Rn, there exist a positive matrix a ∈ Mn(R), b ∈ Rn and a positive measure on Rn\ {0} with

• Rn\{0} min(y2 , 1) dν(y) < ∞ s.t. for

all f ∈ C2

0(G),

γ(f) =

n

• i,j=1

aij ∂2f ∂xi∂xj (0) +

n

• k=1

bk ∂f ∂xk (0) +

• Rn\{0}

      f − f(0) − χB1(0)(y)

n

• ℓ=1

∂f ∂xℓ (0)yℓ        dν(y).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25

Convolution semigroups of measures

Elementary C0-semigroup tools D(γ) is dense in C0(G).

Theorem (Hunt, Hazod–Siebert)

D(γ) contains a dense subalgebra of C0(G). G – Lie group: C2

0(G) ⊆ D(γ).

Generally: C∞

c (G) ⊆ D(γ).

The form of γ is completely understood – the Lévy–Khintchine formula: When G = Rn, there exist a positive matrix a ∈ Mn(R), b ∈ Rn and a positive measure on Rn\ {0} with

• Rn\{0} min(y2 , 1) dν(y) < ∞ s.t. for

all f ∈ C2

0(G),

γ(f) =

n

• i,j=1

aij ∂2f ∂xi∂xj (0) +

n

• k=1

bk ∂f ∂xk (0) +

• Rn\{0}

      f − f(0) − χB1(0)(y)

n

• ℓ=1

∂f ∂xℓ (0)yℓ        dν(y).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 5 / 25

Semigroups of positive-definite functions

Definition

A family (ϕt)t≥0 of normalized positive-definite functions on G is a semigroup if ϕ0 ≡ 1 and ϕs · ϕt = ϕs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+ w.r.t. the compact-open topology. symmetric: invariant under inversion = real valued.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 6 / 25

Semigroups of positive-definite functions

Definition

A family (ϕt)t≥0 of normalized positive-definite functions on G is a semigroup if ϕ0 ≡ 1 and ϕs · ϕt = ϕs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+ w.r.t. the compact-open topology. symmetric: invariant under inversion = real valued.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 6 / 25

Semigroups of positive-definite functions

Definition

A family (ϕt)t≥0 of normalized positive-definite functions on G is a semigroup if ϕ0 ≡ 1 and ϕs · ϕt = ϕs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+ w.r.t. the compact-open topology. symmetric: invariant under inversion = real valued.

Example (G – locally compact abelian group)

(µt)t≥0 – w∗-cont. convolution semigroup of prob. measures on G Fourier–Stieltjes transform (ˆ µt)t≥0 – w∗-cont. semigroup of normalized pos.-def. functions on ˆ G.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 6 / 25

Semigroups of positive-definite functions

Definition

A family (ϕt)t≥0 of normalized positive-definite functions on G is a semigroup if ϕ0 ≡ 1 and ϕs · ϕt = ϕs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+ w.r.t. the compact-open topology. symmetric: invariant under inversion = real valued. It has the form (ϕt)t≥0 = (e−tθ)t≥0 for some continuous θ : G → [0, ∞). Characterize θ?

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 6 / 25

Semigroups of positive-definite functions

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R.

Definition

A continuous θ : G → R is conditionally negative definite if:

1

θ(e) = 0;

2

• θ(gi) + θ(gj) − θ(g−1

j

gi)

• 1≤i,j≤n is positive semi-definite for all

n ∈ N and g1, . . . , gn ∈ G.

Remark

θ is non-negative-valued, and symmetric: θ(g−1) = θ(g) for all g ∈ G.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 7 / 25

Semigroups of positive-definite functions

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R.

Definition

A continuous θ : G → R is conditionally negative definite if:

1

θ(e) = 0;

2

• θ(gi) + θ(gj) − θ(g−1

j

gi)

• 1≤i,j≤n is positive semi-definite for all

n ∈ N and g1, . . . , gn ∈ G.

Remark

θ is non-negative-valued, and symmetric: θ(g−1) = θ(g) for all g ∈ G.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 7 / 25

Semigroups of positive-definite functions

Examples of conditionally negative-definite functions

Let n ∈ N.

1

For 0 < α ≤ 2, the function Rn → [0, ∞) given by x → xα is CND.

2

The function Fn → [0, ∞) given by s → |s| is CND (Haagerup, Invent. Math., 1978/79). Consequences:

1

the C∗-algebra C∗

r(Fn) has the metric approximation property;

2

Fn is weakly amenable;

3

Fn has the Haagerup property.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 8 / 25

Semigroups of positive-definite functions

Examples of conditionally negative-definite functions

Let n ∈ N.

1

For 0 < α ≤ 2, the function Rn → [0, ∞) given by x → xα is CND.

2

The function Fn → [0, ∞) given by s → |s| is CND (Haagerup, Invent. Math., 1978/79). Consequences:

1

the C∗-algebra C∗

r(Fn) has the metric approximation property;

2

Fn is weakly amenable;

3

Fn has the Haagerup property.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 8 / 25

Semigroups of positive-definite functions

Examples of conditionally negative-definite functions

Let n ∈ N.

1

For 0 < α ≤ 2, the function Rn → [0, ∞) given by x → xα is CND.

2

The function Fn → [0, ∞) given by s → |s| is CND (Haagerup, Invent. Math., 1978/79). Consequences:

1

the C∗-algebra C∗

r(Fn) has the metric approximation property;

2

Fn is weakly amenable;

3

Fn has the Haagerup property.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 8 / 25

Semigroups of positive-definite functions

Examples of conditionally negative-definite functions

Let n ∈ N.

1

For 0 < α ≤ 2, the function Rn → [0, ∞) given by x → xα is CND.

2

The function Fn → [0, ∞) given by s → |s| is CND (Haagerup, Invent. Math., 1978/79). Consequences:

1

the C∗-algebra C∗

r(Fn) has the metric approximation property;

2

Fn is weakly amenable;

3

Fn has the Haagerup property.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 8 / 25

Comic relief

Examples for G = Rn

• Conv. semigr. (µt)t≥0

Dual semigr.

• f pos.-def.
• funcs. (ˆ

µt)t≥0 θ Name

µt = δ0

(e−tθ)t≥0

Trivial dµt = (4πt)−n/2 exp

• − x2

4t

• dx

y → y2 The Brownian semigroup dµt = Γ( n+1

2 )t

• π(x2 + t2)

− n+1

2 dx

y → y Cauchy’s semigroup        |y| , |y| ≤ 1 1, |y| ≥ 1

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 9 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

Semigroups of positive-definite functions

Positive-definite functions on G ←→ positive functionals on C∗(G). ⊲ Consider again (ϕt)t≥0 = (e−tθ)t≥0. ⊲ View it as a w∗-cont. semigroup of states of C∗(G). ⊲ Its generating functional γ on C∗(G) is given by γ(x) := lim

t→0+

ϕt(x) − 1(x) t for x ∈ C∗(G) s.t. this limit exists. ⊲ Dense algebra in D(γ)? ⊲ We have L1(G) ֒→ C∗(G) algebraically. ⊲ If f ∈ L1(G), then f ∈ D(γ) ⇐⇒ fθ ∈ L1(G), in which case γ(f) = −

• G

fθ d(Haar). As a result, Cc(G)֒→ C∗(G) is contained in D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 10 / 25

A group as a quantum group

G – locally compact group.

1

L∞(G) – a von Neumann algebra.

2

Co-multiplication: the ∗-homomorphism ∆ : L∞(G) → L∞(G) ⊗ L∞(G) L∞(G × G) defined by (∆(f))(t, s) := f(ts) (f ∈ L∞(G)). By associativity, we have (∆ ⊗ id)∆ = (id ⊗ ∆)∆.

3

Left and right Haar measures. View them as functions ϕ, ψ : L∞(G)+ → [0, ∞] given by integrating w.r.t. the Haar measures.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 11 / 25

Locally compact quantum groups

Definition (Kustermans–Vaes, ’00)

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, unital ∗-homomorphism which is co-associative, i.e., (∆ ⊗ id)∆ = (id ⊗ ∆)∆.

3

Haar weights: there exist two n.s.f. weights on M that are left and right invariant, resp. Denote L∞(G) := M.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 12 / 25

Locally compact quantum groups

Features

Duality G → ˆ G within the category satisfying ˆ ˆ G = G. Three “faces” (algebras): the von Neumann algebra L∞(G), the “reduced” C∗-algebra C0(G), and the “universal” C∗-algebra Cu

0 (G).

The conjugate space Cu

0 (G)∗ carries a convolution ⋆ turning it into a

Banach algebra with unit ǫ (the co-unit): µ ⋆ ν := (µ ⊗ ν) ◦ ∆u (µ, ν ∈ Cu

0 (G)∗).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 13 / 25

Locally compact quantum groups

Features

Duality G → ˆ G within the category satisfying ˆ ˆ G = G. Three “faces” (algebras): the von Neumann algebra L∞(G), the “reduced” C∗-algebra C0(G), and the “universal” C∗-algebra Cu

0 (G).

The conjugate space Cu

0 (G)∗ carries a convolution ⋆ turning it into a

Banach algebra with unit ǫ (the co-unit): µ ⋆ ν := (µ ⊗ ν) ◦ ∆u (µ, ν ∈ Cu

0 (G)∗).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 13 / 25

Locally compact quantum groups

Features

Duality G → ˆ G within the category satisfying ˆ ˆ G = G. Three “faces” (algebras): the von Neumann algebra L∞(G), the “reduced” C∗-algebra C0(G), and the “universal” C∗-algebra Cu

0 (G).

The conjugate space Cu

0 (G)∗ carries a convolution ⋆ turning it into a

Banach algebra with unit ǫ (the co-unit): µ ⋆ ν := (µ ⊗ ν) ◦ ∆u (µ, ν ∈ Cu

0 (G)∗).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 13 / 25

Locally compact quantum groups

Features

Duality G → ˆ G within the category satisfying ˆ ˆ G = G. Three “faces” (algebras): the von Neumann algebra L∞(G), the “reduced” C∗-algebra C0(G), and the “universal” C∗-algebra Cu

0 (G).

The conjugate space Cu

0 (G)∗ carries a convolution ⋆ turning it into a

Banach algebra with unit ǫ (the co-unit): µ ⋆ ν := (µ ⊗ ν) ◦ ∆u (µ, ν ∈ Cu

0 (G)∗).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 13 / 25

Locally compact quantum groups

Features

Duality G → ˆ G within the category satisfying ˆ ˆ G = G. Three “faces” (algebras): the von Neumann algebra L∞(G), the “reduced” C∗-algebra C0(G), and the “universal” C∗-algebra Cu

0 (G).

The conjugate space Cu

0 (G)∗ carries a convolution ⋆ turning it into a

Banach algebra with unit ǫ (the co-unit): µ ⋆ ν := (µ ⊗ ν) ◦ ∆u (µ, ν ∈ Cu

0 (G)∗).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 13 / 25

Convolution semigroups on LCQGs

Definition

A convolution semigroup of states on a LCQG G is a family (µt)t≥0 of states of Cu

0 (G) such that

µ0 = ǫ and µs ⋆ µt = µs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+. symmetric: invariant under the unitary antipode.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 14 / 25

Convolution semigroups on LCQGs

Definition

A convolution semigroup of states on a LCQG G is a family (µt)t≥0 of states of Cu

0 (G) such that

µ0 = ǫ and µs ⋆ µt = µs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+. symmetric: invariant under the unitary antipode.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 14 / 25

Convolution semigroups on LCQGs

Definition

A convolution semigroup of states on a LCQG G is a family (µt)t≥0 of states of Cu

0 (G) such that

µ0 = ǫ and µs ⋆ µt = µs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+. symmetric: invariant under the unitary antipode.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 14 / 25

Convolution semigroups on LCQGs

Definition

A convolution semigroup of states on a LCQG G is a family (µt)t≥0 of states of Cu

0 (G) such that

µ0 = ǫ and µs ⋆ µt = µs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+. symmetric: invariant under the unitary antipode.

Examples

• f w∗-continuous, symmetric, convolution semigroups of states on G:

G = G: w∗-cont., symm., conv. semigr. of prob. meas. on G. G = ˆ G: w∗-cont., symm., semigr. of norm. pos.-def. functs. on G.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 14 / 25

Convolution semigroups on LCQGs

Definition

A convolution semigroup of states on a LCQG G is a family (µt)t≥0 of states of Cu

0 (G) such that

µ0 = ǫ and µs ⋆ µt = µs+t (∀s, t ≥ 0). Adjectives: w∗-continuous: at 0+. symmetric: invariant under the unitary antipode. The associated generating functional γ is given by γ(x) := lim

t→0+

µt(x) − ǫ(x) t for x ∈ Cu

0 (G) s.t. this limit exists.

Dense algebra in D(γ)?

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 14 / 25

Domains of generating functionals

Theorem 1 (Skalski–V)

For every LCQG G, the domain of the generating functional γ of every w∗-continuous, symmetric, convolution semigroup of states on G contains a dense ∗-subalgebra.

Remarks

1

We define D+ as some universal cone of “twisted pos-def funcs”.

2

The subalgebra is span(D+ ∩ D(γ)).

3

Typically, γ does not have reasonable closability properties.

4

A core-like property: γ “is determined on” D+ ∩ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 15 / 25

Domains of generating functionals

Theorem 1 (Skalski–V)

For every LCQG G, the domain of the generating functional γ of every w∗-continuous, symmetric, convolution semigroup of states on G contains a dense ∗-subalgebra.

Remarks

1

We define D+ as some universal cone of “twisted pos-def funcs”.

2

The subalgebra is span(D+ ∩ D(γ)).

3

Typically, γ does not have reasonable closability properties.

4

A core-like property: γ “is determined on” D+ ∩ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 15 / 25

Domains of generating functionals

Theorem 1 (Skalski–V)

For every LCQG G, the domain of the generating functional γ of every w∗-continuous, symmetric, convolution semigroup of states on G contains a dense ∗-subalgebra.

Remarks

1

We define D+ as some universal cone of “twisted pos-def funcs”.

2

The subalgebra is span(D+ ∩ D(γ)).

3

Typically, γ does not have reasonable closability properties.

4

A core-like property: γ “is determined on” D+ ∩ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 15 / 25

Domains of generating functionals

Theorem 1 (Skalski–V)

For every LCQG G, the domain of the generating functional γ of every w∗-continuous, symmetric, convolution semigroup of states on G contains a dense ∗-subalgebra.

Remarks

1

We define D+ as some universal cone of “twisted pos-def funcs”.

2

The subalgebra is span(D+ ∩ D(γ)).

3

Typically, γ does not have reasonable closability properties.

4

A core-like property: γ “is determined on” D+ ∩ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 15 / 25

Domains of generating functionals

Theorem 1 (Skalski–V)

For every LCQG G, the domain of the generating functional γ of every w∗-continuous, symmetric, convolution semigroup of states on G contains a dense ∗-subalgebra.

Remarks

1

We define D+ as some universal cone of “twisted pos-def funcs”.

2

The subalgebra is span(D+ ∩ D(γ)).

3

Typically, γ does not have reasonable closability properties.

4

A core-like property: γ “is determined on” D+ ∩ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 15 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

Earlier results

∗ G = G

• Hunt’s theorem: C2

0(G) when G is Lie.

• Our algebra is span (Pλ(G) ∩ D(γ)). Generally it does not contain

C2

0(G), but it does contain C4 c (G) for G := R.

∗ G = ˆ G

• Cc(G) ֒→ C∗(G).
• Our algebra is
• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G).

Why is it an algebra?

∗ G is compact

• Pol(G) := the Hopf ∗-alg of “coefficients of all finite-dim. reps”.
• Our algebra contains Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 16 / 25

Domains of generating functionals

The case G = ˆ G

Why is

• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G) an algebra?

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 17 / 25

Domains of generating functionals

The case G = ˆ G

Why is

• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G) an algebra?

∗ The product in L1(G) ֒→ C∗(G) is convolution.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 17 / 25

Domains of generating functionals

The case G = ˆ G

Why is

• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G) an algebra?

∗ The product in L1(G) ֒→ C∗(G) is convolution. ∗ Every CND function θ : G → [0, ∞) satisfies

• θ(ts) ≤
• θ(t) +
• θ(s)

(∀t, s ∈ G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 17 / 25

Domains of generating functionals

The case G = ˆ G

Why is

• f ∈ L1(G) : fθ ∈ L1(G)
• ֒→ C∗(G) an algebra?

∗ The product in L1(G) ֒→ C∗(G) is convolution. ∗ Every CND function θ : G → [0, ∞) satisfies

• θ(ts) ≤
• θ(t) +
• θ(s)

(∀t, s ∈ G). ∗ Hence,

• (f ⋆ g)θ
• 1 ≤ fθ1 g1 + f1 gθ1

(∀f, g ∈ L1(G)).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 17 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1)

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ For t ≥ 0, take µ := µt/2 in (1). Then µt(f) = (µt/2 ⋆ µt/2)(f) ≥ 0. Also, µt(f) ≤ f∞ = µ0(f). Thus, µ0(f) − µt(f) ≥ 0. ∗ One can show that (0, ∞) ∋ t → µ0(f) − µt(f) t is decreasing. For instance: setting µ := µ0 − µt in (1) gives (µ0 − 2µt + µ2t) (f) ≥ 0, thus µ0(f)−µt(f)

t

≥ µ0(f)−µ2t(f)

2t

.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ For t ≥ 0, take µ := µt/2 in (1). Then µt(f) = (µt/2 ⋆ µt/2)(f) ≥ 0. Also, µt(f) ≤ f∞ = µ0(f). Thus, µ0(f) − µt(f) ≥ 0. ∗ One can show that (0, ∞) ∋ t → µ0(f) − µt(f) t is decreasing. For instance: setting µ := µ0 − µt in (1) gives (µ0 − 2µt + µ2t) (f) ≥ 0, thus µ0(f)−µt(f)

t

≥ µ0(f)−µ2t(f)

2t

.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ For t ≥ 0, take µ := µt/2 in (1). Then µt(f) = (µt/2 ⋆ µt/2)(f) ≥ 0. Also, µt(f) ≤ f∞ = µ0(f). Thus, µ0(f) − µt(f) ≥ 0. ∗ One can show that (0, ∞) ∋ t → µ0(f) − µt(f) t is decreasing. For instance: setting µ := µ0 − µt in (1) gives (µ0 − 2µt + µ2t) (f) ≥ 0, thus µ0(f)−µt(f)

t

≥ µ0(f)−µ2t(f)

2t

.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ (decreasing – cont.)

• For general t > s > 0, WLOG t

s = m n with n, m ∈ 2N, n > m.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ (decreasing – cont.)

• For general t > s > 0, WLOG t

s = m n with n, m ∈ 2N, n > m.

• The real polynomial mxn − nxm + n − m is non-negative, thus a sos.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ (decreasing – cont.)

• For general t > s > 0, WLOG t

s = m n with n, m ∈ 2N, n > m.

• The real polynomial mxn − nxm + n − m is non-negative, thus a sos.
• Applying (1) gives the desired inequality.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 1/2

(µt)t≥0 – w∗-cont. symm. conv. semigroup of prob. measures on G. Let f : G → C be positive definite: (µ∗ ⋆ µ)(f) ≥ 0 (∀complex measure µ). (1) ∗ (decreasing – cont.)

• For general t > s > 0, WLOG t

s = m n with n, m ∈ 2N, n > m.

• The real polynomial mxn − nxm + n − m is non-negative, thus a sos.
• Applying (1) gives the desired inequality.

∗ In conclusion: f ∈ D(γ)

def

⇐⇒ lim

t→0+

µ0(f) − µt(f) t exists ⇐⇒ sup

t>0

µ0(f) − µt(f) t < ∞.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 18 / 25

Domains of generating functionals

Proof when G = G: part 2/2

∗ For normalized positive-definite functions f, g, we have

• 1 − µt(fg) ≤
• 1 − µt(f) +
• 1 − µt(g)

(∀t ≥ 0) by simple measure theory. ∗ So if f, g ∈ D(γ), namely sup

t>0

µ0(f) − µt(f) t < ∞ and sup

t>0

µ0(g) − µt(g) t < ∞, then also sup

t>0

µ0(fg) − µt(fg) t < ∞ namely fg ∈ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 19 / 25

Domains of generating functionals

Proof when G = G: part 2/2

∗ For normalized positive-definite functions f, g, we have

• 1 − µt(fg) ≤
• 1 − µt(f) +
• 1 − µt(g)

(∀t ≥ 0) by simple measure theory. ∗ So if f, g ∈ D(γ), namely sup

t>0

µ0(f) − µt(f) t < ∞ and sup

t>0

µ0(g) − µt(g) t < ∞, then also sup

t>0

µ0(fg) − µt(fg) t < ∞ namely fg ∈ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 19 / 25

Domains of generating functionals

Proof when G = G: part 2/2

∗ For normalized positive-definite functions f, g, we have

• 1 − µt(fg) ≤
• 1 − µt(f) +
• 1 − µt(g)

(∀t ≥ 0) by simple measure theory. ∗ So if f, g ∈ D(γ), namely sup

t>0

µ0(f) − µt(f) t < ∞ and sup

t>0

µ0(g) − µt(g) t < ∞, then also sup

t>0

µ0(fg) − µt(fg) t < ∞ namely fg ∈ D(γ).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 19 / 25

Domains of generating functionals

Consequence and future directions

∗ Associate cocycles to the generating functionals. Let A be our dense ∗-subalgebra. There exist:

• an inner product space H;
• a rep π of A on H;
• a π-ǫ derivative c : A → H s.t.

γ(b∗a) = c(a), c(b) + γ(a)ǫ(b) + ǫ(a)γ(b) for all a, b ∈ A .

∗ Introduce and study notions such as Gaussianity and Lévy–Khintchine decompositions of convolution semigroups (the compact setting: Franz–Gerhold–Thom). ∗ Establish a 1 − 1 correspondence between generating functionals and cocycles.

• When G = ˆ

G, this would be the correspondence γ(g) =

• c(g)
• 2

(γ – CND, c – cocycle).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 20 / 25

Domains of generating functionals

Consequence and future directions

∗ Associate cocycles to the generating functionals. Let A be our dense ∗-subalgebra. There exist:

• an inner product space H;
• a rep π of A on H;
• a π-ǫ derivative c : A → H s.t.

γ(b∗a) = c(a), c(b) + γ(a)ǫ(b) + ǫ(a)γ(b) for all a, b ∈ A .

∗ Introduce and study notions such as Gaussianity and Lévy–Khintchine decompositions of convolution semigroups (the compact setting: Franz–Gerhold–Thom). ∗ Establish a 1 − 1 correspondence between generating functionals and cocycles.

• When G = ˆ

G, this would be the correspondence γ(g) =

• c(g)
• 2

(γ – CND, c – cocycle).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 20 / 25

Domains of generating functionals

Consequence and future directions

∗ Associate cocycles to the generating functionals. Let A be our dense ∗-subalgebra. There exist:

• an inner product space H;
• a rep π of A on H;
• a π-ǫ derivative c : A → H s.t.

γ(b∗a) = c(a), c(b) + γ(a)ǫ(b) + ǫ(a)γ(b) for all a, b ∈ A .

∗ Introduce and study notions such as Gaussianity and Lévy–Khintchine decompositions of convolution semigroups (the compact setting: Franz–Gerhold–Thom). ∗ Establish a 1 − 1 correspondence between generating functionals and cocycles.

• When G = ˆ

G, this would be the correspondence γ(g) =

• c(g)
• 2

(γ – CND, c – cocycle).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 20 / 25

Domains of generating functionals

Consequence and future directions

∗ Associate cocycles to the generating functionals. Let A be our dense ∗-subalgebra. There exist:

• an inner product space H;
• a rep π of A on H;
• a π-ǫ derivative c : A → H s.t.

γ(b∗a) = c(a), c(b) + γ(a)ǫ(b) + ǫ(a)γ(b) for all a, b ∈ A .

∗ Introduce and study notions such as Gaussianity and Lévy–Khintchine decompositions of convolution semigroups (the compact setting: Franz–Gerhold–Thom). ∗ Establish a 1 − 1 correspondence between generating functionals and cocycles.

• When G = ˆ

G, this would be the correspondence γ(g) =

• c(g)
• 2

(γ – CND, c – cocycle).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 20 / 25

Domains of generating functionals

Consequence and future directions

∗ Associate cocycles to the generating functionals. Let A be our dense ∗-subalgebra. There exist:

• an inner product space H;
• a rep π of A on H;
• a π-ǫ derivative c : A → H s.t.

γ(b∗a) = c(a), c(b) + γ(a)ǫ(b) + ǫ(a)γ(b) for all a, b ∈ A .

∗ Introduce and study notions such as Gaussianity and Lévy–Khintchine decompositions of convolution semigroups (the compact setting: Franz–Gerhold–Thom). ∗ Establish a 1 − 1 correspondence between generating functionals and cocycles.

• When G = ˆ

G, this would be the correspondence γ(g) =

• c(g)
• 2

(γ – CND, c – cocycle).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 20 / 25

Domains of generating functionals

Consequence and future directions

∗ Associate cocycles to the generating functionals. Let A be our dense ∗-subalgebra. There exist:

• an inner product space H;
• a rep π of A on H;
• a π-ǫ derivative c : A → H s.t.

γ(b∗a) = c(a), c(b) + γ(a)ǫ(b) + ǫ(a)γ(b) for all a, b ∈ A .

∗ Introduce and study notions such as Gaussianity and Lévy–Khintchine decompositions of convolution semigroups (the compact setting: Franz–Gerhold–Thom). ∗ Establish a 1 − 1 correspondence between generating functionals and cocycles.

• When G = ˆ

G, this would be the correspondence γ(g) =

• c(g)
• 2

(γ – CND, c – cocycle).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 20 / 25

A reconstruction theorem

Recall:

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R. Quantum analog? The generating functional γ of a w∗-cont. convolution semigroup of states (µt)t≥0 on G is hermitian. If x ∈ D(γ) ∩ ker ǫ is positive, then −γ(x) := lim

t→0+

ǫ(x) − µt(x) t = lim

t→0+

−µt(x) t ≤ 0. This means that −γ is conditionally negative. Also, if G is compact (Cu

0 (G) is unital), then γ(1) = 0.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 21 / 25

A reconstruction theorem

Recall:

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R. Quantum analog? The generating functional γ of a w∗-cont. convolution semigroup of states (µt)t≥0 on G is hermitian. If x ∈ D(γ) ∩ ker ǫ is positive, then −γ(x) := lim

t→0+

ǫ(x) − µt(x) t = lim

t→0+

−µt(x) t ≤ 0. This means that −γ is conditionally negative. Also, if G is compact (Cu

0 (G) is unital), then γ(1) = 0.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 21 / 25

A reconstruction theorem

Recall:

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R. Quantum analog? The generating functional γ of a w∗-cont. convolution semigroup of states (µt)t≥0 on G is hermitian. If x ∈ D(γ) ∩ ker ǫ is positive, then −γ(x) := lim

t→0+

ǫ(x) − µt(x) t = lim

t→0+

−µt(x) t ≤ 0. This means that −γ is conditionally negative. Also, if G is compact (Cu

0 (G) is unital), then γ(1) = 0.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 21 / 25

A reconstruction theorem

Recall:

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R. Quantum analog? The generating functional γ of a w∗-cont. convolution semigroup of states (µt)t≥0 on G is hermitian. If x ∈ D(γ) ∩ ker ǫ is positive, then −γ(x) := lim

t→0+

ǫ(x) − µt(x) t = lim

t→0+

−µt(x) t ≤ 0. This means that −γ is conditionally negative. Also, if G is compact (Cu

0 (G) is unital), then γ(1) = 0.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 21 / 25

A reconstruction theorem

Recall:

Corollary of Schönberg’s Theorem

The w∗-cont. semigroups of symmetric, normalized, positive-definite functions on G are exactly (e−tθ)t≥0 for a CND function θ : G → R. Quantum analog? The generating functional γ of a w∗-cont. convolution semigroup of states (µt)t≥0 on G is hermitian. If x ∈ D(γ) ∩ ker ǫ is positive, then −γ(x) := lim

t→0+

ǫ(x) − µt(x) t = lim

t→0+

−µt(x) t ≤ 0. This means that −γ is conditionally negative. Also, if G is compact (Cu

0 (G) is unital), then γ(1) = 0.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 21 / 25

A reconstruction theorem

Theorem (Schürmann’s reconstruction theorem)

On a ∗-bialgebra, a functional γ satisfying these 3 conditions generates a convolution semigroup of states. It is given by µt := exp⋆(tγ) =

∞

• n=0

tn n!γ⋆n (t ≥ 0).

Corollary

Schürmann’s result holds for compact quantum groups G on the ∗-bialgebra Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 22 / 25

A reconstruction theorem

Theorem (Schürmann’s reconstruction theorem)

On a ∗-bialgebra, a functional γ satisfying these 3 conditions generates a convolution semigroup of states. It is given by µt := exp⋆(tγ) =

∞

• n=0

tn n!γ⋆n (t ≥ 0).

Corollary

Schürmann’s result holds for compact quantum groups G on the ∗-bialgebra Pol(G).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 22 / 25

A reconstruction theorem

Theorem 2 (Skalski–V)

A reconstruction theorem for arbitrary LCQGs holds under several assumptions including symmetry. In particular, this is an “exponentiation” result: exp⋆(tγ) is meaningless here.

Corollary 1

If G admits an unbounded functional for which Theorem 2 applies, then ˆ G does not have property (T).

Corollary 2

Theorem 2 can be used to give a different proof of the compact case (assuming symmetry).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 23 / 25

A reconstruction theorem

Theorem 2 (Skalski–V)

A reconstruction theorem for arbitrary LCQGs holds under several assumptions including symmetry. In particular, this is an “exponentiation” result: exp⋆(tγ) is meaningless here.

Corollary 1

If G admits an unbounded functional for which Theorem 2 applies, then ˆ G does not have property (T).

Corollary 2

Theorem 2 can be used to give a different proof of the compact case (assuming symmetry).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 23 / 25

A reconstruction theorem

Theorem 2 (Skalski–V)

A reconstruction theorem for arbitrary LCQGs holds under several assumptions including symmetry. In particular, this is an “exponentiation” result: exp⋆(tγ) is meaningless here.

Corollary 1

If G admits an unbounded functional for which Theorem 2 applies, then ˆ G does not have property (T).

Corollary 2

Theorem 2 can be used to give a different proof of the compact case (assuming symmetry).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 23 / 25

A reconstruction theorem

Theorem 2 (Skalski–V)

A reconstruction theorem for arbitrary LCQGs holds under several assumptions including symmetry. In particular, this is an “exponentiation” result: exp⋆(tγ) is meaningless here.

Corollary 1

If G admits an unbounded functional for which Theorem 2 applies, then ˆ G does not have property (T).

Corollary 2

Theorem 2 can be used to give a different proof of the compact case (assuming symmetry).

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 23 / 25

Closing remarks

The proofs of Theorems 1 & 2 are based on previous work about the non-commutative Dirichlet forms associated to symmetric convolution semigroups.

Further remarks on Theorem 2

The technical assumptions involve:

◮ a special density condition for D(γ); ◮ a lower semi-continuity property.

Several variants, one yields uniqueness.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 24 / 25

Closing remarks

The proofs of Theorems 1 & 2 are based on previous work about the non-commutative Dirichlet forms associated to symmetric convolution semigroups.

Further remarks on Theorem 2

The technical assumptions involve:

◮ a special density condition for D(γ); ◮ a lower semi-continuity property.

Several variants, one yields uniqueness.

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 24 / 25

• A. Skalski and A. Viselter,

Generating functionals for locally compact quantum groups. Preprint.

• A. Skalski and A. Viselter, Convolution semigroups on locally compact quantum groups and

noncommutative Dirichlet forms.

• J. Math. Pures Appl. 124 (2019), 59–105.
• U. Franz, M. Gerhold and A. Thom,

On the Lévy–Khinchin decomposition of generating functionals.

• Commun. Stoch. Anal. 9 (2015), no. 4, 529–544.
• M. Schürmann,

Positive and conditionally positive linear functionals on coalgebras. Quantum prob. and appl. II, Lecture Notes in Math. 1136 (1985).

• U. Haagerup,

An example of a non nuclear C∗-algebra, which has the metric approximation property.

• Invent. Math. 50 (1978/79), no. 3, 279–293.
• G. A. Hunt,

Semi-groups of measures on Lie groups.

• Trans. Amer. Math. Soc. 81 (1956), no. 2, 264–293.

Thank you for your attention!

Ami Viselter (University of Haifa) Generating functionals on quantum groups QGs and their Analysis, Oslo, ’19 25 / 25