Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Group C*-algebras

Ying-Fen Lin

Pure Mathematics Research Centre Queen’s University Belfast

Summer School in Operator Algebras, Athens July 2016

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

If G is a group with a topology such that

(s, t) 7→ st and s 7→ s−1 are continuous, then G is called a topological group. A locally compact group is a topological group which is locally compact and Hausdorff. Examples: R, Z, T, GL(n, R)...

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

If G is a group with a topology such that

(s, t) 7→ st and s 7→ s−1 are continuous, then G is called a topological group. A locally compact group is a topological group which is locally compact and Hausdorff. Examples: R, Z, T, GL(n, R)...

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

If G is a group with a topology such that

(s, t) 7→ st and s 7→ s−1 are continuous, then G is called a topological group. A locally compact group is a topological group which is locally compact and Hausdorff. Examples: R, Z, T, GL(n, R)...

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Haar measure

A left Haar measure on G is a nonzero Radon measure µ on G such that

µ(xE) = µ(E) for x ∈ G, Borel set E ⊂ G,

where xE = {xy : y ∈ E}.

Similar definition for right Haar measure.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Haar measure

A left Haar measure on G is a nonzero Radon measure µ on G such that

µ(xE) = µ(E) for x ∈ G, Borel set E ⊂ G,

where xE = {xy : y ∈ E}.

Similar definition for right Haar measure.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Proposition

1 µ is a left Haar measure if and only if µ˜(E) := µ(E −1) is a right Haar measure. R R 2 µ is a left Haar measure if and only if Ly fdµ = fdµ, for all + −1 y ∈ G, f ∈ Cc (G), where Ly f (x) := f (y x) is the left translation of the function f on G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Proposition

1 µ is a left Haar measure if and only if µ˜(E) := µ(E −1) is a right Haar measure. R R 2 µ is a left Haar measure if and only if Ly fdµ = fdµ, for all + −1 y ∈ G, f ∈ Cc (G), where Ly f (x) := f (y x) is the left translation of the function f on G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Theorem Every locally compact group G has a left Haar measure.

Theorem (Uniqueness) If λ, µ are left Haar measures on G, then there exists c ∈ (0, ∞) such that λ = cµ.

Examples: dx on R δz on Z −n | det T | dT on GL(n, R), where dT is Lebesgue measure on Mn(R).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Theorem Every locally compact group G has a left Haar measure.

Theorem (Uniqueness) If λ, µ are left Haar measures on G, then there exists c ∈ (0, ∞) such that λ = cµ.

Examples: dx on R δz on Z −n | det T | dT on GL(n, R), where dT is Lebesgue measure on Mn(R).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Theorem Every locally compact group G has a left Haar measure.

Theorem (Uniqueness) If λ, µ are left Haar measures on G, then there exists c ∈ (0, ∞) such that λ = cµ.

Examples: dx on R δz on Z −n | det T | dT on GL(n, R), where dT is Lebesgue measure on Mn(R).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Modular function

Let G be a locally compact group, λ be a left Haar measure. For any x ∈ G, define λx (E) := λ(Ex), =⇒ λx is again a left Haar measure, =⇒ ∃∆(x) > 0 such that λx = ∆(x)λ.

Definition The function ∆ : G → (0, ∞) is called a modular function of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Modular function

Let G be a locally compact group, λ be a left Haar measure. For any x ∈ G, define λx (E) := λ(Ex), =⇒ λx is again a left Haar measure, =⇒ ∃∆(x) > 0 such that λx = ∆(x)λ.

Definition The function ∆ : G → (0, ∞) is called a modular function of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Modular function

Let G be a locally compact group, λ be a left Haar measure. For any x ∈ G, define λx (E) := λ(Ex), =⇒ λx is again a left Haar measure, =⇒ ∃∆(x) > 0 such that λx = ∆(x)λ.

Definition The function ∆ : G → (0, ∞) is called a modular function of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Modular function

Let G be a locally compact group, λ be a left Haar measure. For any x ∈ G, define λx (E) := λ(Ex), =⇒ λx is again a left Haar measure, =⇒ ∃∆(x) > 0 such that λx = ∆(x)λ.

Definition The function ∆ : G → (0, ∞) is called a modular function of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Proposition ∆ is a continuous homomorphism. 1 R −1 R For every f ∈ L (λ), Ry fdλ = ∆(y ) fdλ, where Ry f (x) := f (xy) is a right translation of f on G. For every left Haar measure λ, the associated right Haar measure ρ satisfies

dρ(x) = ∆(x−1)dλ(x).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Proposition ∆ is a continuous homomorphism. 1 R −1 R For every f ∈ L (λ), Ry fdλ = ∆(y ) fdλ, where Ry f (x) := f (xy) is a right translation of f on G. For every left Haar measure λ, the associated right Haar measure ρ satisfies

dρ(x) = ∆(x−1)dλ(x).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Unitary representations

Definition

Let π : G → U(Hπ) be a homomorphism and continuous in the SOT, i.e. −1 ∗ 1 π(xy) = π(x)π(y), π(x−1) = π(x) = π(x) and

2 x 7→ π(x)u is continuous from G to Hπ.

We call π a unitary representation of G on Hπ and denote dim π := dim Hπ.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Unitary representations

Definition

Let π : G → U(Hπ) be a homomorphism and continuous in the SOT, i.e. −1 ∗ 1 π(xy) = π(x)π(y), π(x−1) = π(x) = π(x) and

2 x 7→ π(x)u is continuous from G to Hπ.

We call π a unitary representation of G on Hπ and denote dim π := dim Hπ.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Unitary representations

Definition

Let π : G → U(Hπ) be a homomorphism and continuous in the SOT, i.e. −1 ∗ 1 π(xy) = π(x)π(y), π(x−1) = π(x) = π(x) and

2 x 7→ π(x)u is continuous from G to Hπ.

We call π a unitary representation of G on Hπ and denote dim π := dim Hπ.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Example: Let G be a locally compact group with a left Haar measure. Then left translations yield a unitary representation πL of G on L2(G), called the left regular representation, i.e. by taking

−1 (πL(x)f )(y) := Lx f (y) = f (x y).

Similarly, one can define right regular representation πR of G on L2(G, ρ), where ρ is a right Haar measure.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Example: Let G be a locally compact group with a left Haar measure. Then left translations yield a unitary representation πL of G on L2(G), called the left regular representation, i.e. by taking

−1 (πL(x)f )(y) := Lx f (y) = f (x y).

Similarly, one can define right regular representation πR of G on L2(G, ρ), where ρ is a right Haar measure.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition

Let π1, π2 be two unitary representations of G. We say that π1 and π2 are unitary equivalent if there is a unitary operator

U : Hπ1 → Hπ2 such that

Uπ1(x) = π2(x)U for all x ∈ G.

Remark: −1 πR is unitary equivalent to πL by taking Uf (x) = f (x ).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition

Let π1, π2 be two unitary representations of G. We say that π1 and π2 are unitary equivalent if there is a unitary operator

U : Hπ1 → Hπ2 such that

Uπ1(x) = π2(x)U for all x ∈ G.

Remark: −1 πR is unitary equivalent to πL by taking Uf (x) = f (x ).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition

Let π1, π2 be two unitary representations of G. We say that π1 and π2 are unitary equivalent if there is a unitary operator

U : Hπ1 → Hπ2 such that

Uπ1(x) = π2(x)U for all x ∈ G.

Remark: −1 πR is unitary equivalent to πL by taking Uf (x) = f (x ).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The group algebra

Let G be a locally compact group, dx be a Haar measure on G. The Banach space L1(G) is of integrable functions on G with respect to dx. For f , g ∈ L1(G), the convolution of f and g is defined by Z f ∗ g(x) := f (y)g(y −1x)dy for x ∈ G. G Moreover, the involution defined on L1(G) is given by

f ∗(x) := ∆(x−1)f (x−1) for f ∈ L1(G), x ∈ G.

=⇒ (L1(G), ∗) is a Banach ∗-algebra, called the group algebra of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The group algebra

Let G be a locally compact group, dx be a Haar measure on G. The Banach space L1(G) is of integrable functions on G with respect to dx. For f , g ∈ L1(G), the convolution of f and g is defined by Z f ∗ g(x) := f (y)g(y −1x)dy for x ∈ G. G Moreover, the involution defined on L1(G) is given by

f ∗(x) := ∆(x−1)f (x−1) for f ∈ L1(G), x ∈ G.

=⇒ (L1(G), ∗) is a Banach ∗-algebra, called the group algebra of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The group algebra

Let G be a locally compact group, dx be a Haar measure on G. The Banach space L1(G) is of integrable functions on G with respect to dx. For f , g ∈ L1(G), the convolution of f and g is defined by Z f ∗ g(x) := f (y)g(y −1x)dy for x ∈ G. G Moreover, the involution defined on L1(G) is given by

f ∗(x) := ∆(x−1)f (x−1) for f ∈ L1(G), x ∈ G.

=⇒ (L1(G), ∗) is a Banach ∗-algebra, called the group algebra of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The group algebra

Let G be a locally compact group, dx be a Haar measure on G. The Banach space L1(G) is of integrable functions on G with respect to dx. For f , g ∈ L1(G), the convolution of f and g is defined by Z f ∗ g(x) := f (y)g(y −1x)dy for x ∈ G. G Moreover, the involution defined on L1(G) is given by

f ∗(x) := ∆(x−1)f (x−1) for f ∈ L1(G), x ∈ G.

=⇒ (L1(G), ∗) is a Banach ∗-algebra, called the group algebra of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Representations of L1(G)

Every unitary representation π of G determines a representation (also denoted by π) of L1(G) in the sense: for f ∈ L1(G), an bounded operator π(f ) on Hπ defined by Z π(f ) = f (x)π(x)dx. G

More precisely, for u ∈ Hπ we define Z hπ(f )u, vi := f (x)hπ(x)u, vidx. (1) G

Since hπ(x)u, vi ∈ Cb(G) and hπ(f )u, vi is linear on u, conjugate linear on v and

|hπ(f )u, vi| ≤ kf k1kukkvk,

π(f ) ∈ B(Hπ) with kπ(f )k ≤ kf k1, it is well defined. Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Representations of L1(G)

Every unitary representation π of G determines a representation (also denoted by π) of L1(G) in the sense: for f ∈ L1(G), an bounded operator π(f ) on Hπ defined by Z π(f ) = f (x)π(x)dx. G

More precisely, for u ∈ Hπ we define Z hπ(f )u, vi := f (x)hπ(x)u, vidx. (1) G

Since hπ(x)u, vi ∈ Cb(G) and hπ(f )u, vi is linear on u, conjugate linear on v and

|hπ(f )u, vi| ≤ kf k1kukkvk,

π(f ) ∈ B(Hπ) with kπ(f )k ≤ kf k1, it is well defined. Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Example: Let πL be the left regular representation of G, i.e. πL(x) = Lx . Then πL(f ) is given by Z πL(f )g = f (y)Ly gdy = f ∗ g. G

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Theorem If π is a unitary representation of G, then f 7→ π(f ) is a 1 nondegenerate ∗-representation of L (G) on Hπ, i.e. π is a ∗-homomorphism and there is no u ∈ Hπ such that π(f )u = 0 for all f ∈ L1(G).

Furthermore, for every irreducible representation π of the algebra L1(G) on a Hilbert space H, there exists a unique irreducible unitary representation (π, H) of G such that π(f ) = π(f ) for f ∈ L1(G).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Theorem If π is a unitary representation of G, then f 7→ π(f ) is a 1 nondegenerate ∗-representation of L (G) on Hπ, i.e. π is a ∗-homomorphism and there is no u ∈ Hπ such that π(f )u = 0 for all f ∈ L1(G).

Furthermore, for every irreducible representation π of the algebra L1(G) on a Hilbert space H, there exists a unique irreducible unitary representation (π, H) of G such that π(f ) = π(f ) for f ∈ L1(G).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The dual space of G

Let π be a representation of G on Hπ. If π has no nontrivial invariant subspace of Hπ, we call π an irreducible representation of G. We denote by [π] the equivalence class of π. Remark: If G is abelian, then every irreducible representation of G is one-dimensional.

Definition Let Gb := {[π] : irreducible unitary representation π of G} be the (unitary) dual space of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The dual space of G

Let π be a representation of G on Hπ. If π has no nontrivial invariant subspace of Hπ, we call π an irreducible representation of G. We denote by [π] the equivalence class of π. Remark: If G is abelian, then every irreducible representation of G is one-dimensional.

Definition Let Gb := {[π] : irreducible unitary representation π of G} be the (unitary) dual space of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The dual space of G

Let π be a representation of G on Hπ. If π has no nontrivial invariant subspace of Hπ, we call π an irreducible representation of G. We denote by [π] the equivalence class of π. Remark: If G is abelian, then every irreducible representation of G is one-dimensional.

Definition Let Gb := {[π] : irreducible unitary representation π of G} be the (unitary) dual space of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The dual space of G

Let π be a representation of G on Hπ. If π has no nontrivial invariant subspace of Hπ, we call π an irreducible representation of G. We denote by [π] the equivalence class of π. Remark: If G is abelian, then every irreducible representation of G is one-dimensional.

Definition Let Gb := {[π] : irreducible unitary representation π of G} be the (unitary) dual space of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups If f ∈ L1(G), we define

kf k∗ := sup kπ(f )k, [π]∈Gb

1 then k · k∗ is a norm on L (G) such that kf k∗ ≤ kf k1. Moreover, we have that

kf ∗ gk∗ = sup kπ(f )π(g)k ≤ kf k∗kgk∗, [π]∈Gb ∗ ∗ kf k∗ = sup kπ(f ) k = kf k∗, [π]∈Gb ∗ ∗ 2 2 kf ∗ f k∗ = sup kπ(f ) π(f )k = sup kπ(f )k = kf k∗. [π]∈Gb [π]∈Gb Hence, the convolution and involution on L1(G) can be extended 1 continuously to the completion of L (G) w.r.t k · k∗. Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups If f ∈ L1(G), we define

kf k∗ := sup kπ(f )k, [π]∈Gb

1 then k · k∗ is a norm on L (G) such that kf k∗ ≤ kf k1. Moreover, we have that

kf ∗ gk∗ = sup kπ(f )π(g)k ≤ kf k∗kgk∗, [π]∈Gb ∗ ∗ kf k∗ = sup kπ(f ) k = kf k∗, [π]∈Gb ∗ ∗ 2 2 kf ∗ f k∗ = sup kπ(f ) π(f )k = sup kπ(f )k = kf k∗. [π]∈Gb [π]∈Gb Hence, the convolution and involution on L1(G) can be extended 1 continuously to the completion of L (G) w.r.t k · k∗. Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups If f ∈ L1(G), we define

kf k∗ := sup kπ(f )k, [π]∈Gb

1 then k · k∗ is a norm on L (G) such that kf k∗ ≤ kf k1. Moreover, we have that

kf ∗ gk∗ = sup kπ(f )π(g)k ≤ kf k∗kgk∗, [π]∈Gb ∗ ∗ kf k∗ = sup kπ(f ) k = kf k∗, [π]∈Gb ∗ ∗ 2 2 kf ∗ f k∗ = sup kπ(f ) π(f )k = sup kπ(f )k = kf k∗. [π]∈Gb [π]∈Gb Hence, the convolution and involution on L1(G) can be extended 1 continuously to the completion of L (G) w.r.t k · k∗. Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition The C*-algebra of G is given by

k·k C ∗(G) := L1(G) ∗ with the C*-norm

kf k∗ := sup kπ(f )k. π∈Gb

Note that the dual space C\∗(G) of the C ∗(G) can be identified with the unitary dual Gb of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition The C*-algebra of G is given by

k·k C ∗(G) := L1(G) ∗ with the C*-norm

kf k∗ := sup kπ(f )k. π∈Gb

Note that the dual space C\∗(G) of the C ∗(G) can be identified with the unitary dual Gb of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Let G = Z, an abelian discrete group. Suppose that π is an irreducible representation of Z. Then π is one-dimensional, and hence acts on the Hilbert space C. Since π(1) is unitary, we have that

π(1) ∈ T = {z ∈ C : |z| = 1},

n n say π(1) = ζ. Then π(n) = π(n · 1) = π(1) = ζ , n ∈ Z. Conversely, every ζ ∈ T gives rise to an irreducible representation n of Z (acting on C) by setting π(n) = ζ , n ∈ Z. Thus, Zˆ = T.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Let G = Z, an abelian discrete group. Suppose that π is an irreducible representation of Z. Then π is one-dimensional, and hence acts on the Hilbert space C. Since π(1) is unitary, we have that

π(1) ∈ T = {z ∈ C : |z| = 1},

n n say π(1) = ζ. Then π(n) = π(n · 1) = π(1) = ζ , n ∈ Z. Conversely, every ζ ∈ T gives rise to an irreducible representation n of Z (acting on C) by setting π(n) = ζ , n ∈ Z. Thus, Zˆ = T.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Let G = Z, an abelian discrete group. Suppose that π is an irreducible representation of Z. Then π is one-dimensional, and hence acts on the Hilbert space C. Since π(1) is unitary, we have that

π(1) ∈ T = {z ∈ C : |z| = 1},

n n say π(1) = ζ. Then π(n) = π(n · 1) = π(1) = ζ , n ∈ Z. Conversely, every ζ ∈ T gives rise to an irreducible representation n of Z (acting on C) by setting π(n) = ζ , n ∈ Z. Thus, Zˆ = T.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Let G = Z, an abelian discrete group. Suppose that π is an irreducible representation of Z. Then π is one-dimensional, and hence acts on the Hilbert space C. Since π(1) is unitary, we have that

π(1) ∈ T = {z ∈ C : |z| = 1},

n n say π(1) = ζ. Then π(n) = π(n · 1) = π(1) = ζ , n ∈ Z. Conversely, every ζ ∈ T gives rise to an irreducible representation n of Z (acting on C) by setting π(n) = ζ , n ∈ Z. Thus, Zˆ = T.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Suppose that ζ ∈ T, and let πζ be the corresponding irreducible 1 representation of Z. Let f ∈ ` (Z), say f = (fn)n∈Z. We have that Z X n πζ (f ) = fnπζ (n)dδZ = fnζ . Z n∈Z Let fˆ : T → C be the function given by ˆ X n f (z) = fnz , z ∈ T. n∈Z It follows that kf k∗ = sup kπζ (f )k = kfˆk∞. ζ∈T Since the Laurent polynomials are dense in C(T), it follows that ∗ ∼ C (Z) = C(T).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Suppose that ζ ∈ T, and let πζ be the corresponding irreducible 1 representation of Z. Let f ∈ ` (Z), say f = (fn)n∈Z. We have that Z X n πζ (f ) = fnπζ (n)dδZ = fnζ . Z n∈Z Let fˆ : T → C be the function given by ˆ X n f (z) = fnz , z ∈ T. n∈Z It follows that kf k∗ = sup kπζ (f )k = kfˆk∞. ζ∈T Since the Laurent polynomials are dense in C(T), it follows that ∗ ∼ C (Z) = C(T).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Suppose that ζ ∈ T, and let πζ be the corresponding irreducible 1 representation of Z. Let f ∈ ` (Z), say f = (fn)n∈Z. We have that Z X n πζ (f ) = fnπζ (n)dδZ = fnζ . Z n∈Z Let fˆ : T → C be the function given by ˆ X n f (z) = fnz , z ∈ T. n∈Z It follows that kf k∗ = sup kπζ (f )k = kfˆk∞. ζ∈T Since the Laurent polynomials are dense in C(T), it follows that ∗ ∼ C (Z) = C(T).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups An example on Z

Suppose that ζ ∈ T, and let πζ be the corresponding irreducible 1 representation of Z. Let f ∈ ` (Z), say f = (fn)n∈Z. We have that Z X n πζ (f ) = fnπζ (n)dδZ = fnζ . Z n∈Z Let fˆ : T → C be the function given by ˆ X n f (z) = fnz , z ∈ T. n∈Z It follows that kf k∗ = sup kπζ (f )k = kfˆk∞. ζ∈T Since the Laurent polynomials are dense in C(T), it follows that ∗ ∼ C (Z) = C(T).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

In general, if G is an abelian locally compact group, then

∗ ∼ C (G) = C0(Gˆ).

Question: For non-abelian G, what is C ∗(G)?

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

In general, if G is an abelian locally compact group, then

∗ ∼ C (G) = C0(Gˆ).

Question: For non-abelian G, what is C ∗(G)?

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Algebra of operator fields

Let T be a locally compact Hausdorff space and At , t ∈ T , be a C*-algebra. A *-algebra A of functions x on T is called an algebra of operator fields on T if

x(t) ∈ At for each t ∈ T ; for each x ∈ A, the mapping t 7→ kx(t)k is continuous on T and vanishes at infinity;

for each t ∈ T , the set {x(t): x ∈ A} is dense in At ;

A is complete in the norm kxk := supt∈T kx(t)k. Then A is a C*-algebra with pointwise operation.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Algebra of operator fields

Let T be a locally compact Hausdorff space and At , t ∈ T , be a C*-algebra. A *-algebra A of functions x on T is called an algebra of operator fields on T if

x(t) ∈ At for each t ∈ T ; for each x ∈ A, the mapping t 7→ kx(t)k is continuous on T and vanishes at infinity;

for each t ∈ T , the set {x(t): x ∈ A} is dense in At ;

A is complete in the norm kxk := supt∈T kx(t)k. Then A is a C*-algebra with pointwise operation.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Algebra of operator fields

Let T be a locally compact Hausdorff space and At , t ∈ T , be a C*-algebra. A *-algebra A of functions x on T is called an algebra of operator fields on T if

x(t) ∈ At for each t ∈ T ; for each x ∈ A, the mapping t 7→ kx(t)k is continuous on T and vanishes at infinity;

for each t ∈ T , the set {x(t): x ∈ A} is dense in At ;

A is complete in the norm kxk := supt∈T kx(t)k. Then A is a C*-algebra with pointwise operation.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Some results

Fell, 1961 the problem of representing an arbitrary C*-algebra as an algebra of operator fields and characterised the C*-algebra of SL(2, C)

Gorbach¨ev,1980 Knowing that the group C*-algebra of Hn, the Heisenberg group of dimension 2n + 1, is an extension of an ideal isomorphic to 2 C0(R \{0}, K) with quotient algebra isomorphic to C0(R ), he 2 ∗ gave an expression for a linear map from C0(R ) to C (Hn).

Wang, 1989 characterised the C*-algebra of two-step solvable Lie groups

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Some results

Fell, 1961 the problem of representing an arbitrary C*-algebra as an algebra of operator fields and characterised the C*-algebra of SL(2, C)

Gorbach¨ev,1980 Knowing that the group C*-algebra of Hn, the Heisenberg group of dimension 2n + 1, is an extension of an ideal isomorphic to 2 C0(R \{0}, K) with quotient algebra isomorphic to C0(R ), he 2 ∗ gave an expression for a linear map from C0(R ) to C (Hn).

Wang, 1989 characterised the C*-algebra of two-step solvable Lie groups

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Recent results

Lin and Ludwig, 2010 C*-algebras of ax + b-like groups, i.e., R n V , where V is a finite-dim real vector space and reals act diagonally on VC with α(1+iθ) eigenvalues e , α ∈ R.

Ludwig and Turowska, 2011 C*-algebras of Heisenberg group and thread-like groups GN ;

Abdelmoula, Mounir and Ludwig, 2011 n C*-algebra of motion group SO(n) n R

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Recent results

Lin and Ludwig, 2010 C*-algebras of ax + b-like groups, i.e., R n V , where V is a finite-dim real vector space and reals act diagonally on VC with α(1+iθ) eigenvalues e , α ∈ R.

Ludwig and Turowska, 2011 C*-algebras of Heisenberg group and thread-like groups GN ;

Abdelmoula, Mounir and Ludwig, 2011 n C*-algebra of motion group SO(n) n R

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Recent results

Inoue, Lin and Ludwig, 2015 The group G6 = exp g6, where g6 is a Lie algebra spanned by the vectors A, B, P, Q, R, S and equipped with the Lie brackets: 1 1 [P, Q] = R, [P, R] = S, [A, P] = P, [B, P] = − P, 2 2 1 1 [B, Q] = Q, [A, R] = R, [B, R] = R, [A, S] = S, 2 2 [A, Q] = 0, [B, S] = 0.

The group G6 := exp g6 is a 6-dimensional exponential Lie group.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Exponential Lie groups

A Lie group G is called exponential if G is a connected, simply connected solvable Lie group and the mapping exp : g → G is a diffeomorphism. The Kirillov-Bernat-Vergne-Pukanszky theory shows that there is a canonical homeomorphism g∗/G 7→ Gb from the space of co-adjoint orbits of G in the linear dual space g∗ onto the unitary dual space Gˆ of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Exponential Lie groups

A Lie group G is called exponential if G is a connected, simply connected solvable Lie group and the mapping exp : g → G is a diffeomorphism. The Kirillov-Bernat-Vergne-Pukanszky theory shows that there is a canonical homeomorphism g∗/G 7→ Gb from the space of co-adjoint orbits of G in the linear dual space g∗ onto the unitary dual space Gˆ of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Exponential Lie groups

A Lie group G is called exponential if G is a connected, simply connected solvable Lie group and the mapping exp : g → G is a diffeomorphism. The Kirillov-Bernat-Vergne-Pukanszky theory shows that there is a canonical homeomorphism g∗/G 7→ Gb from the space of co-adjoint orbits of G in the linear dual space g∗ onto the unitary dual space Gˆ of G.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Fourier transform

To analyze a C*-algebra A, one can use the Fourier transform F to decompose A over its unitary dual Ab. To be able to define this transform, consider the algebra `∞(Ab) of bounded operator fields defined over Ab by

`∞(A) := {A = (A(π) ∈ B(H )) ; b π π∈Ab kAk∞ := sup kA(π)kop < ∞}. π

The space `∞(Ab) is a C*-algebra itself. The Fourier transform F defined by

F(a) :=a ˆ := (π(a)) for a ∈ A π∈Ab

is an injective, hence isometric, homomorphism of A into `∞(Ab). Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Fourier transform

To analyze a C*-algebra A, one can use the Fourier transform F to decompose A over its unitary dual Ab. To be able to define this transform, consider the algebra `∞(Ab) of bounded operator fields defined over Ab by

`∞(A) := {A = (A(π) ∈ B(H )) ; b π π∈Ab kAk∞ := sup kA(π)kop < ∞}. π

The space `∞(Ab) is a C*-algebra itself. The Fourier transform F defined by

F(a) :=a ˆ := (π(a)) for a ∈ A π∈Ab

is an injective, hence isometric, homomorphism of A into `∞(Ab). Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Fourier transform

To analyze a C*-algebra A, one can use the Fourier transform F to decompose A over its unitary dual Ab. To be able to define this transform, consider the algebra `∞(Ab) of bounded operator fields defined over Ab by

`∞(A) := {A = (A(π) ∈ B(H )) ; b π π∈Ab kAk∞ := sup kA(π)kop < ∞}. π

The space `∞(Ab) is a C*-algebra itself. The Fourier transform F defined by

F(a) :=a ˆ := (π(a)) for a ∈ A π∈Ab

is an injective, hence isometric, homomorphism of A into `∞(Ab). Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Fourier transform

To analyze a C*-algebra A, one can use the Fourier transform F to decompose A over its unitary dual Ab. To be able to define this transform, consider the algebra `∞(Ab) of bounded operator fields defined over Ab by

`∞(A) := {A = (A(π) ∈ B(H )) ; b π π∈Ab kAk∞ := sup kA(π)kop < ∞}. π

The space `∞(Ab) is a C*-algebra itself. The Fourier transform F defined by

F(a) :=a ˆ := (π(a)) for a ∈ A π∈Ab

is an injective, hence isometric, homomorphism of A into `∞(Ab). Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The C*-algebra of ax + b-(like) group (L., Ludwig)

The C*-algebra of ax + b-group G is isomorphic to the space D∗(G) of all operator fields A defined over the set Γ := D ∪ S ∪ {0n} such that 2 2 A(f ) ∈ K(L (R)) for f ∈ D, A(f ) ∈ B(L (R)) for f ∈ S and ∗ A(0n) ∈ C (R × V0) with 1 the mapping f 7→ A(f ) is continuous on D ∪ S, 2 A satisfies the infinity condition, 3 A satisfies the generic property, 4 A satisfies the compact condition. Indeed, the Fourier transform F defined by

F(a) :=a ˆ = (π(a)) π∈C\∗(G) maps C ∗(G) onto D∗(G).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The C*-algebra of ax + b-(like) group (L., Ludwig)

The C*-algebra of ax + b-group G is isomorphic to the space D∗(G) of all operator fields A defined over the set Γ := D ∪ S ∪ {0n} such that 2 2 A(f ) ∈ K(L (R)) for f ∈ D, A(f ) ∈ B(L (R)) for f ∈ S and ∗ A(0n) ∈ C (R × V0) with 1 the mapping f 7→ A(f ) is continuous on D ∪ S, 2 A satisfies the infinity condition, 3 A satisfies the generic property, 4 A satisfies the compact condition. Indeed, the Fourier transform F defined by

F(a) :=a ˆ = (π(a)) π∈C\∗(G) maps C ∗(G) onto D∗(G).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups The C*-algebra of ax + b-(like) group (L., Ludwig)

The C*-algebra of ax + b-group G is isomorphic to the space D∗(G) of all operator fields A defined over the set Γ := D ∪ S ∪ {0n} such that 2 2 A(f ) ∈ K(L (R)) for f ∈ D, A(f ) ∈ B(L (R)) for f ∈ S and ∗ A(0n) ∈ C (R × V0) with 1 the mapping f 7→ A(f ) is continuous on D ∪ S, 2 A satisfies the infinity condition, 3 A satisfies the generic property, 4 A satisfies the compact condition. Indeed, the Fourier transform F defined by

F(a) :=a ˆ = (π(a)) π∈C\∗(G) maps C ∗(G) onto D∗(G).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Infinity condition)

We say that A satisfies the infinity condition if kA(f )kop → 0 if f → ∞ in D, and if for every pair of real sequences (Rk )k , (Sk )k such that Rk > Sk → +∞, limk (Rk − Sk ) = ∞ we have that

lim sup kM(−∞,−Rk ) ◦ A(f ) ◦ M(−Sk ,∞)kop = 0 k f ∈D∪S and

lim sup kM(Rk ,∞) ◦ A(f ) ◦ M(−∞,Sk )kop = 0, k f ∈D∪S and the same conditions hold for the field A∗. 2 2 Here M(s,t) : L (R) → L (R) is the multiplication operator given by M(s,t)f = f χ(s,t) of the interval (s, t).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Infinity condition)

We say that A satisfies the infinity condition if kA(f )kop → 0 if f → ∞ in D, and if for every pair of real sequences (Rk )k , (Sk )k such that Rk > Sk → +∞, limk (Rk − Sk ) = ∞ we have that

lim sup kM(−∞,−Rk ) ◦ A(f ) ◦ M(−Sk ,∞)kop = 0 k f ∈D∪S and

lim sup kM(Rk ,∞) ◦ A(f ) ◦ M(−∞,Sk )kop = 0, k f ∈D∪S and the same conditions hold for the field A∗. 2 2 Here M(s,t) : L (R) → L (R) is the multiplication operator given by M(s,t)f = f χ(s,t) of the interval (s, t).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Infinity condition)

We say that A satisfies the infinity condition if kA(f )kop → 0 if f → ∞ in D, and if for every pair of real sequences (Rk )k , (Sk )k such that Rk > Sk → +∞, limk (Rk − Sk ) = ∞ we have that

lim sup kM(−∞,−Rk ) ◦ A(f ) ◦ M(−Sk ,∞)kop = 0 k f ∈D∪S and

lim sup kM(Rk ,∞) ◦ A(f ) ◦ M(−∞,Sk )kop = 0, k f ∈D∪S and the same conditions hold for the field A∗. 2 2 Here M(s,t) : L (R) → L (R) is the multiplication operator given by M(s,t)f = f χ(s,t) of the interval (s, t).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Generic condition

2 Let A = (A(f ) ∈ B(L (R)), f ∈ Γ) be a field of bounded operators. We say that A satisfies the generic condition if for every properly

converging sequence (πfk )k ⊂ Gb with fk ∈ D for every k ∈ N,

which admits limit points π(f0,0,f−), π(f0,f+,0) and for every pair of sequences (ρk , σk )k adapted to the sequence (fk )k we have that

lim kU(rk )◦A(fk )◦U(−rk )◦M(−∞,σ )−A(f0, f+, 0)◦M(−∞,σ )kop = 0, k→∞ k k

lim kU(sk )◦A(fk )◦U(−sk )◦M(−ρ ,∞)−A(f0, 0, f−)◦M(−ρ ,∞)kop = 0, k→∞ k k 2 where U(r) is a unitary operator on L (R) defined by

2 U(r)ξ(s) := ξ(s + r) for all ξ ∈ L (R), s ∈ R.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Compact condition) 2 Let A = (A(f ) ∈ B(L (R)), f ∈ Γ) be a field of bounded operators. We say that A satisfies the compact condition if

2 A(f ) − πf (β(A(0n))) ∈ K(L (R)) for every f ∈ S, where β : Cc (R × V0) → Cc (G) is given by

β(ϕ)(s, v0, vn) := ϕ(s, v0)q(vn) for ϕ ∈ Cc (R × V0), vn ∈ Vn, where q ∈ C (V ) with q ≥ 0 and R q(v)dv = 1. c n Vn

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Compact condition) 2 Let A = (A(f ) ∈ B(L (R)), f ∈ Γ) be a field of bounded operators. We say that A satisfies the compact condition if

2 A(f ) − πf (β(A(0n))) ∈ K(L (R)) for every f ∈ S, where β : Cc (R × V0) → Cc (G) is given by

β(ϕ)(s, v0, vn) := ϕ(s, v0)q(vn) for ϕ ∈ Cc (R × V0), vn ∈ Vn, where q ∈ C (V ) with q ≥ 0 and R q(v)dv = 1. c n Vn

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups A 6-dimensional exponential Lie group

Let g be a Lie algebra spanned by the vectors A, B, P, Q, R, S and equipped with the Lie brackets: 1 1 [P, Q] = R, [P, R] = S, [A, P] = P, [B, P] = − P, 2 2 1 1 [B, Q] = Q, [A, R] = R, [B, R] = R, [A, S] = S, 2 2 [A, Q] = 0, [B, S] = 0.

The group G6 := exp g is a 6-dimensional exponential Lie group.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Theorem (Inoue, L. and Ludwig, 2015) ∗ The group C*-algebra C (G6) is an almost C0(K)-C*-algebra.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Inoue, L. and Ludwig, 2015) Let A be a C*-algebra satisfying the conditions:

1 The mappings γ 7→ F(a)(γ) are norm continuous on each Γi .

2 For each i = 1, ··· , d, there is a sequence (σi,k )k of linear mappings σi,k : CB(Si−1) → CB(Si ) which are uniformly bounded in k such that   lim dis (σi,k (F(a)|S ) − F(a)|Γ ), C0(Γi , K(Hi )) = 0, k→∞ i−1 i  ∗ ∗  lim dis (σi,k (F(a)|S ) − F(a )|Γi ), C0(Γi , K(Hi )) = 0. k→∞ i−1

We call such a C*-algebra almost C0(K).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

Definition (Inoue, L. and Ludwig, 2015) Let A be a C*-algebra satisfying the conditions:

1 The mappings γ 7→ F(a)(γ) are norm continuous on each Γi .

2 For each i = 1, ··· , d, there is a sequence (σi,k )k of linear mappings σi,k : CB(Si−1) → CB(Si ) which are uniformly bounded in k such that   lim dis (σi,k (F(a)|S ) − F(a)|Γ ), C0(Γi , K(Hi )) = 0, k→∞ i−1 i  ∗ ∗  lim dis (σi,k (F(a)|S ) − F(a )|Γi ), C0(Γi , K(Hi )) = 0. k→∞ i−1

We call such a C*-algebra almost C0(K).

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Another example

Let Gn,µ be an exponential Lie group with Lie algebra gn,µ s.t.

gn,µ is an extension of Heisenberg Lie algebra hn by reals; and

the quotient group Gn,µ/Z(H) is an ax + b-like group, where Z(H) is the centre of the Heisenberg Lie group H = exp hn.

Theorem (Inoue, L. and Ludwig, 2016) ∗ The C*-algebra C (Gn,µ) of Gn,µ is an almost C0(K)-C*-algebra.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups Another example

Let Gn,µ be an exponential Lie group with Lie algebra gn,µ s.t.

gn,µ is an extension of Heisenberg Lie algebra hn by reals; and

the quotient group Gn,µ/Z(H) is an ax + b-like group, where Z(H) is the centre of the Heisenberg Lie group H = exp hn.

Theorem (Inoue, L. and Ludwig, 2016) ∗ The C*-algebra C (Gn,µ) of Gn,µ is an almost C0(K)-C*-algebra.

Ying-Fen Lin Group C*-algebras Locally compact groups The group algebra The group C*-algebra C*-algebra of operator fields On exponential Lie groups

other examples!?...

Ying-Fen Lin Group C*-algebras