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 2 G; Borel set E ⊂ G; where xE = fxy : y 2 Eg. 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 2 G; Borel set E ⊂ G; where xE = fxy : y 2 Eg. 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 2 G, f 2 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 2 G, f 2 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 2 (0; 1) such that λ = cµ. Examples: dx on R δz on Z −n j det T j 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 2 (0; 1) such that λ = cµ. Examples: dx on R δz on Z −n j det T j 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 2 (0; 1) such that λ = cµ. Examples: dx on R δz on Z −n j det T j 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 2 G, define λx (E) := λ(Ex), =) λx is again a left Haar measure, =) 9∆(x) > 0 such that λx = ∆(x)λ. Definition The function ∆ : G ! (0; 1) 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 2 G, define λx (E) := λ(Ex), =) λx is again a left Haar measure, =) 9∆(x) > 0 such that λx = ∆(x)λ. Definition The function ∆ : G ! (0; 1) 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 2 G, define λx (E) := λ(Ex), =) λx is again a left Haar measure, =) 9∆(x) > 0 such that λx = ∆(x)λ. Definition The function ∆ : G ! (0; 1) 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 2 G, define λx (E) := λ(Ex), =) λx is again a left Haar measure, =) 9∆(x) > 0 such that λx = ∆(x)λ. Definition The function ∆ : G ! (0; 1) 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 2 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 2 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.