Operator valued Fourier transforms on nilpotent Lie groups Daniel Beltit ¸˘ a Institute of Mathematics of the Romanian Academy *** Joint work with Ingrid Beltit ¸˘ a (IMAR) and Jean Ludwig (UL) Hamburg, 16.02.2015 Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 1 / 14
Reference a, D. B., J. Ludwig , Fourier transforms of C ∗ -algebras ◮ I. Beltit ¸˘ of nilpotent Lie groups. Preprint arXiv:1411.3254 [math.OA]. Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 2 / 14
Plan 1 Motivation: continuity properties of the Kirillov correspondence 2 Operator-valued Fourier transforms: continuity of operator fields 3 Tools from C ∗ -algebra extension theory: Busby invariant, completely positive lifting 4 C ∗ -algebras of nilpotent Lie groups: stratifications of the dual, C ∗ -solvability 5 Application to Heisenberg groups Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 3 / 14
Motivation (1): Lie group representations • Nilpotent Lie group G = ( g , · ): finite-dim. R -linear space g with polynomial group law satisfying ( sx ) · ( tx ) = ( s + t ) x for s , t ∈ R , x ∈ g • � G := unitary equivalence classes [ π ] of unirreps π : G → U ( H π ) G ∼ • Kirillov correspondence: κ : � → g ∗ / Ad ∗ G (=the coadjoint G -orbits) G : G × g ∗ → g ∗ where Ad ∗ � κ ⇒ ( ∀ ϕ ∈ C ∞ Recall: [ π ] ← → O ⇐ c ( g )) Tr π ( ϕ ) = ϕ � O Goal continuity properties of the bijection κ • Regular representation λ : L 1 ( G ) → B ( L 2 ( G )), λ ( f ) ϕ = f ∗ ϕ �·� ⊆ B ( L 2 ( G )) ❀ � G ≃ � • C ∗ ( G ) := λ ( L 1 ( G )) C ∗ ( G ) Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 4 / 14
Motivation (2): C ∗ -algebra representations • C ∗ -alg. A ❀ spectrum � A := { [ π ] | π : A → B ( H π ) irred. ∗ -repres. } ❀ topology with open sets { [ π ] ∈ � A | π | J �≡ 0 } for closed 2-sided ideals J ⊆ A • A 0 := { a ∈ A | � A → [0 , ∞ ) , [ π ] �→ Tr ( π ( a ) π ( a ) ∗ ) well-def. & cont } • A has continuous trace ⇐ ⇒ A 0 = A ⇒ � A is loc. comp. Hausdorff and π ( A ) = K ( H π ) for all [ π ] ∈ � A \ { [0] } Example : A = C 0 (Γ , K ( H )) with Γ loc. comp. Hausdorff ⇒ � A ≃ Γ and A has continuous trace Theorem (N.V. Pedersen, 1984) If G is a nilpotent Lie group then there exist closed 2-sided ideals of C ∗ ( G ) { 0 } = J 0 ⊆ J 1 ⊆ · · · ⊆ J n = C ∗ ( G ) with J j / J j − 1 having continuous trace for j = 1 , . . . , n Question : Can we always arrange to have J j / J j − 1 ≃ C 0 (Γ j , K ( H j )) j = 1 , . . . , n ? Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 5 / 14
Operator valued Fourier transforms of a C ∗ -algebra A • let Γ ⊆ � A • select π γ : A → B ( H γ ) with [ π γ ] = γ for all γ ∈ Γ ❀ Fourier transform � F Γ : A → ℓ ∞ (Γ , B ( H γ )) , a �→ { π γ ( a ) } γ ∈ Γ γ ∈ Γ Problem : What is the range of F Γ , particularly for A = C ∗ ( G ) & Γ = � A ? Continuity of operator fields • Γ Hausdorff • ( ∀ γ ∈ Γ) H γ = H • total subset V ⊆ H , dense ∗ -subalg. S ⊆ A satisfying 1. ( ∀ a ∈ S )( ∀ v 1 , v 2 ∈ V ) Γ → C , γ �→ � π γ ( a ) v 1 , v 2 � is continuous Γ → C , γ �→ Tr π γ ( a ) is well-defined & continuous 2. ( ∀ a ∈ S ) = ⇒ ( ∀ a ∈ A ) F Γ ( a ) ∈ C b (Γ , K ( H )) Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 6 / 14
Tools from C ∗ -algebra extension theory q • Extension of C ∗ -algebras: 0 → J ֒ → A − → B → 0 classified by Busby’s ∗ -morphism β : B → M ( J ) / J Example J = C 0 (Γ , K ( H )) ⇒ M ( J ) = { ϕ : Γ → B ( H ) | ϕ bounded strong ∗ -cont. } • If the points of Γ = � J are closed separated in � A , then β : B → C b (Γ , K ( H )) / C 0 (Γ , K ( H )) ( ⊆ M ( J ) / J ) Choi-Effros completely positive lifting theorem If B nuclear separable, then there exists ν : B → C b (Γ , K ( H )) linear, completely positive, � ν � ≤ 1, satisfying ( ∀ b ∈ B ) β ( b ) − ν ( b ) ∈ C 0 (Γ , K ( H )) . Also ν ( b 1 b 2 ) − ν ( b 1 ) ν ( b 2 ) ∈ C 0 (Γ , K ( H )) for b 1 , b 2 ∈ B . Examples: C ∗ ( G ) is nuclear separable. The class of nuclear separable C ∗ -algebras is closed under closed 2-sided ideals and quotients. Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 7 / 14
Boundary values of operator fields • A separable C ∗ -algebra • open sets ∅ = V 0 ⊆ V 1 ⊆ · · · ⊆ V n = � A • ideals { 0 } = J 0 ⊆ J 1 ⊆ · · · ⊆ J n = A with � J ℓ = V ℓ , satisfying 1 Γ ℓ := V ℓ \ V ℓ − 1 is dense in � A \ V ℓ − 1 ; 2 there exist a complex Hibert space H ℓ and π γ : A → K ( H ℓ ) with [ π γ ] = γ for all γ ∈ Γ ℓ such that for every a ∈ A the mapping Γ ℓ → K ( H ℓ ), γ �→ π γ ( a ) is norm continuous. Define L ℓ := { f : � A \ V ℓ → K ( H ℓ +1 ) ⊕ · · · ⊕ K ( H n ) | f ( γ ) ∈ K ( H j ) if γ ∈ Γ j } and F A / J ℓ : A / J ℓ → L ℓ , ( F A / J ℓ ( a + J ℓ ))( γ ) := π γ ( a ), ℓ = 0 , . . . , n . There exist linear maps ν ℓ : F A / J ℓ ( A / J ℓ ) → C b (Γ ℓ , K ( H ℓ )), which are completely positive, completely isometric, almost ∗ -morphisms, with F A ( A ) = { f ∈ L 0 | f | Γ ℓ − ν ℓ ( f | ˆ A\ V ℓ ) ∈ C 0 (Γ ℓ , K ( H ℓ )) , ℓ = 1 , . . . , n − 1 } Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 8 / 14
C ∗ -algebras of nilpotent Lie groups are solvable (1) ⇒ C ∗ ( G ) has a special solving series. G nilpotent Lie group = That is, a finite series of ideals { 0 } = J 0 ⊆ J 1 ⊆ · · · ⊆ J n = A := C ∗ ( G ) with J j / J j − 1 ≃ C 0 (Γ j , K ( H j )) for j = 1 , . . . , n , and moreover A is a topological R -space, Γ j are R -subspaces, � � A = Γ 1 ⊔ · · · ⊔ Γ n 1 2 dim H n = 1 and Γ n ≃ [ g , g ] ⊥ as topological R -spaces 3 dim H j = ∞ if j < n , Γ j is open dense, having closed and separated points in � A \ � J j − 1 4 Γ j ≃ semi-algebraic cone in a finite-dimensional vector space, which is a Zariski open set for j = 1, and its dimension is the index of G , denoted by ind G . 5 there exists a homogeneous function ϕ j : � A → R such that ϕ j | Γ 1 is a polynomial function and Γ j = { γ ∈ � A | ϕ j ( γ ) � = 0 and ϕ i ( γ ) = 0 if i < j } . Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 9 / 14
C ∗ -algebras of nilpotent Lie groups are solvable (2) A topological R -space is a topological space X with a continuous map R × X → X , ( t , x ) �→ t · x , and with a distinguished point x 0 ∈ X satisfying 1 ( ∀ x ∈ X ) 0 · x = x 0 2 ( ∀ t , s ∈ R )( ∀ x ∈ X ) t · ( s · x ) = ts · x 3 For every x ∈ X \ { x 0 } the map R → X , t �→ t · x is a homeomorphism onto its image. An R -subspace is any Γ ⊆ X with R · Γ ⊆ Γ ∪ { x 0 } , so Γ ∪ { x 0 } is a topological R -space. Examples : 1. Finite-dimensional R -linear spaces are topological R -spaces. 2. G = ( g , · ) ❀ � G ≃ � C ∗ ( G ) ≃ g ∗ / Ad ∗ G topological R -space via t · O ξ := O t ξ G ( G ) ξ . The linear space [ g , g ] ⊥ ( ≃ the singleton orbits) is where O ξ = Ad ∗ an R -subspace of g ∗ / Ad ∗ G . Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 10 / 14
C ∗ -algebras of nilpotent Lie groups are solvable (3) • nilpotent Lie group G = ( g , · ) G = ( g , · ) • Jordan-H¨ older sequence { 0 } = g 0 ⊆ · · · ⊆ g m = g • duality pairing �· , ·� : g ∗ × g → R • coadjoint isotropy at ξ ∈ g ∗ : g ( ξ ) := { x ∈ g | � ξ, [ x , g ] � = 0 } • jump set at ξ ∈ g ∗ : J ξ := { j ∈ { 1 , . . . , m } | g j �⊂ g ( ξ ) + g j − 1 } • E the set of all subsets of { 1 , . . . , m } Piecewise continuity of trace wrt the coarse stratification Define Ω e := { ξ ∈ g ∗ | J ξ = e } for e ∈ E . Coarse stratification : g ∗ = � e ∈E Ω e , finite partition into G -invariant sets G = � ❀ � G ≃ g ∗ / Ad ∗ e ∈E Ξ e where Ξ e := Ω e / Ad ∗ G For every e ∈ E one has: 1 The relative topology of Ξ e ⊆ g ∗ / Ad ∗ G is Hausdorff. 2 For every ϕ ∈ C ∞ 0 ( G ) the function Ξ e → C , O �→ Tr ( π O ( ϕ )) is well defined and continuous, where [ π O ] ← → O . Daniel Beltit ¸˘ a (IMAR) Nilpotent Lie groups Hamburg, 16.02.2015 11 / 14
Recommend
More recommend
