Exotic Ideals in the Fourier-Stieltjes Algebra of a Locally Compact group.

Transcription

1 Exotic Ideals in the Fourier-Stieltjes Algebra of a Locally Compact group. Brian Forrest Department of Pure Mathematics University of Waterloo May 21, / 1

2 Set up: G-locally compact group (LCG) with fixed left Haar measure µ. 2 / 1

3 Set up: G-locally compact group (LCG) with fixed left Haar measure µ. G: LCG, H: Hilbert space, π : G U(H): unitary rep. 2 / 1

4 Set up: G-locally compact group (LCG) with fixed left Haar measure µ. G: LCG, H: Hilbert space, π : G U(H): unitary rep. π ξ,η (x) := π(x)ξ, η is a coefficient function associated with π. 2 / 1

5 Set up: G-locally compact group (LCG) with fixed left Haar measure µ. G: LCG, H: Hilbert space, π : G U(H): unitary rep. π ξ,η (x) := π(x)ξ, η is a coefficient function associated with π. π is continuous if all π ξ,η (x) s are continuous. 2 / 1

6 Set up: G-locally compact group (LCG) with fixed left Haar measure µ. G: LCG, H: Hilbert space, π : G U(H): unitary rep. π ξ,η (x) := π(x)ξ, η is a coefficient function associated with π. π is continuous if all π ξ,η (x) s are continuous. Σ G = all equivalence classes of continuous unitary reps. 2 / 1

7 Set up: G-locally compact group (LCG) with fixed left Haar measure µ. G: LCG, H: Hilbert space, π : G U(H): unitary rep. π ξ,η (x) := π(x)ξ, η is a coefficient function associated with π. π is continuous if all π ξ,η (x) s are continuous. Σ G = all equivalence classes of continuous unitary reps. Definition (Group C -algebra) For f L 1 (G) define and f u = sup{ π(f ) B(Hπ) π Σ G } C (G) = completion of L 1 (G) wrt u 2 / 1

8 Definition ( C π(g)-algebra) Let π Σ G. Let N π be the kernel of π : C (G) B(H π ). Then C π(g) = C (G)/N π 3 / 1

9 Definition ( C π(g)-algebra) Let π Σ G. Let N π be the kernel of π : C (G) B(H π ). Then C π(g) = C (G)/N π Example The left-regular representation λ : G U(L 2 (G)) is defined as (λ(x)f )(y) = f (x 1 y). Cλ (G) is called the reduced C -algebra of G. 3 / 1

10 Definition ( C π(g)-algebra) Let π Σ G. Let N π be the kernel of π : C (G) B(H π ). Then C π(g) = C (G)/N π Example The left-regular representation λ : G U(L 2 (G)) is defined as (λ(x)f )(y) = f (x 1 y). Cλ (G) is called the reduced C -algebra of G. Theorem (Hulanicki, 1964) C (G) = Cλ (G) if and only if G is amenable. 3 / 1

11 Question: If G is non-amenable do there exist intermediate C -algebras between C (G) and C λ(g)? 4 / 1

12 Question: If G is non-amenable do there exist intermediate C -algebras between C (G) and C λ(g)? Definition (L p-representations) π is an L p representation for 1 p < if π ξ,ξ (x) := π(x)ξ, ξ L p(g) for a dense set of vectors ξ H π. π p = { π π is an L P representation} 4 / 1

13 Question: If G is non-amenable do there exist intermediate C -algebras between C (G) and C λ(g)? Definition (L p-representations) π is an L p representation for 1 p < if π ξ,ξ (x) := π(x)ξ, ξ L p(g) for a dense set of vectors ξ H π. π p = { π π is an L P representation} Theorem (Brown and Guentner: Okayasu) If G = F 2 and 2 < p < q <, then C λ(g), C π p (G), C π q (G) and C (G) are all distinct. 4 / 1

14 Definition (Eymard: Fourier-Stieltjes Algebra of G) with B(G) = {π ξ,η π Σ G } f B(G) = min{ ξ η u = π ξ,η }. is a commutative Banach algebra called the Fourier-Stieltjes algebra of G. 5 / 1

15 Definition (Eymard: Fourier-Stieltjes Algebra of G) B(G) = {π ξ,η π Σ G } with f B(G) = min{ ξ η u = π ξ,η }. is a commutative Banach algebra called the Fourier-Stieltjes algebra of G. Key Facts: 1) B(G) = (C (G)) 5 / 1

16 Definition (Eymard: Fourier-Stieltjes Algebra of G) B(G) = {π ξ,η π Σ G } with f B(G) = min{ ξ η u = π ξ,η }. is a commutative Banach algebra called the Fourier-Stieltjes algebra of G. Key Facts: 1) B(G) = (C (G)) 2) B(G) completely determines G. 5 / 1

17 Definition (Arsac: A π (G) and B π (G)) Given π Σ G, let A π (G) = span{π ξ,η ξ, η H π } B(G) and B π (G) = A π (G) w = (Cπ(G)) 6 / 1

18 Definition (Arsac: A π (G) and B π (G)) Given π Σ G, let A π (G) = span{π ξ,η ξ, η H π } B(G) and B π (G) = A π (G) w = (Cπ(G)) Example For 1 p 2 A πp (G) = A λ (G) = A(G) is called the Fourier Algebra of G. A(G) is a closed ideal of B(G) with (A(G)) = G. 6 / 1

19 Note: For G = F 2 and 1 < p <, the B πp (G) s are distinct closed ideals in B(G) containing A(G), but for G amenable for all 1 p <. B πp (G) = B(G) 7 / 1

20 Note: For G = F 2 and 1 < p <, the B πp (G) s are distinct closed ideals in B(G) containing A(G), but for G amenable for all 1 p <. B πp (G) = B(G) Question: What can we say about A πp (G) for 2 < p <? 7 / 1

21 Note: For G = F 2 and 1 < p <, the B πp (G) s are distinct closed ideals in B(G) containing A(G), but for G amenable for all 1 p <. B πp (G) = B(G) Question: What can we say about A πp (G) for 2 < p <? Theorem (Wiersma) For 2 p, A πp (G) is a closed ideal of B(G) with (A π (G)) = G containing A(G) which completely determines G. 7 / 1

22 A πp (G)) for 2 p Note: 1) For G = F 2, we have that for 2 < p < q < that A(G) A πp (G) A πq (G) B 0 (G) where B 0 (G) = B(G) C 0 (G) is the Rajchman algebra of G. 8 / 1

23 A πp (G)) for 2 p Note: 1) For G = F 2, we have that for 2 < p < q < that A(G) A πp (G) A πq (G) B 0 (G) where B 0 (G) = B(G) C 0 (G) is the Rajchman algebra of G. 2) If G is compact then A(G) = A πp (G) = B(G) for all 1 p <. 8 / 1

24 A πp (G)) for 2 p Note: 1) For G = F 2, we have that for 2 < p < q < that A(G) A πp (G) A πq (G) B 0 (G) where B 0 (G) = B(G) C 0 (G) is the Rajchman algebra of G. 2) If G is compact then for all 1 p <. A(G) = A πp (G) = B(G) Question: For which locally compact groups are the A πp (G) ideals distinct for 2 < p <? 8 / 1

25 A πp (G) for 2 p Theorem (Wiersma) If G is non-compact and abelian, then for 2 < p < q < we have that A(G) A πp (G) A πq (G) B 0 (G) 9 / 1

26 A πp (G) for 2 p < Theorem (F, Tanko, Wiersma) If G is an infinite discrete group which satisfies one of the following conditions: 1) G is locally finite, 2) G is elementary amenable, 3) G is a linear group, 4) G has polynomial growth, then for 2 < p < q < we have and A πp (G) is non-separable. A(G) A πp (G) A πq (G) B 0 (G) 10 / 1

27 A πp (G) for 2 p < Theorem (F, Tanko, Wiersma) If G is an infinite discrete group which satisfies one of the following conditions: 1) G is locally finite, 2) G is elementary amenable, 3) G is a linear group, 4) G has polynomial growth, then for 2 < p < q < we have and A πp (G) is non-separable. A(G) A πp (G) A πq (G) B 0 (G) Question: Are the A πp (G) ideals distinct for 2 < p < q < for every infinite discrete group? 10 / 1

28 A πp (G) for 2 p < Definition 1) G is [IN] is G has a neighborhood of the identity that is invariant under inner automorphisms., 2) G is [MAP] if the finite dimensional reps. separate points. 3) G almost connected if G/G 0 is compact. 11 / 1

29 A πp (G) for 2 p < Definition 1) G is [IN] is G has a neighborhood of the identity that is invariant under inner automorphisms., 2) G is [MAP] if the finite dimensional reps. separate points. 3) G almost connected if G/G 0 is compact. Theorem (F, Tanko, Wiersma) If G is a non-compact almost connected [IN]-group, then then for 2 < p < q < we have and A πp (G) is non-separable. A(G) A πp (G) A πq (G) B 0 (G) 11 / 1

30 A πp (G) for 2 p < Theorem (F, Tanko, Wiersma) If G is either an [IN]-group or a [MAP]-group and if either 1) A πp (G) = A πq (G) for some 2 < p < q or 2) A πp (G) is separable, then G has a compact, open subgroup. 12 / 1

31 A πp (G) for 2 p < Question: If G is non-compact and 2 < p < q < must A(G) A πp (G) A πq (G) B 0 (G)? 13 / 1

32 A πp (G) for 2 p < Question: If G is non-compact and 2 < p < q < must A(G) A πp (G) A πq (G) B 0 (G)? Example If ( a b G = { 0 1 ) a 0, b R} is the ax + b group then for any 1 p <, we have A(G) = A πp (G) = B 0 (G) 13 / 1

33 A πp (G) for 2 p < For Garth!!! Theorem (F, Tanko, Wiersma) 1) A πp (F 2 ) is not BSE for any 1 p < 14 / 1

34 A πp (G) for 2 p < For Garth!!! Theorem (F, Tanko, Wiersma) 1) A πp (F 2 ) is not BSE for any 1 p < 2) A (F 2 ) = ( Aπ p (F 2 )) B(F 2) is BSE. 2<p< 14 / 1

35 A πp (G) for 2 p < THANK YOU 15 / 1