Primitivity and unital full free product of residually finite dimensional C*-algebras

Transcription

1 Primitivity and unital full free product of residually finite dimensional C*-algebras Francisco Torres-Ayala, joint work with Ken Dykema 2013 JMM, San Diego

2 Definition (Push out) Let A 1, A 2 and D be C -algebras and assume there are injective -homomorphisms γ i : D A i, i = 1, 2. The push out of (A 1, A 2, D, γ 1, γ 2 ) is the unique C -algebra, denoted A 1 D A 2, together with -homomorphisms ι i : A i A 1 D A 2, i = 1, 2, satisfying ι 1 γ 1 = ι 2 γ 2 and the following universal property D γ 1 A 1 γ 2 A 2 ι 2 ι 1 = A 1 D A 2 ϕ = ϕ 2 ϕ 1 B

3 Definition (Push out) Let A 1, A 2 and D be C -algebras and assume there are injective -homomorphisms γ i : D A i, i = 1, 2. The push out of (A 1, A 2, D, γ 1, γ 2 ) is the unique C -algebra, denoted A 1 D A 2, together with -homomorphisms ι i : A i A 1 D A 2, i = 1, 2, satisfying ι 1 γ 1 = ι 2 γ 2 and the following universal property D γ 1 A 1 γ 2 A 2 ι 2 ι 1 = A 1 D A 2 ϕ = ϕ 2 ϕ 1 B Note: we restrict to D = C and unital C -algebras and unital -homomorphisms.

4 Intuition As with groups, what one usually does is to think A 1 and A 2 as alphabets and construct words, a 1 a 2 a 3 a n, where neighboring elements lie in different algebras. Then take linear combinations of words and define a multiplication and adjoint. Regarding multiplication, in situations like (a 1 a n )(b 1 b m ) one have to take care weather a n and b n lie on the same algebra. Lastly, one completes taking supremum of norms of -homomorphism into bounded linear operators in Hilbert spaces.

5 C 2 C 2 Unital Full Free Products are highly abstract and difficult to identify.

6 C 2 C 2 Unital Full Free Products are highly abstract and difficult to identify. Fortunately C*-algebra C 2 C 2 is very well studied (Pedersen, Raerburn, Sinclair).

7 C 2 C 2 Unital Full Free Products are highly abstract and difficult to identify. Fortunately C*-algebra C 2 C 2 is very well studied (Pedersen, Raerburn, Sinclair). All irreducible finite dimensional -representations of C 2 C 2 are of dimension 1 or 2. Even more C 2 C 2 { } f is continuous, f(0) and f(1) f : [0, 1] M 2 (C) : are diagonal matrices where a -isomorphism ϕ is given by ϕ(ι 1 (1, 0))(t) = [ ] [, ϕ(ι 2 (1, 0))(t) = ] t t(1 t) t(1 t) 1 t

8 Definition A C*-algebra A is called primitive if it admits a faithful and irreducible -representation. In other words, there is (faithful -representation) an isometric -homomorphism into the bounded operators of a Hilbert space with the property that (irreducible) the only closed invariant subspaces for the image are the trivial ones

9 Definition A C*-algebra A is called primitive if it admits a faithful and irreducible -representation. In other words, there is (faithful -representation) an isometric -homomorphism into the bounded operators of a Hilbert space with the property that (irreducible) the only closed invariant subspaces for the image are the trivial ones ( or equivalently its commutant consists of just scalar operators).

10 Definition A C*-algebra A is called primitive if it admits a faithful and irreducible -representation. In other words, there is (faithful -representation) an isometric -homomorphism into the bounded operators of a Hilbert space with the property that (irreducible) the only closed invariant subspaces for the image are the trivial ones ( or equivalently its commutant consists of just scalar operators). Note: The center of any primitive C -algebra is trivial.

11 Examples of Primitive C -algebras Hereditary C -subalgebras of primitive ones are primitive.

12 Examples of Primitive C -algebras Hereditary C -subalgebras of primitive ones are primitive. Any simple C*-algebra is primitive.

13 Examples of Primitive C -algebras Hereditary C -subalgebras of primitive ones are primitive. Any simple C*-algebra is primitive. (Choi 80) For 2 n, the full group C*-algebra C (F n ) is primitive.

14 Examples of Primitive C -algebras Hereditary C -subalgebras of primitive ones are primitive. Any simple C*-algebra is primitive. (Choi 80) For 2 n, the full group C*-algebra C (F n ) is primitive. (Omland, 12 ) If G 1 and G 2 are countable amenable discrete groups with ( G 1 1)( G 2 1) 2 and σ is a 2-cocycle in G 1 G 2. Then C (G 1 G 2, σ) is primitive.

15 Main result Theorem Let A 1 and A 2 denote two nontrivial separable residually finite dimensional C -algebras. If (dim(a 1 ) 1)(dim(A 2 ) 1) 2 then A 1 A 2 is primitive.

16 Main result Theorem Let A 1 and A 2 denote two nontrivial separable residually finite dimensional C -algebras. If (dim(a 1 ) 1)(dim(A 2 ) 1) 2 then A 1 A 2 is primitive. In other words, C 2 C 2 is the only (non trivial) separable unital full free product of residually finite dimensional C*-algebras that fails to be primitive.

17 Some consequences Assume A 1 and A 2 are non trivial separable C -algebras with (dim(a 1 ) 1)(dim(A 2 ) 1) 2.

18 Some consequences Assume A 1 and A 2 are non trivial separable C -algebras with (dim(a 1 ) 1)(dim(A 2 ) 1) 2. If A 1 and A 2 are AF algebras then A 1 A 2 is primitive.

19 Some consequences Assume A 1 and A 2 are non trivial separable C -algebras with (dim(a 1 ) 1)(dim(A 2 ) 1) 2. If A 1 and A 2 are AF algebras then A 1 A 2 is primitive. If A 1 and A 2 are residually finite dimensional then A 1 A 2 is antilimial.

20 Some consequences Assume A 1 and A 2 are non trivial separable C -algebras with (dim(a 1 ) 1)(dim(A 2 ) 1) 2. If A 1 and A 2 are AF algebras then A 1 A 2 is primitive. If A 1 and A 2 are residually finite dimensional then A 1 A 2 is antilimial. Assume D is a matrix algebra, A 1 and A 2 are separable and residually finite dimensional. If (dim(a 1 ) dim(d))(dim(a 2 ) dim(d)) 2 dim(d) 2 then A 1 D A 2 is primitive.

21 Perturbations of representations Definition For a -representation π : A 1 A 2 B(H), let π (i) denote the composition of π with the natural inclusion ι i : A i A 1 A 2, i = 1, 2. For a unitary operator u of H, a perturbation of π by u is a -representation of the form π (1) (Ad u π (2) ), (Ad u(x) := uxu ). Remark π (1) (Ad u π (2) ) is irreducible if and only if where B i = π (i) (A i ), i = 1, 2. B 1 Ad u(b 2 ) = C

22 DPI representations Definition A -representation π : A 1 A 2 B(H) is said to be densely perturbable to an irreducible -representation (DPI) if the set (π) := {u U(H) : π (1) (A 1 ) Ad u(π (2) (A 2 ) ) = C} is norm dense in U(H), the unitary operators in H (commutants are relative to B(H)).

23 DPI representations Definition A -representation π : A 1 A 2 B(H) is said to be densely perturbable to an irreducible -representation (DPI) if the set (π) := {u U(H) : π (1) (A 1 ) Ad u(π (2) (A 2 ) ) = C} is norm dense in U(H), the unitary operators in H (commutants are relative to B(H)). Remark DPI representations are stable under perturbations.

24 Rank of Central Projections condition Definition Assume A 1 and A 2 are finite dimensional and let π : A 1 A 2 B(H) denote a unital finite dimensional -representation. We say π satisfies the Rank of Central Projections condition (in short RCP condition) if for both i = 1, 2, the rank of π (i) (p) is the same for all minimal projections p of the center C(A i ) of A i.

25 Rank of Central Projections condition Definition Assume A 1 and A 2 are finite dimensional and let π : A 1 A 2 B(H) denote a unital finite dimensional -representation. We say π satisfies the Rank of Central Projections condition (in short RCP condition) if for both i = 1, 2, the rank of π (i) (p) is the same for all minimal projections p of the center C(A i ) of A i. For instance, on (M 2 M 3 ) (M 4 M 4 M 4 ) take π equal to a π (1) (a b) = a a b b c π(2) (c d e) = d e

26 RCP condition Remarks Assume A 1 and A 2 are finite dimensional and let π, ρ, σ : A 1 A 2 B(H) be unital finite dimensional -representations.

27 RCP condition Remarks Assume A 1 and A 2 are finite dimensional and let π, ρ, σ : A 1 A 2 B(H) be unital finite dimensional -representations. there is a finite dimensional Hilbert space Ĥ and a unital -representation σ : A 1 A 2 B(Ĥ) such that σ ˆσ satisfies the RCP condition.

28 RCP condition Remarks Assume A 1 and A 2 are finite dimensional and let π, ρ, σ : A 1 A 2 B(H) be unital finite dimensional -representations. there is a finite dimensional Hilbert space Ĥ and a unital -representation σ : A 1 A 2 B(Ĥ) such that σ ˆσ satisfies the RCP condition. if π and ρ satisfy the RCP condition then π ρ and π (1) (Ad u π (2) ) also satisfy the RCP.

29 RCP condition Remarks Assume A 1 and A 2 are finite dimensional and let π, ρ, σ : A 1 A 2 B(H) be unital finite dimensional -representations. there is a finite dimensional Hilbert space Ĥ and a unital -representation σ : A 1 A 2 B(Ĥ) such that σ ˆσ satisfies the RCP condition. if π and ρ satisfy the RCP condition then π ρ and π (1) (Ad u π (2) ) also satisfy the RCP. if π satisfies the RCP then it is DPI.

30 Fundamental Difference with C 2 C 2 Proposition Assume A 1 and A 2 are nontrivial separable residually finite dimensional C -algebras with (dim(a 1 ) 1)(dim(A 2 ) 1) 2 and let π : A 1 A 2 B(H) be a unital finite dimensional -representation. There is a finite dimensional Hilbert space Ĥ and a unital -representation ˆπ : A 1 A 2 B(Ĥ) such that π ˆπ is DPI.

31 Take C B 1, B 2 M N. Denote Main technical theorem (B 1, B 2 ) := {u U(M N ) : B 1 Ad u(b 2 ) = C}. Assume one of the following holds 1. B 1 and B 2 are simple. 2. B 2 is simple and for some k 2, B 1 M N/k M N/k. }{{} k times 3. B 1 M N/2 M N/2 and B 2 M N/2 M N/(2k) where k For i = 1, 2, there are integers k 1 2, k 2 3 such that B i M N/ki M N/ki. }{{} k i times Then (B 1, B 2 ) is dense in U(M N ).

32 Under RCP condition Why so many cases? A 1 is simple B 1 = π (1) (A 1 ) is simple A 2 is simple B 2 = π (2) (A 2 ) is simple k := dim(c(a 1 )) 2 B 1 = π (1) (C(A 1 )) B 1 M N/k M N/k A 2 is simple B 2 = π (2) (A 2 ) is simple A 1 M n1 (1) M n1 (2) B 1 = π (1) (C(A 1 )) B 1 M N/2 M N/2 A 2 M n2 (1) M n2 (2), B 2 = π (2) (C M n2 (2)) n 2 (2) 2 B 2 M N/2 M N/2k, k = n 2 (2) 2 k 1 := dim(c(a 1 )) 2 B 1 = π (1) (C(A 2 )) B 1 M N/k1 M N/k1, k 1 2 k 2 := dim(c(a 2 )) 3 B 2 = π (2) (C(A 2 )) B 2 M N/k2 M N/k2, k 2 3

33 Sketch Assume (dim(a 1 ) 1)(dim(A 2 ) 1) 2, and let A := A 1 A 2. Let {π n : A B(H n )} n 1 be a separating family of unital finite dimensional -representations (Exel and Loring).

34 Sketch Assume (dim(a 1 ) 1)(dim(A 2 ) 1) 2, and let A := A 1 A 2. Let {π n : A B(H n )} n 1 be a separating family of unital finite dimensional -representations (Exel and Loring). We may assume that for each k 1, k n=1 π n is DPI and the image of their direct sum, denoted by π, contains no nonzero compact operators.

35 Sketch Assume (dim(a 1 ) 1)(dim(A 2 ) 1) 2, and let A := A 1 A 2. Let {π n : A B(H n )} n 1 be a separating family of unital finite dimensional -representations (Exel and Loring). We may assume that for each k 1, k n=1 π n is DPI and the image of their direct sum, denoted by π, contains no nonzero compact operators. For each k 1, we can perturb k n=1 π n into θ k, an irreducible -representation, such that σ k := θ k n>k π n is faithful.

36 Sketch Assume (dim(a 1 ) 1)(dim(A 2 ) 1) 2, and let A := A 1 A 2. Let {π n : A B(H n )} n 1 be a separating family of unital finite dimensional -representations (Exel and Loring). We may assume that for each k 1, k n=1 π n is DPI and the image of their direct sum, denoted by π, contains no nonzero compact operators. For each k 1, we can perturb k n=1 π n into θ k, an irreducible -representation, such that σ k := θ k n>k π n is faithful. If we control each perturbation on a suitable set of the unit ball, we can find a -representation σ such that lim k σ k (x) = σ(x) for all x in A 1 A 2.

37 Sketch Assume (dim(a 1 ) 1)(dim(A 2 ) 1) 2, and let A := A 1 A 2. Let {π n : A B(H n )} n 1 be a separating family of unital finite dimensional -representations (Exel and Loring). We may assume that for each k 1, k n=1 π n is DPI and the image of their direct sum, denoted by π, contains no nonzero compact operators. For each k 1, we can perturb k n=1 π n into θ k, an irreducible -representation, such that σ k := θ k n>k π n is faithful. If we control each perturbation on a suitable set of the unit ball, we can find a -representation σ such that lim k σ k (x) = σ(x) for all x in A 1 A 2. we show σ is faithful and irreducible.

38 Thank you