Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Primitivity and unital full free product of
residually finite dimensional C*-algebras
Francisco Torres-Ayala,
joint work with Ken Dykema
2013 JMM, San Diego
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Definition (Push out)
Let A1 , A2 and D be C ∗ -algebras and assume there are injective
∗-homomorphisms γi : D → Ai , i = 1, 2. The push out of
(A1 , A2 , D, γ1 , γ2 ) is the unique C ∗ -algebra , denoted A1 ∗D A2 ,
together with ∗-homomorphisms ιi : Ai → A1 ∗D A2 , i = 1, 2,
satisfying ι1 ◦ γ1 = ι2 ◦ γ2 and the following universal property
D
γ2
γ1
A2
/ A1
ι2
ι1
=
ϕ1
/ A1 ∗D A2
=
ϕ2
ϕ
$ 4B
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Definition (Push out)
Let A1 , A2 and D be C ∗ -algebras and assume there are injective
∗-homomorphisms γi : D → Ai , i = 1, 2. The push out of
(A1 , A2 , D, γ1 , γ2 ) is the unique C ∗ -algebra , denoted A1 ∗D A2 ,
together with ∗-homomorphisms ιi : Ai → A1 ∗D A2 , i = 1, 2,
satisfying ι1 ◦ γ1 = ι2 ◦ γ2 and the following universal property
D
γ2
γ1
A2
/ A1
ι2
ι1
=
ϕ1
/ A1 ∗D A2
=
ϕ2
ϕ
$ 4B
Note: we restrict to D = C and unital C ∗ -algebras and unital
∗-homomorphisms.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Intuition
As with groups, what one usually does is to think A1 and A2 as
alphabets and construct words, a1 a2 a3 · · · an , where neighboring
elements lie in different algebras. Then take linear combinations of
words and define a multiplication and adjoint. Regarding
multiplication, in situations like
(a1 · · · an )(b1 · · · bm )
one have to take care weather an and bn lie on the same algebra.
Lastly, one completes taking supremum of norms of
∗-homomorphism into bounded linear operators in Hilbert spaces.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
C2 ∗ C2
Unital Full Free Products are highly abstract and difficult to
identify.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
C2 ∗ C2
Unital Full Free Products are highly abstract and difficult to
identify. Fortunately C*-algebra C2 ∗ C2 is very well studied
(Pedersen, Raerburn, Sinclair).
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
C2 ∗ C2
Unital Full Free Products are highly abstract and difficult to
identify. Fortunately C*-algebra C2 ∗ C2 is very well studied
(Pedersen, Raerburn, Sinclair).
All irreducible finite dimensional ∗-representations of C2 ∗ C2
are of dimension 1 or 2.
Even more
n
C2 ∗ C2 ' f : [0, 1] → M2 (C) : f
is continuous, f (0) and f (1)
are diagonal matrices
o
where a ∗-isomorphism ϕ is given by
p
t(1 − t)
1 0
t
p
ϕ(ι1 (1, 0))(t) =
, ϕ(ι2 (1, 0))(t) =
0 0
t(1 − t)
1−t
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
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
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
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).
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
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.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Examples of Primitive C ∗ -algebras
• Hereditary C ∗ -subalgebras of primitive ones are primitive.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Examples of Primitive C ∗ -algebras
• Hereditary C ∗ -subalgebras of primitive ones are primitive.
• Any simple C*-algebra is primitive.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
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 ∗ (Fn )
is primitive.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
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 ∗ (Fn )
is primitive.
• (Omland, ’12 ) If G1 and G2 are countable amenable discrete
groups with (|G1 | − 1)(|G2 | − 1) ≥ 2 and σ is a 2-cocycle in
G1 ∗ G2 . Then C ∗ (G1 ∗ G2 , σ) is primitive.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main result
Theorem
Let A1 and A2 denote two nontrivial separable residually finite
dimensional C ∗ -algebras.
If (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2 then A1 ∗ A2 is primitive.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Main result
Theorem
Let A1 and A2 denote two nontrivial separable residually finite
dimensional C ∗ -algebras.
If (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2 then A1 ∗ A2 is primitive.
In other words, C2 ∗ C2 is the only (non trivial) separable unital full
free product of residually finite dimensional C*-algebras that fails
to be primitive.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Some consequences
Assume A1 and A2 are non trivial separable C ∗ -algebras with
(dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Some consequences
Assume A1 and A2 are non trivial separable C ∗ -algebras with
(dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2.
• If A1 and A2 are AF algebras then A1 ∗ A2 is primitive.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Some consequences
Assume A1 and A2 are non trivial separable C ∗ -algebras with
(dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2.
• If A1 and A2 are AF algebras then A1 ∗ A2 is primitive.
• If A1 and A2 are residually finite dimensional then A1 ∗ A2 is
antilimial.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Some consequences
Assume A1 and A2 are non trivial separable C ∗ -algebras with
(dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2.
• If A1 and A2 are AF algebras then A1 ∗ A2 is primitive.
• If A1 and A2 are residually finite dimensional then A1 ∗ A2 is
antilimial.
• Assume D is a matrix algebra, A1 and A2 are separable and
residually finite dimensional.
If (dim(A1 ) − dim(D))(dim(A2 ) − dim(D)) ≥ 2 dim(D)2
then A1 ∗D A2 is primitive.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Perturbations of representations
Definition
For a ∗-representation π : A1 ∗ A2 → B(H), let π (i) denote the
composition of π with the natural inclusion ιi : Ai → A1 ∗ A2 ,
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
B1 ∩ Ad u(B2 ) = C
where Bi = π (i) (Ai )0 , i = 1, 2.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
DPI representations
Definition
A ∗-representation π : A1 ∗ A2 → B(H) is said to be densely
perturbable to an irreducible ∗-representation (DPI) if the set
∆(π) := {u ∈ U (H) : π (1) (A1 )0 ∩ Ad u(π (2) (A2 )0 ) = C}
is norm dense in U (H), the unitary operators in H (commutants
are relative to B(H)).
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
DPI representations
Definition
A ∗-representation π : A1 ∗ A2 → B(H) is said to be densely
perturbable to an irreducible ∗-representation (DPI) if the set
∆(π) := {u ∈ U (H) : π (1) (A1 )0 ∩ Ad u(π (2) (A2 )0 ) = 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.
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Rank of Central Projections condition
Definition
Assume A1 and A2 are finite dimensional and let
π : A1 ∗ A2 → 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(Ai ) of Ai .
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Rank of Central Projections condition
Definition
Assume A1 and A2 are finite dimensional and let
π : A1 ∗ A2 → 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(Ai ) of Ai .
For instance, on (M2 ⊕ M3 ) ∗ (M4 ⊕ M4 ⊕ M4 ) take π equal to


a




a
c


 π (2) (c ⊕ d ⊕ e) = 

d
a
π (1) (a ⊕ b) = 




b
e
b
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
RCP condition
Remarks
Assume A1 and A2 are finite dimensional and let
π, ρ, σ : A1 ∗ A2 → B(H) be unital finite dimensional
∗-representations.
Main Theorem
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
RCP condition
Remarks
Assume A1 and A2 are finite dimensional and let
π, ρ, σ : A1 ∗ A2 → B(H) be unital finite dimensional
∗-representations.
• there is a finite dimensional Hilbert space Ĥ and a unital
∗-representation σ : A1 ∗ A2 → B(Ĥ) such that σ ⊕ σ̂ satisfies
the RCP condition.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
RCP condition
Remarks
Assume A1 and A2 are finite dimensional and let
π, ρ, σ : A1 ∗ A2 → B(H) be unital finite dimensional
∗-representations.
• there is a finite dimensional Hilbert space Ĥ and a unital
∗-representation σ : A1 ∗ A2 → B(Ĥ) such that σ ⊕ σ̂ 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.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
RCP condition
Remarks
Assume A1 and A2 are finite dimensional and let
π, ρ, σ : A1 ∗ A2 → B(H) be unital finite dimensional
∗-representations.
• there is a finite dimensional Hilbert space Ĥ and a unital
∗-representation σ : A1 ∗ A2 → B(Ĥ) such that σ ⊕ σ̂ 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.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Fundamental Difference with C2 ∗ C2
Proposition
Assume A1 and A2 are nontrivial separable residually finite
dimensional C ∗ -algebras with (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2
and let π : A1 ∗ A2 → B(H) be a unital finite dimensional
∗-representation. There is a finite dimensional Hilbert space Ĥ and
a unital ∗-representation π̂ : A1 ∗ A2 → B(Ĥ) such that π ⊕ π̂ is
DPI.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Main technical theorem
Take C
B1 , B2 ( MN . Denote
∆(B1 , B2 ) := {u ∈ U(MN ) : B1 ∩ Ad u(B2 ) = C}.
Assume one of the following holds
1. B1 and B2 are simple.
2. B2 is simple and for some k ≥ 2, B1 ' MN/k ⊕ · · · ⊕ MN/k .
|
{z
}
k−times
3. B1 ' MN/2 ⊕ MN/2 and B2 ' MN/2 ⊕ MN/(2k) where k ≥ 2.
4. For i = 1, 2, there are integers k1 ≥ 2, k2 ≥ 3 such that
Bi ' MN/ki ⊕ · · · ⊕ MN/ki .
|
{z
}
ki times
Then ∆(B1 , B2 ) is dense in U(MN ).
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Why so many cases ?
Under RCP condition
A1 is simple
A2 is simple
k := dim(C(A1 )) ≥ 2
A2 is simple
A1 ' Mn1 (1) ⊕ Mn1 (2)
A2 ' Mn2 (1) ⊕ Mn2 (2) ,
n2 (2) ≥ 2
k1 := dim(C(A1 )) ≥ 2
k2 := dim(C(A2 )) ≥ 3
B1
B2
B1
B1
B2
B1
B1
B2
B2
B1
B1
B2
B2
= π (1) (A1 )0 is simple
= π (2) (A2 )0 is simple
= π (1) (C(A1 ))0
' MN/k ⊕ · · · ⊕ MN/k
= π (2) (A2 )0 is simple
= π (1) (C(A1 ))0
' MN/2 ⊕ MN/2
= π (2) (C ⊕ Mn2 (2) )0
' MN/2 ⊕ MN/2k , k = n2 (2) ≥ 2
= π (1) (C(A2 ))0
' MN/k1 ⊕ · · · ⊕ MN/k1 , k1 ≥ 2
= π (2) (C(A2 ))0
' MN/k2 ⊕ · · · ⊕ MN/k2 , k2 ≥ 3
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Sketch
Assume (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2, and let A := A1 ∗ A2 .
• Let {πn : A → B(Hn )}n≥1 be a separating family of unital
finite dimensional ∗-representations (Exel and Loring).
Main idea
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Sketch
Assume (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2, and let A := A1 ∗ A2 .
• Let {πn : A → B(Hn )}n≥1 be a separating family of unital
finite dimensional ∗-representations (Exel and Loring).
• We may assume that for each k ≥ 1, ⊕kn=1 πn is DPI and the
image of their direct sum, denoted by π, contains no nonzero
compact operators.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Sketch
Assume (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2, and let A := A1 ∗ A2 .
• Let {πn : A → B(Hn )}n≥1 be a separating family of unital
finite dimensional ∗-representations (Exel and Loring).
• We may assume that for each k ≥ 1, ⊕kn=1 πn is DPI and the
image of their direct sum, denoted by π, contains no nonzero
compact operators.
• For each k ≥ 1, we can perturb ⊕kn=1 πn into θk , an irreducible
∗-representation, such that σk := θk ⊕ ⊕n>k πn is faithful.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Sketch
Assume (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2, and let A := A1 ∗ A2 .
• Let {πn : A → B(Hn )}n≥1 be a separating family of unital
finite dimensional ∗-representations (Exel and Loring).
• We may assume that for each k ≥ 1, ⊕kn=1 πn is DPI and the
image of their direct sum, denoted by π, contains no nonzero
compact operators.
• For each k ≥ 1, we can perturb ⊕kn=1 πn into θk , an irreducible
∗-representation, such that σk := θk ⊕ ⊕n>k πn is faithful.
• If we control each perturbation on a suitable set of the unit
ball, we can find a ∗-representation σ such that
limk σk (x) = σ(x) for all x in A1 ∗ A2 .
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Main Theorem
Main idea
Sketch
Assume (dim(A1 ) − 1)(dim(A2 ) − 1) ≥ 2, and let A := A1 ∗ A2 .
• Let {πn : A → B(Hn )}n≥1 be a separating family of unital
finite dimensional ∗-representations (Exel and Loring).
• We may assume that for each k ≥ 1, ⊕kn=1 πn is DPI and the
image of their direct sum, denoted by π, contains no nonzero
compact operators.
• For each k ≥ 1, we can perturb ⊕kn=1 πn into θk , an irreducible
∗-representation, such that σk := θk ⊕ ⊕n>k πn is faithful.
• If we control each perturbation on a suitable set of the unit
ball, we can find a ∗-representation σ such that
limk σk (x) = σ(x) for all x in A1 ∗ A2 .
• we show σ is faithful and irreducible.
Full free products
Definition Primitive C ∗ -algebras
Examples of Primitive C ∗ -algebras
Thank you
Main Theorem
Main idea