residual finite-dimensionality for general operator algebrasbanach2019/pdf/clouatre.pdf ·...

Residual finite-dimensionality for general operator algebras

Raphael Clouatre

University of Manitoba

Banach Algebras and ApplicationsUniversity of Manitoba

July 15, 2019

R. Clouatre (University of Manitoba) RFD operator algebras CMS Summer 2019 1 / 11

Joint work with Christopher Ramsey.

A natural definition

Definition

Let A ⇢ B(H) be an operator algebra (not necessarily self-adjoint). We say that A isresidually finite-dimensional (RFD) if there is a completely isometric homomorphism⇡ : A !

Q� Mn� . Furthermore, we say that A is normed in finite dimensions (NFD)

if for every A 2 Mn(A) there is a completely contractive homomorphism ✓ : A ! Mn

such that k✓(n)(A)k = kAk.

Theorem (Courtney–Shulman, 2019)

Let A be a C⇤-algebra and let

N = {a 2 A : kak = k⇡(a)k for some finite-dimensional representation ⇡}.

Consider the following statements.

(a) The algebra A is NFD, i.e N = A.

(b) All irreducible representations of A are finite-dimensional.

(c) The set N is dense in A.

(d) The algebra A is RFD.

Then, we have that (a) , (b) ) (c) , (d).

The full group C⇤-algebra of F2

is RFD but not NFD (Choi 1980).

Definition

is RFD but not NFD (Choi 1980).

Definition

is RFD but not NFD (Choi 1980).R. Clouatre (University of Manitoba) RFD operator algebras CMS Summer 2019 3 / 11

Finite-dimensional operator algebras

A C⇤-algebra A is RFD if one of the following equivalent conditions hold.

(a) There is a completely isometric homomorphism ⇡ : A !Q

� Mn� .

(b) There is a completely isometric homomorphism ⇡ : A !Q

� F�, where each F�

is a finite-dimensional C⇤-algebra.

Are properties (a) and (b) equivalent in the non self-adjoint context?

Example

The finite-dimensional unital operator algebra

⇢a b+ cz

�: a, b, c, d 2 C

�⇢ M

(C(T))

cannot be represented (completely isometrically) on a finite-dimensional Hilbertspace.

Theorem (C.-Ramsey 2019)

A finite-dimensional operator algebra is NFD.

� Mn� .

Example

⇢a b+ cz

�: a, b, c, d 2 C

�⇢ M

(C(T))

� Mn� .

Example

⇢a b+ cz

�: a, b, c, d 2 C

�⇢ M

(C(T))

� Mn� .

Example

⇢a b+ cz

�: a, b, c, d 2 C

�⇢ M

(C(T))

Some examples

Example (The disc algebra)

Let A(D) ⇢ C(D) be the subalgebra of functions that are holomorphic on the interior.This is a uniform algebra, so it is completely normed by its characters, and we seethat A(D) is NFD.

Example (A higher dimensional analogue)

Let Ad denote the norm closure of the polynomial multipliers of the Drury-Arvesonspace on Bd. For each finite subset F ⇢ Bd, the subspace KF = span{kz : z 2 F} iscoinvariant for Ad and [F⇢BdKF is dense. Thus, Ad is RFD.

Recall that a C⇤-correspondence X over a C⇤-algebra A is a right Hilbert A-moduleequipped with a left action of A. The tensor algebra T +

X is a non self-adjointoperator algebra acting on some Fock-type construction associated to X.

If A is finite-dimensional, then T +

X is RFD.

Example (Popescu’s non-commutative disc algebra)

It is RFD since it is the tensor algebra of the C⇤-correspondence Cd over C.

Some examples

X is RFD.

Some examples

X is RFD.

Some examples

X is RFD.

Some examples

X is RFD.

It is RFD since it is the tensor algebra of the C⇤-correspondence Cd over C.R. Clouatre (University of Manitoba) RFD operator algebras CMS Summer 2019 5 / 11

Connection with the self-adjoint world

Question

Let A be an RFD unital operator algebra. Let ⇡ : A ! B(H⇡) be a unital completely

isometric homomorphism. Is C⇤(⇡(A)) RFD?

C⇤max

(A) = C⇤

✓ cc rep

✓(a) : a 2 A

(“biggest” C⇤-algebra generated by a completely isometric copy of A)

C⇤e(A) = C⇤(A)/Shilov ideal of A

(“smallest” C⇤-algebra generated by a completely isometric copy of A)

Question

C⇤max

(A) = C⇤

✓ cc rep

✓(a) : a 2 A

Question

C⇤max

(A) = C⇤

✓ cc rep

✓(a) : a 2 A

The smallest C

⇤-algebra generated by A

Example

Consider the unital operator algebra

⇢a f

�: a, b 2 C, f 2 span{1, x

, . . . xd}�

⇢ M2

(C⇤(Ad)).

Then, T is finite-dimensional, and hence NFD (and RFD). On the other hand, acalculation shows that C⇤

e(T ) ⇠= M2

(C⇤e(Ad)), which contains the compact operators

(when d > 1) and hence cannot be RFD.

Let A ⇢Q1

Mrn be a unital operator algebra and let K ⇢ C⇤(A) denote the ideal of

compact operators in C⇤(A). Then, C⇤e(A) is RFD provided that one of the following

conditions hold.

(a) Every C⇤-algebra which is a quotient of C⇤(A)/K is RFD. For instance, this

occurs if C⇤(A)/K is commutative or finite-dimensional.

(b) The quotient map : C⇤(A) ! C⇤(A)/K strictly decreases the norm of a dense

subset of elements in A. For instance, this occurs if the algebra A contains

�1n=1

The smallest C

⇤-algebra generated by A

Example

Consider the unital operator algebra

⇢a f

�: a, b 2 C, f 2 span{1, x

, . . . xd}�

⇢ M2

(C⇤(Ad)).

Then, T is finite-dimensional, and hence NFD (and RFD). On the other hand, acalculation shows that C⇤

e(T ) ⇠= M2

(C⇤e(Ad)), which contains the compact operators

(when d > 1) and hence cannot be RFD.

Let A ⇢Q1

Mrn be a unital operator algebra and let K ⇢ C⇤(A) denote the ideal of

compact operators in C⇤(A). Then, C⇤e(A) is RFD provided that one of the following

conditions hold.

(a) Every C⇤-algebra which is a quotient of C⇤(A)/K is RFD. For instance, this

occurs if C⇤(A)/K is commutative or finite-dimensional.

(b) The quotient map : C⇤(A) ! C⇤(A)/K strictly decreases the norm of a dense

subset of elements in A. For instance, this occurs if the algebra A contains

�1n=1

The free C

⇤-algebra of an operator space

X operator space

C⇤hX i = C⇤

� cc linear

�(x) : x 2 X)!

(“biggest” C⇤-algebra generated by a completely isometric copy of X )

Theorem (Pestov 1994)

The algebra C⇤hX i is RFD for any operator space X .

Proof.

Compressions of X to arbitrary subspaces lift to ⇤-homomorphisms on C⇤hX i.

The free C

⇤-algebra of an operator space

X operator space

C⇤hX i = C⇤

� cc linear

�(x) : x 2 X)!

(“biggest” C⇤-algebra generated by a completely isometric copy of X )

Theorem (Pestov 1994)

The algebra C⇤hX i is RFD for any operator space X .

Proof.

Compressions of X to arbitrary subspaces lift to ⇤-homomorphisms on C⇤hX i.

Back to RFD operator algebras: the biggest C

⇤-algebra

Let A be an operator algebra.

(a) If A is finite-dimensional, then C⇤max

(A) is RFD.

(b) Assume that A = �1n=1

An, where An is a unital finite-dimensional operator

algebra for each n 2 N. Then, C⇤max

(A) is RFD.

Example (Popescu’s non-commutative disc algebra revisited)

We saw before that Ad is RFD. We also have that C⇤max

(Ad) is RFD. Indeed,C⇤

(Ad) coincides with the universal row contraction C⇤-algebra, which is known tobe projective (Loring-Shulman 2012) and hence RFD (Loring-Pedersen 1998).

Let A be an operator algebra. For every n 2 N, let Jn ⇢ A be a closed two-sided ideal

of A with a contractive approximate identity (e(n)

� )�2⇤n . Assume that for every

a 2 A we have limn!1 lim inf�2⇤n max{kae(n)

� k, ke(n)

� ak} = 0. Assume also that

A/Jn is finite-dimensional for every n 2 N. Then, C⇤max

(A) is RFD.

⇤-algebra

(A) is RFD.

� k, ke(n)

(A) is RFD.

⇤-algebra

(A) is RFD.

� k, ke(n)

(A) is RFD.

⇤-algebra

(A) is RFD.

� k, ke(n)

(A) is RFD.

A related problem

The bidisc algebra A(D2) is a nice uniform algebra, and hence it is NFD.

A consequence of Ando’s theorem is that C⇤max

(A(D2)) is the universal C⇤-algebra ofa pair of commuting contractions.

Theorem (Courtney-Sherman 2019)

The universal C⇤-algebra of a pair of doubly commuting contractions is RFD if and

only the Connes embedding conjecture holds.

A related problem

The bidisc algebra A(D2) is a nice uniform algebra, and hence it is NFD.A consequence of Ando’s theorem is that C⇤

A related problem

The bidisc algebra A(D2) is a nice uniform algebra, and hence it is NFD.A consequence of Ando’s theorem is that C⇤

Thank you!

residual finite-dimensionality for general operator algebrasbanach2019/pdf/clouatre.pdf ·...

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