Unitarizable representations and amenable operator algebras Yemon Choi Lancaster University QOP Network Meeting Lancaster University, 25th September 2014 0 / 21
Today’s menu Unitarizable representations 1 Amenable operator algebras? 2 Non-unitarizable representations 3 Unitarizable representations (reprise) 4 Open questions 5 0 / 21
Today’s menu Unitarizable representations 1 Amenable operator algebras? 2 Non-unitarizable representations 3 Unitarizable representations (reprise) 4 Open questions 5 0 / 21
Unitarizable representations and fixed points What do I mean by “a representation ( θ, Γ , A ) ”? In this talk: a discrete group Γ , a unital C ∗ -algebra A , and a HM θ : Γ → A inv . We say the representation is bounded if sup x ∈ Γ � θ ( x ) � < ∞ . 1 / 21
Unitarizable representations and fixed points What do I mean by “a representation ( θ, Γ , A ) ”? In this talk: a discrete group Γ , a unital C ∗ -algebra A , and a HM θ : Γ → A inv . We say the representation is bounded if sup x ∈ Γ � θ ( x ) � < ∞ . A (bounded) representation ( θ, Γ , A ) is unitarizable, or similar to a ∗ -representation, if there exists s ∈ A inv such that sθ ( x ) s − 1 ∈ U ( A ) for all x ∈ Γ . We say that s is a similarity element for θ . 1 / 21
Let A + inv = A inv ∩ A + . Then Γ acts on A + inv , as follows; θ + ( x ) : h �→ θ ( x ) hθ ( x ) ∗ . Exercise ( θ, Γ , A ) is unitarizable if and only if θ + : Γ � A + inv has a fixed point. 2 / 21
Let A + inv = A inv ∩ A + . Then Γ acts on A + inv , as follows; θ + ( x ) : h �→ θ ( x ) hθ ( x ) ∗ . Exercise ( θ, Γ , A ) is unitarizable if and only if θ + : Γ � A + inv has a fixed point. Exercise Suppose Γ is finite and let θ : Γ → A inv be a HM. Show that the action θ + : Γ � A + inv has a fixed point. ( Hint: average over orbits.) Thus every finite subgroup of A inv is similar to a subgroup of U ( A ) . 2 / 21
Example 1. Let ε > 0 and consider � 1 � � 1 � 0 ε x = , y = . 0 − 1 0 − 1 These give a pair of representations θ x , θ y : Z / 2 Z → ( M 2 ) inv . y act on ( M 2 ) + θ + x and θ + inv , but have no common fixed point. Hence there is no s ∈ ( M 2 ) inv which simultaneously unitarizes θ x and θ y . 3 / 21
A theorem of Day and Dixmier Theorem ( Day , 1950; Dixmier , 1950) Let Γ be an amenable discrete group and M a von Neumann algebra. Then every bounded representation ( θ, Γ , M ) is unitarizable. The case Γ = Z , M = B ( H ) was proved by Sz.-Nagy (1947) and contains the essential ideas for the general case. 4 / 21
A theorem of Day and Dixmier Theorem ( Day , 1950; Dixmier , 1950) Let Γ be an amenable discrete group and M a von Neumann algebra. Then every bounded representation ( θ, Γ , M ) is unitarizable. The case Γ = Z , M = B ( H ) was proved by Sz.-Nagy (1947) and contains the essential ideas for the general case. Theorem ( Pisier , 2007) If Γ is a discrete non-amenable group, then there is some von Neumann algebra M and some bounded, non-unitarizable rep ( θ, Γ , M ) . Unknown if we can always take M = B ( H ) ! 4 / 21
Today’s menu Unitarizable representations 1 Amenable operator algebras? 2 Non-unitarizable representations 3 Unitarizable representations (reprise) 4 Open questions 5 4 / 21
Amenable Banach algebras Quick definition: a Banach algebra A is amenable if it has a bounded approximate diagonal, i.e. a bounded net ( m α ) ∈ A � ⊗ A satisfying a · m α − m α · a → 0 and aπ ( m α ) → a for each a ∈ A . Example 2. [ Johnson , 1972] If Γ is a discrete amenable group, then ℓ 1 (Γ) is amenable. In particular, ℓ 1 (Γ) is amenable whenever Γ is abelian. 5 / 21
Amenable Banach algebras Quick definition: a Banach algebra A is amenable if it has a bounded approximate diagonal, i.e. a bounded net ( m α ) ∈ A � ⊗ A satisfying a · m α − m α · a → 0 and aπ ( m α ) → a for each a ∈ A . Example 2. [ Johnson , 1972] If Γ is a discrete amenable group, then ℓ 1 (Γ) is amenable. In particular, ℓ 1 (Γ) is amenable whenever Γ is abelian. Some hereditary properties if A is amenable and φ : A → B is a HM with dense range, then B is amenable; if A is a Banach algebra, J is a closed ideal in A , and J and A / J are both amenable, then so is A . 5 / 21
Example 3. [ Johnson , 1972] C ( X ) and K ( H ) are amenable. Both examples are closures of HM’ic images of ℓ 1 (Γ) , for some choice of amenable Γ . 6 / 21
Example 3. [ Johnson , 1972] C ( X ) and K ( H ) are amenable. Both examples are closures of HM’ic images of ℓ 1 (Γ) , for some choice of amenable Γ . Remark Johnson went on to show (1972) that every GCR (i.e. Type I ) C ∗ -algebra is amenable. (In fact, strongly amenable.) Also, the algebras O n , 2 ≤ n ≤ ∞ , are amenable (but not strongly amenable). [ Rosenberg , 1977] None of these proofs need the word “nuclear” 6 / 21
Amenable operator algebras? (“Operator algebra” = norm closed subalg of B ( H ) .) We might wish to study amenable operator algebras. But how can we find examples? 7 / 21
Amenable operator algebras? (“Operator algebra” = norm closed subalg of B ( H ) .) We might wish to study amenable operator algebras. But how can we find examples? Question. Let A be an amenable operator algebra. Must A be isomorphic to (the underlying Banach algebra of) some C ∗ -algebra? In the finite-dimensional setting, the answer is YES , by Wedderburn’s theorem. This was pushed further by Gifford in his PhD thesis. 7 / 21
Amenable operator algebras? (“Operator algebra” = norm closed subalg of B ( H ) .) We might wish to study amenable operator algebras. But how can we find examples? Question. Let A be an amenable operator algebra. Must A be isomorphic to (the underlying Banach algebra of) some C ∗ -algebra? In the finite-dimensional setting, the answer is YES , by Wedderburn’s theorem. This was pushed further by Gifford in his PhD thesis. Theorem ( Gifford , 1997/2006) Amenable, closed subalgebras of K ( H ) are isomorphic to C ∗ -algebras. In full generality, this question resisted attempts over many years. . . 7 / 21
A question and idea of Ozawa Let Q ( H ) := B ( H ) / K ( H ) denote the Calkin algebra and q : B ( H ) → Q ( H ) the quotient HM. Question. Is there a bounded, non-unitarizable rep ( θ, Z , Q ( H )) ? What if we replace Z by some other discrete abelian group? 8 / 21
A question and idea of Ozawa Let Q ( H ) := B ( H ) / K ( H ) denote the Calkin algebra and q : B ( H ) → Q ( H ) the quotient HM. Question. Is there a bounded, non-unitarizable rep ( θ, Z , Q ( H )) ? What if we replace Z by some other discrete abelian group? The point of Ozawa’s question: suppose Γ is abelian; then each bounded rep ( θ, Γ , Q ( H )) gives an amenable A ⊂ B ( H ) ; if A is isomorphic to a C ∗ -algebra then θ is unitarizable. 8 / 21
A question and idea of Ozawa Let Q ( H ) := B ( H ) / K ( H ) denote the Calkin algebra and q : B ( H ) → Q ( H ) the quotient HM. Question. Is there a bounded, non-unitarizable rep ( θ, Z , Q ( H )) ? What if we replace Z by some other discrete abelian group? The point of Ozawa’s question: suppose Γ is abelian; then each bounded rep ( θ, Γ , Q ( H )) gives an amenable A ⊂ B ( H ) ; if A is isomorphic to a C ∗ -algebra then θ is unitarizable. So, a bounded non-unitarizable ( θ, Γ , Q ( H )) gives rise to an amenable operator algebra not isomorphic to any C ∗ -algebra. 8 / 21
Details Given θ : Γ → Q ( H ) define B = lin { θ ( x ) : x ∈ Γ } . B is amenable. Let A = q − 1 ( B ) . There is a short exact sequence q 0 → K ( H ) → A − → B → 0 By hereditary properties, A is an amenable operator algebra. 9 / 21
Details Given θ : Γ → Q ( H ) define B = lin { θ ( x ) : x ∈ Γ } . B is amenable. Let A = q − 1 ( B ) . There is a short exact sequence q 0 → K ( H ) → A − → B → 0 By hereditary properties, A is an amenable operator algebra. Now suppose A is also isomorphic to a C ∗ -algebra. Then there exists R ∈ B ( H ) inv such that R A R − 1 is a self-adjoint subalgebra of B ( H ) . Put s := q ( R ) . Then s B s − 1 is a commutative and self-adjoint subalgebra of Q ( H ) . Observe: if x ∈ Γ , then sθ ( x ) s − 1 is normal with spectrum contained in T , hence is unitary. So s unitarizes θ . 9 / 21
Today’s menu Unitarizable representations 1 Amenable operator algebras? 2 Non-unitarizable representations 3 Unitarizable representations (reprise) 4 Open questions 5 9 / 21
A construction of Farah and Ozawa Theorem (see arXiv:1309.2415v1 ) There is a set T of bounded HMs Z ⊕ c → Q ( ℓ 2 ) , with | T | = 2 c , such that T is parametrized by certain “ 1 -cocycles” Let θ ∈ T ; then ( θ, Z ⊕ c , Q ( ℓ 2 )) is unitarizable iff θ corresponds to an “inner” cocycle . 10 / 21
A construction of Farah and Ozawa Theorem (see arXiv:1309.2415v1 ) There is a set T of bounded HMs Z ⊕ c → Q ( ℓ 2 ) , with | T | = 2 c , such that T is parametrized by certain “ 1 -cocycles” Let θ ∈ T ; then ( θ, Z ⊕ c , Q ( ℓ 2 )) is unitarizable iff θ corresponds to an “inner” cocycle . But |{ inner cocycles }| ≤ Q ( ℓ 2 ) = c < 2 c = | T | . Therefore there is a bounded, non-unitarizable ( θ, Z ⊕ c , Q ( ℓ 2 )) 10 / 21
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