Dual Banach algebras: an overview Uniqueness of the predual - PowerPoint PPT Presentation

Dual Banach algebras: an overview Volker Runde Dual Banach algebras Dual Banach algebras: an overview Uniqueness of the predual Representation theory Amenability Volker Runde Virtual diagonals Connes- amenability University of Alberta . . . for von Neumann algebras G¨ oteborg, August 4, 2013 . . . and in general Normal, virtual diagonals Injectivity

Dual Banach algebras: the definition Dual Banach algebras: an overview Volker Runde Dual Banach Definition algebras Uniqueness of the predual A dual Banach algebra is a pair ( A , A ∗ ) of Banach spaces such Representation theory that: Amenability 1 A = ( A ∗ ) ∗ ; Virtual diagonals Connes- 2 A is a Banach algebra, and multiplication in A is amenability separately σ ( A , A ∗ ) continuous. . . . for von Neumann algebras . . . and in general Normal, virtual diagonals Injectivity

Dual Banach algebras: some examples Dual Banach Examples algebras: an overview 1 Every W ∗ -algebra; Volker Runde 2 ( M ( G ) , C 0 ( G )) for every locally compact group G ; Dual Banach algebras Uniqueness of 3 ( M ( S ) , C ( S )) for every compact, semitopological the predual Representation semigroup S ; theory Amenability 4 ( B ( G ) , C ∗ ( G )) for every locally compact group G ; Virtual diagonals 5 ( B ( E ) , E ⊗ γ E ∗ ) for every reflexive Banach space E ; Connes- amenability 6 Let A be a Banach algebra and let A ∗∗ be equipped with . . . for von Neumann either Arens product. Then ( A ∗∗ , A ∗ ) is a dual Banach algebras . . . and in general algebra if and only if A is Arens regular; Normal, virtual diagonals Injectivity 7 If ( A , A ∗ ) is a dual Banach algebra and B is a σ ( A , A ∗ ) closed subalgebra of A , then ( B , A ∗ / ⊥ B ) is a dual Banach algebra.

Uniqueness of the predual, I Dual Banach algebras: an overview Question Volker Runde Given a dual Banach algebra ( A , A ∗ ), is A ∗ unique, i.e., if E 1 Dual Banach algebras and E 2 are Banach spaces such that ( A , E 1 ) and ( A , E 2 ) are Uniqueness of the predual dual Banach algebras, do σ ( A , E 1 ) and σ ( A , E 2 ) coincide on A ? Representation theory Amenability Theorem (S. Sakai, 1956) Virtual diagonals Connes- The predual space of a W ∗ -algebra is unique. amenability . . . for von Neumann algebras Theorem (M. Daws, H. L. Pham, S. White, 2009) . . . and in general Normal, virtual Let A be an Arens regular Banach algebra such that A ∗∗ is diagonals Injectivity unital. Then A ∗ is the unique predual of A .

Uniqueness of the predual, II Dual Banach algebras: an But. . . overview Volker Runde Example Dual Banach Let A = ℓ 1 with trivial multiplication, i.e., fg = 0 for all algebras Uniqueness of f , g ∈ ℓ 1 . Then ( ℓ 1 , c 0 ) and ( ℓ 1 , c ) are both dual Banach the predual Representation algebras. If σ ( ℓ 1 , c 0 ) and σ ( ℓ 1 , c ) coincided on ℓ 1 , then the theory Amenability induced images of c 0 and c in ℓ ∞ would have to coincide. This Virtual diagonals is impossible because the unit ball of c has extreme points Connes- amenability whereas the one of c 0 has none. . . . for von Neumann algebras . . . and in Question general Normal, virtual diagonals Are there “more natural” examples of dual Banach algebras Injectivity with non-unique preduals?

Uniqueness of the predual, III Indeed. . . Dual Banach algebras: an overview Theorem (M. Daws, et al., 2012) Volker Runde There is a family ( E t ) t ∈ R of Banach spaces such that: Dual Banach algebras 1 ( ℓ 1 ( Z ) , E t ) is a dual Banach algebra for each t ∈ R where Uniqueness of ℓ 1 ( Z ) is equipped with the convolution product; the predual Representation theory 2 E t ∼ = c 0 for each t ∈ R ; Amenability Virtual 3 σ ( ℓ 1 ( Z ) , E t ) � = σ ( ℓ 1 ( Z ) , E s ) for t � = s. diagonals Connes- amenability Still, . . . . . . for von Neumann algebras . . . and in Even though the predual A ∗ of a dual Banach algebra ( A , A ∗ ) general Normal, virtual need not be unique, there is in many cases a canonical choice diagonals Injectivity for A ∗ , e.g., M ( G ) ∗ = C 0 ( G ). In such cases, we usually suppose tacitly that we are dealing with that particular A ∗ , and simply call A a dual Banach algebra.

Daws’ representation theorem Dual Banach algebras: an overview Recall. . . Volker Runde ( B ( E ) , E ⊗ γ E ∗ ) is a dual Banach algebra for reflexive E , as is each of its weak ∗ closed subalgebras. Dual Banach algebras Uniqueness of the predual Representation Theorem (M. Daws, 2007) theory Amenability Let ( A , A ∗ ) be a dual Banach algebra. Then there are a Virtual diagonals reflexive Banach space E and an isometric, σ ( A , A ∗ ) -weak ∗ Connes- amenability continuous algebra homomorphism π : A → B ( E ) . . . . for von Neumann algebras . . . and in In short. . . general Normal, virtual diagonals Every dual Banach algebra “is” a weak ∗ closed subalgebra of Injectivity B ( E ) for some reflexive E .

A bicommutant theorem Dual Banach Question algebras: an overview Does von Neumann’s bicommutant theorem extend to general Volker Runde dual Banach algebras? Dual Banach algebras Example Uniqueness of the predual Representation theory Let �� a b Amenability � � A := : a , b , c ∈ C . Virtual 0 c diagonals Then A ⊂ B ( C 2 ) is a dual Banach algebra, but A ′′ = B ( C 2 ). Connes- amenability . . . for von Neumann algebras Theorem (M. Daws, 2010) . . . and in general Normal, virtual Let A be a unital dual Banach algebra. Then there are a diagonals Injectivity reflexive Banach space E and a unital, isometric, weak ∗ -weak ∗ continuous algebra homomorphism π : A → B ( E ) such that π ( A ) = π ( A ) ′′ .

Banach A -bimodules and derivations Dual Banach algebras: an overview Volker Runde Definition Dual Banach Let A be a Banach algebra, and let E be a Banach A -bimodule. algebras Uniqueness of A bounded linear map D : A → E is called a derivation if the predual Representation theory D ( ab ) := a · Db + ( Da ) · b ( a , b ∈ A ) . Amenability Virtual diagonals If there is x ∈ E such that Connes- amenability . . . for von Neumann Da = a · x − x · a ( a ∈ A ) , algebras . . . and in general Normal, virtual we call D an inner derivation. diagonals Injectivity

Amenable Banach algebras Dual Banach algebras: an Remark overview Volker Runde If E is a Banach A -bimodule, then so is E ∗ : Dual Banach algebras ( a ∈ A , φ ∈ E ∗ , x ∈ E ) � x , a · φ � := � x · a , φ � Uniqueness of the predual Representation theory and Amenability Virtual diagonals ( a ∈ A , φ ∈ E ∗ , x ∈ E ) . � x , φ · a � := � a · x , φ � Connes- amenability We call E ∗ a dual Banach A -bimodule. . . . for von Neumann algebras . . . and in general Definition (B. E. Johnson, 1972) Normal, virtual diagonals Injectivity A is called amenable if, for every Banach A -bimodule E , every derivation D : A → E ∗ , is inner.

Amenability for groups and Banach algebras Dual Banach algebras: an overview Theorem (B. E. Johnson, 1972) Volker Runde The following are equivalent for a locally compact group G: Dual Banach algebras 1 L 1 ( G ) is an amenable Banach algebra; Uniqueness of the predual Representation 2 the group G is amenable. theory Amenability Virtual diagonals Theorem (H. G. Dales, F. Ghahramani, & A. Ya. Helemski˘ ı, Connes- amenability 2002) . . . for von Neumann algebras The following are equivalent: . . . and in general 1 M ( G ) is amenable; Normal, virtual diagonals Injectivity 2 G is amenable and discrete.

Amenable C ∗ -algebras Dual Banach Theorem (A. Connes, U. Haagerup, et al.) algebras: an overview The following are equivalent for a C ∗ -algebra A : Volker Runde 1 A is nuclear; Dual Banach algebras Uniqueness of 2 A is amenable. the predual Representation theory Amenability Theorem (S. Wasserman, 1976) Virtual diagonals The following are equivalent for a von Neumann algebra M : Connes- amenability 1 M is nuclear; . . . for von Neumann algebras 2 M is subhomogeneous, i.e., . . . and in general Normal, virtual diagonals M ∼ = M n 1 ( M 1 ) ⊕ · · · ⊕ M n k ( M k ) Injectivity with n 1 , . . . , n k ∈ N and M 1 , . . . , M k abelian.

Virtual digaonals Dual Banach algebras: an overview Definition (B. E. Johnson, 1972) Volker Runde An element D ∈ ( A ⊗ γ A ) ∗∗ is called a virtual diagonal for A if Dual Banach algebras Uniqueness of the predual a · D = D · a ( a ∈ A ) Representation theory Amenability and Virtual diagonals a ∆ ∗∗ D = a ( a ∈ A ) , Connes- amenability where ∆ : A ⊗ γ A → A denotes multiplication. . . . for von Neumann algebras . . . and in general Theorem (B. E. Johnson, 1972) Normal, virtual diagonals Injectivity A is amenable if and only if A has a virtual diagonal.