Projections in Eberlein compactifications Nico Spronk (U. Waterloo) - PowerPoint PPT Presentation

Projections in Eberlein compactifications Nico Spronk (U. Waterloo) Fields Institute, COSy 2014

A classical decomposition G – locally compact group π : G → U ( H ) continuous unitary representation Theorem [Jacobs–de Leeuw–Glicksberg] π = π wm ⊕ π ret on p wm H ⊕ 2 p ret H where � w � H wm = ξ ∈ H : 0 ∈ π ( G ) ξ � w � w whenever η ∈ π ( G ) ξ H ret = ξ ∈ H : ξ ∈ π ( G ) η .

Semigroup perspective ( ball ( B ( H )) , w . o . t . ) – semitopological semigroup i.e. x �→ xy , yx each continuous for each fixed y w . o . t . – compact semitopological semigroup G π = π ( G ) E.g.: λ : G → U ( L 2 ( G )) left reg. rep’n, G λ = G ∞ Theorem [de Leeuw–Glicksberg, Troallic] • p ret minimal projection (idempotent) in G π ret = p ret G π compact group & ideal in G π • G π

Eberlein compactification S – compact semitop’l semigroup called Eberlein if S ֒ → ( ball ( B ( H )) , w . o . t . ) homeo’lly ̟ : G → U ( H ) – universal representation Theorem [Megrelishvili, S.–Stokke] G E := G ̟ universal Eberlein compactification of G S Eberlein semigroup, η : G → S homo’m w. dense range (i.e. ( η, S ) is an Eberlein compactification of G ) η : G E ։ S ⇒ ∃ extension ˜ Can be done for non-locally compact G as well.

Eberlein groups & topologies ( G , τ G ) – (complete) topological group � � s �→ � π ( s ) ξ | η � : π : G → U ( H ) τ G - w . o . t . -cts. B ( G ) = ξ, η ∈ H , H Hil. space ( G , τ G ) is Eberlein if τ G = σ ( G , B ( G )). → ̟ ( G ) ⊂ G E is a homeomorphism. Equivalently, ̟ : G ֒ E.g. ( G , τ G ) locally compact, or discrete. Coarser Eberlein topologies: � T ( G ) = { τ ⊆ τ G : ( G , τ ) top’l group, τ = σ ( G , B τ ( G )) } where B τ ( G ) = B ( G ) ∩ C ( G , τ ).

... Eberlein topologies τ ∈ � T ( G ) N τ = � { U : U τ -nbhd. of e } is a τ -closed normal subgroup ¯ τ – (Hausdorff) toplogy induced on G / N τ U ¯ τ – two-sided uniformity on G / N τ generated by ¯ τ . Facts τ is an Eberlein group U ¯ • G τ = ( G / N τ , ¯ τ ) • ∃ cts. homo’m η τ : G → G τ w. dense range η τ : G E ։ G E • ∃ unique cts. ext’n ˜ τ

Relations to central projections ZE ( G E ) = { z ∈ G E : z 2 = z & tz = zt ∀ t ∈ G E } Theorem (after [Ruppert] for abelian G ) (i) ∃ map T : ZE ( G E ) → � T ( G ): • define for z , η z : G → G E by η z ( s ) = z ̟ ( s ) • let T ( z ) = σ ( G , { η z } ) (ii) ∃ map E : � T ( G ) → ZE ( G E ): τ ( { e τ } ) ⊂ G E admits a η − 1 • given τ , the compact semigroup ˜ unique min’l idempotent, z = E ( τ ) [Ruppert, Troallic] • E ( τ ) is central in G E Notes. • E ◦ T = id ZE ( G E ) , T ◦ E ( τ ) ⊇ τ . = G E ( z ) := { t ∈ G E : tz = t & tt ∗ = z = t ∗ t } • G T ( z ) ∼ • τ ⊆ τ ′ ⇒ E ( τ ) ≤ E ( τ ′ ), z ≤ z ′ ⇒ T ( z ) ⊆ T ( z ′ )

� � When is T ◦ E ( τ ) = τ ? η τ ′ � τ ⊆ τ ′ in � T ( G ) G G τ ′ ❆ get cts. homo’ms w. dense range ❆ ❆ η τ ′ ❆ ❆ η τ ′ η τ ❆ τ ◦ η τ ′ = η τ τ ❆ ❆ G τ Co-compact/Cauchy containment τ ⊆ c τ ′ in � T ( G ) if τ ⊆ τ ′ & • ker η τ ′ τ compact & η τ ′ τ open. • Eq’ly, each τ -Cauchy net in G admits τ ′ -Cauchy refinement. Theorem τ ⊆ τ ′ in � τ ⊆ c τ ′ ⇔ E ( τ ) = E ( τ ′ ) T ( G ): & τ ⊆ c T ◦ E ( τ ) “Reasonable” Eberlein topologies: T ( G ) = T ( ZE ( G E )) ⊆ � T ( G )

Jacobs–de Leeuw–Glicksberg revisted B ( G ) ∼ = ( ̟ ( G ) ′′ ) ∗ , Banach algebra of functions on G : � π ( · ) ξ | η � + � π ′ ( · ) ξ ′ | η ′ � = � π ⊕ π ′ ( · ) ξ ⊕ ξ ′ | η ⊕ η ′ � � π ( · ) ξ | η �� π ′ ( · ) ξ ′ | η ′ � = � π ⊗ π ′ ( · ) ξ ⊗ ξ ′ | η ⊗ η ′ � Almost periodic (Bohr) topology τ ap = T ( p ret ) satisfies ap ⊕ π τ ap , p ret = π ′′ ( E ( τ ap )) • π : G → U ( H ) rep’n, π = π τ ⊥ • B ( G ) = I τ ap ( G ) ⊕ B τ ap ( G ), B τ ap ( G ) = E ( τ ap ) · B ( G ), Theorem Let τ ∈ T ( G ). Then • π : G → U ( H ) rep’n, π = π τ ⊥ ⊕ π τ , π τ = π ′′ ( E ( τ )) π • B ( G ) = I τ ( G ) ⊕ B τ ( G ) where B τ ( G ) = E ( τ ) · B ( G ), I τ ( G ) ⊳ B ( G )

Operator amenability of B ( G ) G locally compact Theorem [Dales–Ghahramani–Helemski˘ ı, Brown–Moran] Measure algebra M ( G ) (op.) amenable ⇔ G discrete & amenable. G abelian: B ( G ) ∼ = M ( � G ) (op.) amenable ⇔ G compact. False conjecture: B ( G ) op. amenable ⇔ G compact. Theorem [Runde-S.] (after [Ilie-S.]) G n , p = Q n p ⋊ GL n ( O p ) has B ( G n , p ) op. amenable. Proposition B ( G ) op. amenable ⇒ | ZE ( G E ) | = |T ( G ) | < ∞ . un] G abelian non-compact, | ZE ( G E ) | ≥ c [Elg¨

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