Amenability of operator algebras on Banach spaces, I PDF
Preview Amenability of operator algebras on Banach spaces, I
Amenabilityof operator algebrason Banach spaces,I VolkerRunde Amenability of operator algebras on Banach Prelude spaces, I Amenable groups Amenable Banach algebras Volker Runde C∗-andvon Neumann UniversityofAlberta algebras Similarity problems NBFAS, Leeds, June 1, 2010 Philosophical musings Amenabilityof operator algebrason Banach spaces,I A quote VolkerRunde Big things are fucking rarely amenable! Prelude Amenable groups N.N., Istanbul, 2004. Amenable Banach algebras C∗-andvon Thus... Neumann algebras AMENABLE ≈ SMALL Similarity problems Mean things Amenabilityof operator algebrason Definition Banach spaces,I Let G be a locally compact group. A mean on L∞(G) is a VolkerRunde functional M ∈ L∞(G)∗ such that (cid:104)1,M(cid:105) = (cid:107)M(cid:107) = 1. Prelude Amenable Definition (J. von Neumann 1929; M. M. Day, 1949) groups Amenable G is amenable if there is a mean on L∞(G) which is left Banach algebras invariant, i.e., C∗-andvon Neumann algebras (cid:104)L φ,M(cid:105) = (cid:104)φ,M(cid:105) (x ∈ G, φ ∈ L∞(G)), x Similarity problems where (L φ)(y) := φ(xy) (y ∈ G). x Some amenable groups... Amenabilityof Examples operator algebrason Banach spaces,I 1 Compact groups amenable: M = Haar measure. VolkerRunde 2 Abelian groups are amenable: use Markov–Kakutani to get Prelude M. Amenable groups Amenable Really nice hereditary properties! Banach algebras C∗-andvon 1 If G is amenable and H < G, then H is amenable. Neumann algebras 2 If G is is amenable and N (cid:67)G, then G/N is amenable. Sprimobillaermitsy 3 If G and N (cid:67)G are such that N and G/N are amenable, then G is amenable. 4 If (Hα)α is a directed union of closed, amenable subgroups (cid:83) of G such that G = H , then G is amenable. α α ...and a non-amenable one, I Amenabilityof operator algebrason Banach spaces,I Example VolkerRunde Let F be the free group in two generators. 2 Prelude Assume that there is a left invariant mean M on (cid:96)∞(F ). 2 Amenable Define groups µ : P(F ) → [0,1], E (cid:55)→ (cid:104)χ ,M(cid:105). Amenable 2 E Banach algebras Then C∗-andvon Neumann µ is finitely additive, algebras µ(F ) = 1, and Similarity 2 problems µ(xE) = µ(E) (x ∈ F , E ⊂ F ). 2 2 ...and a non-amenable one, II Amenabilityof operator Example (continued...) algebrason Banach spaces,I For x ∈ {a,b,a−1,b−1} set VolkerRunde W(x) := {w ∈ F : w starts with x}. Prelude 2 Amenable groups Let w ∈ F2\W(a). Then a−1w ∈ W(a−1), therefore Amenable Banach w ∈ aW(a−1), algebras C∗-andvon Neumann and thus algebras F = W(a)∪aW(a−1). Similarity 2 problems Similarly, F = W(b)∪bW(b−1) 2 holds. ...and a non-amenable one, III Amenabilityof operator Example (continued even further) algebrason Banach spaces,I Since VolkerRunde · · · · F = {(cid:15)} ∪ W(a) ∪ W(a−1) ∪ W(b) ∪ W(b−1), Prelude 2 Amenable groups we have Amenable Banach 1 = µ(F ) algebras 2 C∗-andvon ≥ µ(W(a))+µ(aW(a−1))+µ(W(b))+µ(bW(b−1)) Neumann algebras ≥ µ(W(a)∪aW(a−1))+µ(W(b)∪bW(b−1)) Similarity problems = µ(F )+µ(F ) 2 2 = 2, which is nonsense. Consequences Amenabilityof operator algebrason Banach Hence, G is amenable if... spaces,I VolkerRunde 1 G is solvable or Prelude 2 G is locally finite. Amenable groups Amenable But G is not amenable if... Banach algebras G contains F as closed subgroup, e.g., if 2 C∗-andvon Neumann G = SL(N,R) with N ≥ 2, algebras Similarity G = GL(N,R) with N ≥ 2, or problems G = SO(N,R) with N ≥ 3 equipped with the discrete topology. Amenable Banach algebras via approximate diagonals Amenabilityof operator algebrason Banach spaces,I Definition (B. E. Johnson, 1972) VolkerRunde A Banach algebra A is said to be amenable if it possesses an Prelude approximate diagonal, i.e., a bounded net (d ) in the α α Amenable groups projective tensor product A⊗ˆA such that Amenable Banach a·d −d ·a → 0 (a ∈ A) algebras α α C∗-andvon Neumann and algebras a∆d → a (a ∈ A) Similarity α problems with ∆ : A⊗ˆA → A denoting multiplication. Amenable Banach algebras and amenable groups Amenabilityof operator algebrason Banach spaces,I Theorem (B. E. Johnson, 1972) VolkerRunde Prelude The following are equivalent for a locally compact group G: Amenable 1 G is amenable; groups Amenable 2 L1(G) is amenable. Banach algebras C∗-andvon Grand theme Neumann algebras Let C be a class of Banach algebras. Characterize the amenable Similarity problems members of C!
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