Operator amenability of the L1-algebras of compact quantum groups

18 Pages·2013·0.06 MB·English

OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Operator amenability of the L1-algebras of
compact quantum groups
Hun Hee Lee, Seoul National University
(jointly with Martijn Caspers and E(cid:19)ric Ricard)
Banach algebras 2013
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Amenability of L1(G) for a locally compact group G
▶ A Banach algebra A is called amenable if every bounded
derivations D : A ! X(cid:3) is inner for any A-bimodule X.
▶ A locally compact group G is called amenable if there is a
1
left invariant mean on L (G) (i.e. there is a state m of
L1(G) s.t. m((cid:21)(x)f) = m(f), x 2 G, f 2 L1(G), where
(cid:21) : G ! B(L2(G)), (cid:21)(x)f(y) = f(x(cid:0)1y), x;y 2 G, the left
regular representation of G.)
▶ (Johnson ’72) L1(G) is amenable , G is amenable.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
The Fourier algebra A(G) and operator amenability
▶ Recall the group VN-alg. VN(G) = f(cid:21)(x)g′′ (cid:18) B(L2(G)).
x2G
▶ The comultiplication
∆:VN(G)!VN(G (cid:2)G); (cid:21)(x)7!(cid:21)(x)(cid:10)(cid:21)(x) is a unital
normal (cid:3)-isomorphism, so that A(G) = VN(G)(cid:3), the predual
of VN(G) can be equipped with a Banach algebra structure
given by ∆(cid:3). We call A(G) the Fourier algebra on G.
▶ Note that if we identify A(G) = ff (cid:3)g(cid:20) : f;g 2 L2(G)g, then
∆(cid:3) is nothing but the pointwise multiplication.
▶ (Ruan ’95) A(G) is operator amenable (i.e. amenable in
the category of operator spaces) , G is amenable.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Extension to quantum groups
▶ (Def) A compact quantum group G is given by (A;∆), a
unital C(cid:3)-algebra A = C(G) and a unital (cid:3)-isomorphism
∆ : A ! A(cid:10) A called the co-multiplication with
min
▶ co-associativity (∆(cid:10)id)∆=(id (cid:10)∆)∆
▶ cancellation law
∆(A)(A(cid:10)1)=spanf∆(a)(b(cid:10)1):a;b 2Ag and ∆(A)(1(cid:10)A)
are dense in A(cid:10) A.
min
▶ For a compact quantum group G = (A;∆) there is a unique
Haar state h on A such that
(h(cid:10)id)∆(a) = h(a)1 = (id (cid:10)h)∆(a); a 2 A:
▶ G is called Kac-type (or a Kac algebra) is h is tracial.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Extension to quantum groups: continued
▶ The reduced version C(G) of C(G) is the C(cid:3)-algebra
red
(cid:26)(C(G)) for the GNS representation (cid:26) of h. We de(cid:12)ne
L1(G) := C(G)′′ . Then the co-multiplication
red
∆ : C(G) ! C(G)(cid:10) C(G) can be naturally extended to a
min
unital normal (cid:3)-isomorphism ∆ : L1(G) ! L1(G)(cid:10)(cid:22)L1(G).
▶ The pre-adjoint map ∆(cid:3) : L1(G)(cid:10)bL1(G) ! L1(G), where
L1(G) = L1(G)(cid:3) gives us a (completely contractive) Banach
algebra structure on L1(G).
▶ Note that (L1(G);∆;h) is a VN-algebraic compact quantum
group in the Kusterman-Vaes sense.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Extension to quantum groups: continued 2
▶ Can we extend the connection between operator amenability
of L1(G) and amenability of G to the quantum case?
▶ (Ruan ’96) Let G be a compact Kac algebra. Then,
L1(G) is operator amenable i(cid:11) G is co-amenable (i.e. L1(G)
has a bounded approximate identity (shortly BAI)).
▶ (Easy implications)
G : a locally compact quantum group
L1(G) is operator amenable ) G is amenable (i.e. having a
left invariant mean on L1(G)) and co-amenable.
▶ (Conjecture by V. Runde) We will actually have equivalence
in the above.
▶ We will answer the above conjecture in a negative way.
▶ Note that a compact quantum group G is always amenable.
Actually, the Haar state does the job.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Biprojectivity and Bi(cid:13)atness
▶ A: completely contractive Banach algebra with the algebra
multiplication m : A(cid:10)bA ! A:
A is called operator biprojective if 9 a cb-A-bimodule map
(cid:26) : A ! A(cid:10)bA such that m(cid:14)(cid:26) = idA.
A is called operator bi(cid:13)at if 9 a cb-A-bimodule map
(cid:18) : (A(cid:10)bA)(cid:3) ! A(cid:3) such that (cid:18)(cid:14)m(cid:3) = idA(cid:3):
▶ In the above cases (cid:26) and (cid:18) are called splitting
homomorphisms.
▶ (Facts)
Operator biprojectivity ) operator bi(cid:13)atness.
Operator amenability , operator bi(cid:13)atness + having a BAI.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Biprojectivity and Bi(cid:13)atness: continued
▶ G : a locally compact group.
(Helemskii, ’85)
L1(G) is operator biprojective , G is compact.
(Aristov, ’02, Wood, ’02)
A(G) operator biprojective , G is discrete.
▶ L1(G) is operator bi(cid:13)at , G is amenable.
(Aristov/Runde/Spronk, ’04)
A(G) operator bi(cid:13)at ( G is a quasi-[SIN] group (i.e. there is
a BAI (e ) in L1(G) s.t. (cid:14) (cid:3)e (cid:0)e (cid:3)(cid:14) ! 0, x 2 G), a
(cid:11) x (cid:11) (cid:11) x
strictly bigger class than amenable groups and discrete groups.
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
Biprojectivity and Bi(cid:13)atness: quantum cases
▶ (Aristov ’04, Daws ’10, Ruan/Xu ’96) Some general theory
▶ G : LCQGp with L1(G) operator biprojective ) G is compact.
▶ G : compact Kac algebra ) L1(G) is operator biprojective.
▶ L1(G) is operator biprojective with completely contractive (or
completely positive) splitting hom. ) G is of Kac-type.
▶ (Question by Aristov) Is L1(G) operator biprojective for any
compact quantum group G?
OperatoramenabilityoftheL1-algebrasofcompactquantumgroups
The main results
Theorem
Let G be a compact quantum group. If L1(G) is operator bi(cid:13)at,
then G is of Kac type.
Corollary
Let G be a compact quantum group. Then, L1(G) is operator
amenable if and only if G is co-amenable and of Kac type.
Note that there are many examples of co-amenable compact
quantum group of non-Kac type, e.g. SU (2), 0 < q < 1.
q
Corollary
Let G be a locally compact quantum group. Then, L1(G) is
operator biprojective if and only if G is compact and of Kac type.