Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

Runde, Volker

doi:10.1007/pl00000490

Cited by 4 publications

(1 citation statement)

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“…In this final section, we apply the methods developed in Section 4 to confirm a conjecture of Galé-Ransford-White [15] for p-convolution algebras. Their conjecture asserts that every reflexive, amenable Banach algebra is automatically finite dimensional, and it has been confirmed in a number of situations; see, for example, the introduction of [36]. We begin with some preparatory results, which are interesting on their own right.…”

Section: The Isomorphism Problem For P-convolution Algebrasmentioning

confidence: 95%

Isomorphisms of Algebras of Convolution Operators

Gardella¹,

Thiel²

2018

Preprint

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For p, q ∈ [1, ∞), we study the isomorphism problem for the pand q-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras CVp(G) of p-convolvers and P Mp(G) of p-pseudomeasures, for p = 2. More generally, we show that if CVp(G) is isometrically isomorphic to CVq(H), with p, q = 2, then G must be isomorphic to H and p and q are either equal or conjugate. This implies that there is no L p -version of Connes' uniqueness of the hyperfinite II 1 -factor. Similar results apply to the algebra P Fp(G) of p-pseudofunctions, generalizing a classical result of Wendel. We also show that other L p -rigidity results for groups can be easily recovered and extended using our main theorem.Our results answer questions originally formulated in the work of Herz in the 70's. Moreover, our methods reveal new information about the Banach algebras in question. As a non-trivial application, we verify the reflexivity conjecture for all Banach algebras lying between P Fp(G) and CVp(G): if any such algebra is reflexive and amenable, then G is finite.EUSEBIO GARDELLA AND HANNES THIEL von Neumann algebra, or even from its reduced group C * -algebra: consider, for example, Z 2 ⊕ Z 2 and Z 4 . More drastically, Connes' celebrated result on uniqueness of the hyperfinite II 1 -factor implies that any two countable amenable ICC groups have isomorphic group von Neumann algebras. There also exist groups that can be recovered from their C * -algebras but not from their von Neumann algebras (such as Z). Positive results in this context (usually referred to as superrigidity results) are difficult to find: for von Neumann algebras, the first one is the groundbreaking work of Ioana-Popa-Vaes ([25]) on wreath product groups, while non-trivial results for C * -algebras are even more recent ([10, 28]). Indeed, the passage from C * λ (G) to L(G) tends to "erase" a lot of information about G. For example, while every countable, torsion-free, abelian group is recovered from its reduced group C * -algebra, all such groups have isomorphic group von Neumann algebras. Similarly, while it is known that F n and F m have non-isomorphic group C * -algebras, whether this is the case for their von Neumann algebras is a notable open problem.In this work, we are interested in the L p -version of the problems described above. Given p ∈ [1, ∞) and a locally compact group G, we let λ p be the representation of L 1 (G) on L p (G) given by left convolution. The algebra of p-pseudofunctions P F p (G) is the Banach algebra generated by λ p (L 1 (G)); the algebra of p-convolvers CV p (G) is the double commutant of P F p (G); and, for p > 1, the algebra of ppseudomeasures P M p (G) is the weak * -closure 1 of P F p (G). We thus obtain a continuously varying family of Banach algebras which for p = 1 give the algebras L 1 (G) and M (G), and for p = 2 agree with C * λ (G) and L(G). These algebras were introduced by Herz [23] several decad...

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Section: The Isomorphism Problem For P-convolution Algebrasmentioning

confidence: 95%