Isomorphisms of Algebras of Convolution Operators

Abstract

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...

Rigidity results for $L^p$-operator algebras and applications

“…The interest in localizable measure algebras stems from the fact that they form the largest class of meausre algebras where the Radon-Nikodym theorem is applicable. Most importantly for us, Lamperti's description of the invertible isometries of an L p -space for p = 2 from [24], which was originally proved only for σ-finite spaces, remains valid in the more general context of localizable measure algebras; see Section 3 in [18].…”

“…Moreover, the resulting map [0, 1] → Isom(L p (µ)) into the group of surjective isometric operators, given by t → u t , is norm-continuous. By Lamperti's theorem (in the form given in Theorem 3.7 in [18]; see [24] for the original statement), for every t ∈ [0, 1] there exist a unique h t ∈ U(L ∞ (µ)), the unitary group of L ∞ (µ), and a unique Boolean automorphism ϕ t of A such that, in the notation of Lemma 3.3 of [18], we have u T = m ht • v ϕt . By the norm computation in equation ( 6) of [18], for s, t ∈ [0, 1] we have…”

A modern look at algebras of operators onLp-spaces

“…This result had been earlier announced (without proof) by Banach for the unit interval with the Lebesgue measure, and for this reason it is also sometimes referred to as the "Banach-Lamperti Theorem". In this section, which is based on Sections 2 and 3 of [19], we generalize Lamperti's result by characterizing the surjective, linear isometries on the L p -space of a localizable measure algebra; see Theorem 2.12. The generalization from σ-finite spaces to localizable ones will allow us in the next sections to deal with locally compact groups that are not σ-compact.…”