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

Abstract

For p ∈ [1, ∞), we show that every unital L p -operator algebra contains a unique maximal C * -subalgebra, which we call the C * -core. When p = 2, the C * -core of an L p -operator algebra is abelian. Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By calculating this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2; the most striking one being that of crossed products by essentially free actions.These rigidity results give answers to questions concerning the existence of homomorphisms or isomorphisms between different algebras. Among others, we show that for the L p -analog O p 2 of the Cuntz algebra, there is no isometric isomorphism between O p 2 and O p 2 ⊗p O p 2 , when p = 2. In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K-theoretic classification of purely infinite simple amenable L p -operator algebras for p = 2.

Reconstructing Etale Groupoids from Semigroups

“…Note that Example 3.5 corresponds to [Exe10]. Also Example 3.6 corresponds to [Ste19] (see the comments after Corollary 7.5), while Example 3.8 with Y = C corresponds to [Kum86], [Ren08] and [CGT19] (see the comments after Corollary 6.10).…”

“…Again, the precise methods of reconstruction differ -instead of using ultrafilters to recover G, [Ren08] and [CGT19] first use characters to recover G 0 and then use germs to recover the rest of G. One advantage of ultrafilters over characters/germs is that they only depend on the product rather than the full algebra structure. Consequently, we only need a semigroup isomorphism between the C*-algebras or L p -algebras rather than an isometric algebra isomorphism.…”

“…Continuous functions on étale groupoids provide a rich array of non-commutative structures like C*-algebras, L p -algebras, Steinberg algebras and inverse semigroups. Under suitable conditions, the groupoid can be even reconstructed from the corresponding algebraic structure, as seen in [Kum86], [Ren08], [Exe10], [Ste19] and [CGT19]. While the algebraic structures involved differ, what they do share is a product defined in essentially the same way.…”

Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras

“…On the other side, the Weyl groupoid also comes with a twist or, equivalently, a Fell line bundle. In recent years there has also been a big push to extend the Weyl groupoid construction in various directions -see [BCaH17], [BCW17], [CRST17], [CR18], [CGT19], [Ste19], [KM20], [BC20], [AdCC + 21] and [Bic21]. From this together with the vast literature on groupoid C*-algebras that has emerged over the past 30-40 years, it seems safe to say that étale groupoids are the best candidate for these long sought after 'non-commutative spaces'.…”

A modern look at algebras of operators onLp-spaces

“…The next four sections are devoted to the study of three very prominent classes of examples: group algebras (Sections 4 and 5); Cuntz and graph algebras (Section 6); and crossed products (Section 7). Finally, in Section 8 we discuss a recent result obtained in [3]: O p 2 ⊗ O p 2 is not isomorphic to O p 2 for p ∈ [1, ∞) \ {2} (while it is well-known that an isomorphism exists for p = 2; see [28]). This answers a question of Phillips. Most of this work is expository, although the exposition given here is quite different from what has appeared elswehere.…”