2019
DOI: 10.48550/arxiv.1909.03612
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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;… Show more

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Cited by 6 publications
(13 citation statements)
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“…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).…”
Section: Bumpy Semigroupsmentioning
confidence: 99%
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“…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).…”
Section: Bumpy Semigroupsmentioning
confidence: 99%
“…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.…”
Section: Note That We Always Havementioning
confidence: 99%
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“…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'.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%