Bounded derivations on uniform Roe algebras. (English) Zbl 1456.46057
This paper studies bounded derivations of uniform Roe algebras, operator algebras capable of detecting phenomena in coarse geometry. These algebras (and their nonuniform analogues) were introduced by J. Roe [Coarse cohomology and index theory on complete Riemannian manifolds. Providence, RI: American Mathematical Society (AMS) (1993; Zbl 0780.58043)] for their connections with (higher) index theory, specifically for their applications to elliptic operators on noncompact manifolds [J. Roe, Index theory, coarse geometry, and topology of manifolds. Providence, RI: AMS, American Mathematical Society (1996; Zbl 0853.58003)]; later the study of these algebras was boosted due to its intrinsic relation with the coarse Baum-Connes conjecture and, consequently, with the Novikov conjecture [G.-L. Yu, Invent. Math. 139, No. 1, 201–240 (2000; Zbl 0956.19004)]. Recently they have also been used as a framework in mathematical physics to study the classification of topological phases and the topology of quantum systems (see, e.g., [E. E. Ewert and R. Meyer, Commun. Math. Phys. 366, No. 3, 1069–1098 (2019; Zbl 1481.46076)]).
This paper focuses on a purely algebraic aspect of uniform Roe algebras: the study of their derivations. A derivation on a \(C^*\)-algebra \(A\) is a linear map \(\delta: A\to A\) with the property that \[ \delta(ab)=a\delta(b)+\delta(a)b \] for all \(a,b\in A\). If \(A\) is unital and \(d\in A\), the map \(a\mapsto da-ad\) is a derivation; these derivations are called inner. The question of which \(C^*\)-algebras have only inner derivations was studied because of its connections with mathematical physics and one-parameters groups. It was shown (see [C. A. Akemann and G. K. Pedersen, Am. J. Math. 101, 1047–1061 (1979; Zbl 0432.46059); G. A. Elliott, Ann. Math. (2) 106, 121–143 (1977; Zbl 0365.46051)]) that the only separable \(C^*\)-algebras having only inner bounded derivations are those that can be written as a direct sum of continuous trace and simple blocks. On the other side of the spectrum, von Neumann algebras only have inner bounded derivations [S. Sakai, Ann. Math. (2) 83, 273–279 (1966; Zbl 0139.30601)].
Uniform Roe algebras of uniformly locally finite spaces are nonseparable, and far from being simple or abelian. The main result of this paper shows that these objects (which, even though they are not von Neumann algebras, can be written as a countable union of weakly closed Banach subspaces) retain some sort of rigidity when it comes to derivations. In particular, they only have inner bounded derivations. This theorem describes a completely new class of objects which only have inner derivations.
The proof has two main ingredients: a basic form of a “reduction of cocycles” argument of A. M. Sinclair and R. R. Smith [Contemp. Math. 365, 383–400 (2004; Zbl 1080.46039)] used in studying Hochschild cohomology of von Neumann algebras, and recent Ramsey-theoretic ideas [B. M. Braga and I. Farah, Trans. Am. Math. Soc. 374, No. 2, 1007–1040 (2021; Zbl 1458.51007)] used in the study of uniform Roe algebras.
This paper focuses on a purely algebraic aspect of uniform Roe algebras: the study of their derivations. A derivation on a \(C^*\)-algebra \(A\) is a linear map \(\delta: A\to A\) with the property that \[ \delta(ab)=a\delta(b)+\delta(a)b \] for all \(a,b\in A\). If \(A\) is unital and \(d\in A\), the map \(a\mapsto da-ad\) is a derivation; these derivations are called inner. The question of which \(C^*\)-algebras have only inner derivations was studied because of its connections with mathematical physics and one-parameters groups. It was shown (see [C. A. Akemann and G. K. Pedersen, Am. J. Math. 101, 1047–1061 (1979; Zbl 0432.46059); G. A. Elliott, Ann. Math. (2) 106, 121–143 (1977; Zbl 0365.46051)]) that the only separable \(C^*\)-algebras having only inner bounded derivations are those that can be written as a direct sum of continuous trace and simple blocks. On the other side of the spectrum, von Neumann algebras only have inner bounded derivations [S. Sakai, Ann. Math. (2) 83, 273–279 (1966; Zbl 0139.30601)].
Uniform Roe algebras of uniformly locally finite spaces are nonseparable, and far from being simple or abelian. The main result of this paper shows that these objects (which, even though they are not von Neumann algebras, can be written as a countable union of weakly closed Banach subspaces) retain some sort of rigidity when it comes to derivations. In particular, they only have inner bounded derivations. This theorem describes a completely new class of objects which only have inner derivations.
The proof has two main ingredients: a basic form of a “reduction of cocycles” argument of A. M. Sinclair and R. R. Smith [Contemp. Math. 365, 383–400 (2004; Zbl 1080.46039)] used in studying Hochschild cohomology of von Neumann algebras, and recent Ramsey-theoretic ideas [B. M. Braga and I. Farah, Trans. Am. Math. Soc. 374, No. 2, 1007–1040 (2021; Zbl 1458.51007)] used in the study of uniform Roe algebras.
Reviewer: Alessandro Vignati (Toronto)
MSC:
| 46L57 | Derivations, dissipations and positive semigroups in \(C^*\)-algebras |
| 46L85 | Noncommutative topology |
Citations:
Zbl 0780.58043; Zbl 0853.58003; Zbl 0956.19004; Zbl 0432.46059; Zbl 0365.46051; Zbl 0139.30601; Zbl 1080.46039; Zbl 1481.46076; Zbl 1458.51007References:
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