Equivariant dimensions of graph C*-algebras
Introduction
Principal bundles are fundamental objects in algebraic topology, and their local triviality is pivotal for applications in physics. One can measure the local triviality of a given principal bundle using the Schwarz genus [36, Definition 5], which is the smallest number of open sets in a trivializing cover. The local-triviality dimension of [22, Definition 3.1] is then a noncommutative generalization of this invariant, which was inspired by the Rokhlin dimension [24, Definition 1.1] used in and around the classification program of unital, simple, separable, nuclear -algebras. In this manuscript, G always denotes a nontrivial compact group, though we note that many of these concepts apply to compact quantum groups as well.
Definition 1.1 [22, Definition 3.1] Let G act on a unital -algebra A, and let t denote the identity function in . The local-triviality dimension is the smallest n for which it is possible to define G-equivariant ⁎-homomorphisms for all in such a way that . If there is no finite n, the local-triviality dimension is ∞. ⧫
Above, G acts on by translation: . The local-triviality dimension is a measure of an action's complexity, and finiteness of the local-triviality dimension implies freeness of the action in the sense of Ellwood [19, Definition 2.4]. For actions of compact abelian groups, Ellwood's freeness condition is equivalent to freeness (or saturation) in the sense of Rieffel, as in [34, Definition 1.6] or [31, Definition 7.1.4].
The local-triviality dimension is useful for proving noncommutative generalizations of the Borsuk-Ulam theorem (see [22, §6]), namely, results which claim that G-equivariant unital ⁎-homomorphisms between certain unital -algebras do not exist. In particular, if G acts on A and B, and there is a G-equivariant unital ⁎-homomorphism from A to B, then . These pursuits revolve around the Type 1 noncommutative Borsuk-Ulam conjecture of [7, Conjecture 2.3], which is stated for coactions of compact quantum groups. While there is significant interest in the quantum case [14], [17], [22], here we will only consider actions of compact groups, so that the appropriate subcase of the conjecture is equivalent to the following.
Conjecture 1.2 Let G be a nontrivial compact group which acts freely on a unital -algebra A. Equip the join with the diagonal action of G. Then there is no equivariant, unital ⁎-homomorphism .
On the other hand, the local-triviality dimension (along with its variants discussed in Definition 2.3) also generalizes an earlier approach to noncommutative Borsuk-Ulam theory. In [39, Main Theorem], Taghavi considers actions of finite abelian groups on unital Banach algebras and places restrictions on the structure of individual elements of spectral subspaces. This culminates in questions such as [39, Question 3] for noncommutative spheres. From this point of view, one seeks to bound how many elements from a spectral subspace are needed to produce an invertible sum-square. In Section 3, we recast the local-triviality dimensions in terms of such computations.
Our results focus on graph -algebras [5], [33]. Many algebraic properties of a graph -algebra, such as simplicity, or classification of certain ideals, can be described purely in terms of the underlying directed graph. Further, every graph -algebra is equipped with a useful action of the circle, called the gauge action. Freeness of this action is again determined by simple conditions on the graph, as in [38, Proposition 2] or [15, Corollary 4.4], so our concern here is to bound the local-triviality dimensions of the gauge action.
The gauge action and its restrictions to finite cyclic subgroups give many examples of locally trivial noncommutative principal bundles. By studying these bundles, we point out phenomena not found in the commutative case, such as Example 4.3. Namely, the tensor product of non-free -actions may be free. Graph -algebras also give a natural framework for element-based noncommutative Borsuk-Ulam theory, as the Vaksman–Soibelman quantum spheres of [41] admit a graph presentation from [25, Proposition 5.1]. The gauge action provides an answer to Taghavi's [39, Question 3], which we examine in Proposition 5.16. These claims follow from a general study of the local-triviality dimension of the gauge action, which we similarly apply to other familiar graph -algebras.
The paper is organized as follows. In Section 2, we recall basic facts on graph -algebras and the local-triviality dimensions. Next, in Section 3, we recast the local-triviality dimensions for actions of or in terms of elements in the spectral subspaces. From this, we generate some bounds on the local-triviality dimensions of the gauge action restricted to , phrased in terms of the adjacency matrix, which we follow with a brief discussion of tensor products and actions. In Section 4, we show that both the local-triviality dimension and its stronger version can only take the value 0 or ∞ for the gauge -action on a -algebra of a finite acyclic graph. For the same actions, however, finiteness of the weak local-triviality dimension is equivalent to freeness. Section 5 contains various examples for which we can obtain additional local-triviality dimension estimates, such as the Cuntz algebras, the Toeplitz algebra, the graph of an n-cycle, and quantum spheres realized as graph -algebras.
Section snippets
Graph algebras and the local-triviality dimensions
We will consider directed graphs , where is a countable set of vertices, is a countable set of edges, and are the range and source maps, respectively. The adjacency matrix , defined by has entries in . In particular, loops and distinct edges with the same source and range are allowed.
Definition 2.1 The graph -algebra is the universal -algebra generated by elements for all and elements for all , subject to
The local-triviality dimensions and spectral subspaces
An action α of a compact abelian group G on a unital -algebra A induces a grading of A by spectral subspaces , which are given by for characters . The action α is free if and only if for each λ [31, Theorem 7.1.15]. A special case of this is the translation action of G on , . If is identified with in the natural way, then any character belongs to its own spectral subspace .
The local-triviality dimensions we
Finite acyclic graphs
In this section, we consider any graph E which is finite and acyclic. By [15, Proposition 4.3], the gauge -action on is free if and only if for every sink v of E, there is a path of length which ends at v. In this case, we seek to compute the local-triviality dimensions using graph properties.
If E is finite and acyclic, then is finite-dimensional, so it decomposes as a direct sum of matrix algebras. These summands are indexed by the sinks of E. However, we caution the reader
Cuntz algebras
The Cuntz algebra is defined as , which corresponds to a graph with a single vertex and n loops (see Fig. 3). We will regard the (free) gauge action α on as a -grading, with degree d component given by .
The -subalgebra is the UHF algebra of type , the direct limit of inclusions where each inclusion is diagonal and unital (see e.g. [16, §1]). As in Proposition 3.1, the local-triviality dimensions are
Acknowledgements
This work is part of the project Quantum Dynamics supported by EU-grant RISE 691246 and Polish Government grant 317281. A.C. was partially supported by NSF grants DMS-1801011 and DMS-2001128. M.T. was partially supported by the project Diamentowy Grant No. DI2015 006945 financed by the Polish Ministry of Science and Higher Education. We are grateful to the referee for helpful comments.
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