Equivariant dimensions of graph C*-algebras

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Abstract

We explore the recently introduced local-triviality dimensions by studying gauge actions on graph C-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For C-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph C-algebras. These include the Toeplitz algebra, Cuntz algebras, and q-deformed spheres.

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 C-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 C-algebra A, and let t denote the identity function in C0((0,1]). The local-triviality dimension dimLTG(A) is the smallest n for which it is possible to define G-equivariant ⁎-homomorphisms ρi:C0((0,1])C(G)A for all 0in in such a way that i=0nρi(t1)=1. If there is no finite n, the local-triviality dimension is ∞.  

Above, G acts on C(G) by translation: (αgf)(h)=f(hg). 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 C-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 dimLTG(A)dimLTG(B). 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 C-algebra A. Equip the joinAC(G)={fC([0,1],AC(G)):f(0)CC(G),f(1)AC} with the diagonal action of G. Then there is no equivariant, unital-homomorphism ϕ:AAC(G).

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 C-algebras [5], [33]. Many algebraic properties of a graph C-algebra, such as simplicity, or classification of certain ideals, can be described purely in terms of the underlying directed graph. Further, every graph C-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 Z/3-actions may be free. Graph C-algebras also give a natural framework for element-based noncommutative Borsuk-Ulam theory, as the Vaksman–Soibelman quantum spheres C(Sq2n1) 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 C-algebras.

The paper is organized as follows. In Section 2, we recall basic facts on graph C-algebras and the local-triviality dimensions. Next, in Section 3, we recast the local-triviality dimensions for actions of Z/k or S1 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 Z/2, phrased in terms of the adjacency matrix, which we follow with a brief discussion of tensor products and Z/k 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 Z/k-action on a C-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 C-algebras.

Section snippets

Graph algebras and the local-triviality dimensions

We will consider directed graphs E=(E0,E1,r,s), where E0 is a countable set of vertices, E1 is a countable set of edges, and r,s:E1E0 are the range and source maps, respectively. The adjacency matrix AE, defined by(AE)vw=# edges with source v and range w, has entries in Z+{}. In particular, loops and distinct edges with the same source and range are allowed.

Definition 2.1

The graph C-algebra C(E) is the universal C-algebra generated by elements Pv for all vE0 and elements Se for all eE1, subject to

The local-triviality dimensions and spectral subspaces

An action α of a compact abelian group G on a unital C-algebra A induces a grading of A by spectral subspaces Aλ, which are given byAλ={aA:for all gG,αg(a)=λ(g)a} for characters λGˆ. The action α is free if and only if 1AλAλ for each λ [31, Theorem 7.1.15]. A special case of this is the translation action of G on C(G), (αgf)(h)=f(hg). If C(G) is identified with C(Gˆ) in the natural way, then any character λGˆ belongs to its own spectral subspace C(G)λ.

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 Z/k-action on C(E) is free if and only if for every sink v of E, there is a path of length k1 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 C(E) 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 On is defined as C(S1,,Sn|SiSi=1,iSiSi=1), which corresponds to a graph with a single vertex and n loops (see Fig. 3). We will regard the (free) gauge action α on On as a Z-grading, with degree d component given by On(d):={xOn|αz(x)=zdx,zS1}.

The C-subalgebra F=On(0)On is the UHF algebra of type n, the direct limit of inclusionsMnkMnk+1, 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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