A metric approach to limit operators

Spakula, Jan; Willett, Rufus

doi:10.1090/tran/6660

Cited by 39 publications

(106 citation statements)

References 20 publications

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“…Thus to prove both of these implications it will suffice to exhibit a quotient of

C_{u}^{*} false(X false)

that contains a proper isometry whenever

X

has positive asymptotic dimension. We will need some machinery from [, Sections 3 and 4]; we refer the reader to that paper for notation and terminology. For compatibility with that paper, note that we may without loss of generality assume that the metric on

X

is integer‐valued by replacing it with

⌈ d ⌉

(where

⌈ \cdot ⌉

is the ceiling function); indeed, this does not affect

C_{u}^{*} false(X false)

, or the asymptotic dimension of

X

.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

“…For each

k ⩾ 2

, let

Y_{k} = false{x_{1}^{false(n false)} ∣ m_{n} ⩾ k false}

; note that this is a cofinite subset of

Y

, and thus

ω (Y_{k}) = 1

for all

k

. Define a partial translation (in the sense of [, Definition 3.1])

t_{k} : Y_{k} \to X

t_{k} false(x_{1}^{false(n false)} false) = x_{k}^{(n)}

. These partial translations are compatible with

ω

in the sense of [, Definition 3.2], so we may define

ω_{k} = t_{k} false(ω false)

.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

“…Define a partial translation (in the sense of [, Definition 3.1])

t_{k} : Y_{k} \to X

t_{k} false(x_{1}^{false(n false)} false) = x_{k}^{(n)}

. These partial translations are compatible with

ω

in the sense of [, Definition 3.2], so we may define

ω_{k} = t_{k} false(ω false)

. Note that if

d_{ω}

is as in [, Proposition 3.7], then

d_{ω} false(ω, ω_{k} false) \in N \cap false[k r, (k + 1) r false)

for each

k

, and in particular, the subset

S = {ω_{k} ∣ k \in N}

of the limit space

X (ω)

of [, Definition 3.10] is a coarsely embedded copy of

double-struckN

.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

“…These partial translations are compatible with

ω

in the sense of [, Definition 3.2], so we may define

ω_{k} = t_{k} false(ω false)

. Note that if

d_{ω}

is as in [, Proposition 3.7], then

d_{ω} false(ω, ω_{k} false) \in N \cap false[k r, (k + 1) r false)

for each

k

, and in particular, the subset

S = {ω_{k} ∣ k \in N}

of the limit space

X (ω)

of [, Definition 3.10] is a coarsely embedded copy of

double-struckN

. Now, define an operator

v

ℓ^{2} false(X false)

v : δ_{x} = \{\begin{matrix} left δ_{x_{k + 1}^{(n)}}, & left x = x_{k}^{false(n false)} for some n and k < m_{n} \\ left 0, & left x = x_{m_{n}}^{false(n false)} for some n \\ left δ_{x}, & left otherwise \end{matrix};

in words

v

shifts right by ‘one unit’ along each

S_{n}

, and acts as the identity elsewhere.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

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Low‐dimensional properties of uniform Roe algebras

Willett

2018

J. London Math. Soc.

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Abstract. The goal of this paper is to study when uniform Roe algebras have certain C * -algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot open.We also establish results about K-theory, showing that all K0-classes come from the inclusion of the canonical Cartan in low dimensional cases, but not in general; in particular, our K-theoretic results answer a question of Elliott and Sierakowski about vanishing of K0 groups for uniform Roe algebras of non-amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our K-theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces. Mathematics Subject Classification (2010): 20F69, 46L05, 46L80, 55M10

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“…Thus to prove both of these implications it will suffice to exhibit a quotient of

C_{u}^{*} false(X false)

that contains a proper isometry whenever

X

X

is integer‐valued by replacing it with

⌈ d ⌉

(where

⌈ \cdot ⌉

is the ceiling function); indeed, this does not affect

C_{u}^{*} false(X false)

, or the asymptotic dimension of

X

.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

“…For each

k ⩾ 2

, let

Y_{k} = false{x_{1}^{false(n false)} ∣ m_{n} ⩾ k false}

; note that this is a cofinite subset of

Y

, and thus

ω (Y_{k}) = 1

for all

k

. Define a partial translation (in the sense of [, Definition 3.1])

t_{k} : Y_{k} \to X

t_{k} false(x_{1}^{false(n false)} false) = x_{k}^{(n)}

. These partial translations are compatible with

ω

in the sense of [, Definition 3.2], so we may define

ω_{k} = t_{k} false(ω false)

.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

“…Define a partial translation (in the sense of [, Definition 3.1])

t_{k} : Y_{k} \to X

t_{k} false(x_{1}^{false(n false)} false) = x_{k}^{(n)}

. These partial translations are compatible with

ω

in the sense of [, Definition 3.2], so we may define

ω_{k} = t_{k} false(ω false)

. Note that if

d_{ω}

is as in [, Proposition 3.7], then

d_{ω} false(ω, ω_{k} false) \in N \cap false[k r, (k + 1) r false)

for each

k

, and in particular, the subset

S = {ω_{k} ∣ k \in N}

of the limit space

X (ω)

of [, Definition 3.10] is a coarsely embedded copy of

double-struckN

.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

“…These partial translations are compatible with

ω

in the sense of [, Definition 3.2], so we may define

ω_{k} = t_{k} false(ω false)

. Note that if

d_{ω}

is as in [, Proposition 3.7], then

d_{ω} false(ω, ω_{k} false) \in N \cap false[k r, (k + 1) r false)

for each

k

, and in particular, the subset

S = {ω_{k} ∣ k \in N}

of the limit space

X (ω)

of [, Definition 3.10] is a coarsely embedded copy of

double-struckN

. Now, define an operator

v

ℓ^{2} false(X false)

v : δ_{x} = \{\begin{matrix} left δ_{x_{k + 1}^{(n)}}, & left x = x_{k}^{false(n false)} for some n and k < m_{n} \\ left 0, & left x = x_{m_{n}}^{false(n false)} for some n \\ left δ_{x}, & left otherwise \end{matrix};

in words

v

shifts right by ‘one unit’ along each

S_{n}

, and acts as the identity elsewhere.…”

Section: Consequences Of Asymptotic Dimension Zeromentioning

confidence: 99%

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Low‐dimensional properties of uniform Roe algebras

Willett

2018

J. London Math. Soc.

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show abstract

“…The

ℓ^{p}

uniform Roe algebra belongs to a class of algebras that we may call

ℓ^{p}

Roe‐type algebras (see Definition ). Such algebras were considered in in connection with criteria for Fredholmness. Other examples of

ℓ^{p}

Roe‐type algebras that may be of particular interest are the following.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of ℓp Roe-type algebras

Chung

2018

Bull. London Math. Soc.

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We investigate the rigidity of the ℓp analog of Roe‐type algebras. In particular, we show that if p∈[1,∞)∖{2}, then an isometric isomorphism between the ℓp uniform Roe algebras of two metric spaces with bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other ℓp Roe‐type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.

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Band‐dominated operators on metric measure spaces

Seifert

2017

Proc Appl Math and Mech

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A metric approach to limit operators

Cited by 39 publications

References 20 publications

Low‐dimensional properties of uniform Roe algebras

Low‐dimensional properties of uniform Roe algebras

Rigidity of ℓp Roe-type algebras

Band‐dominated operators on metric measure spaces

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