Transference and preservation of uniqueness

Todorov, Ivan G.; Turowska, Lyudmila

doi:10.1007/s11856-018-1817-7

Cited by 7 publications

(11 citation statements)

References 20 publications

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“…By Theorem 2.8,

Bim (I^{⊥})

contains a non‐zero compact operator. Thus, again by [17, Corollary 1.5 ],

I

is an ideal of multiplicity.

□

…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 94%

“…A closed set

E \subseteq H

is called an

M

‐set (respectively, an

M_{1}

‐set) if the ideal

J_{H} false(E false)

(respectively,

I_{H} false(E false)

) is an ideal of multiplicity. Corollaries 2.7(i) and 2.9 together with [17, Corollary 3.6] imply the following: Corollary If

E \subseteq H

is a closed set such that

θ^{- 1} false(E false)

is an

M

‐set (respectively, an

M_{1}

‐set), then

E

is an

M

‐set (respectively, an

M_{1}

‐set).…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 97%

“…For the following theorem, we recall from [17] that a closed ideal

J \subseteq A (G)

is an ideal of multiplicity if

J^{⊥} \cap C_{r}^{*} false(G false) \neq false{0 false}

, where

C_{r}^{*} false(G false)

is the reduced

C^{*}

‐algebra of

G

. Theorem Let

I

be a closed ideal of

A (H)

.…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 99%

“…Proof By [17, Corollary 1.5 ], if

J

is an ideal of multiplicity, then

Bim (J^{⊥})

contains a non‐zero compact operator. By Theorem 2.8,

Bim (I^{⊥})

contains a non‐zero compact operator.…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 99%

“…In [13], Shulman, Todorov and Turowska proved that if

G

is a locally compact second countable group and

E \subseteq G

, then

E

is an

M

‐set (respectively,

M_{1}

‐set) if and only if

E^{*}

is an

M

‐set (respectively,

M_{1}

‐set). Subsequently, Todorov and Turowska in [17] proved that an ideal

J \subseteq A (G)

is an ideal of multiplicity if and only if

Bim (J^{⊥})

is an operator space of multiplicity.…”

Section: Introductionmentioning

confidence: 99%

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On synthetic and transference properties of group homomorphisms

Eleftherakis

2020

Bull. London Math. Soc.

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We study Borel homomorphisms θ:G→H for arbitrary locally compact second countable groups G and H for which the measure θ∗false(μfalse)false(αfalse)=μfalse(θ−1(α)false)forα⊆HaBorelsetis absolutely continuous with respect to ν, where μ (respectively, ν) is a Haar measure for G, (respectively, H). We define a natural mapping scriptG from the class of maximal abelian selfadjoint algebra bimodules (masa bimodules) in B(L2false(Hfalse)) into the class of masa bimodules in B(L2false(Gfalse)) and we use it to prove that if k⊆G×G is a set of operator synthesis, then (θ×θ)−1false(kfalse) is also a set of operator synthesis and if E⊆H is a set of local synthesis for the Fourier algebra A(H), then θ−1false(Efalse)⊆G is a set of local synthesis for A(G). We also prove that if θ−1false(Efalse) is an M‐set (respectively, M1‐set), then E is an M‐set (respectively, M1‐set) and if Bim(I⊥) is the masa bimodule generated by the annihilator of the ideal I in VN(G), then there exists an ideal J such that Gfalse(Bim(I⊥)false)=Bimfalse(J⊥false). If this ideal J is an ideal of multiplicity, then I is an ideal of multiplicity. In case θ∗false(μfalse) is a Haar measure for θ(G), we show that J is equal to the ideal ρ∗false(Ifalse) generated by ρ(I), where ρ(u)=u∘θ,∀u∈I.

show abstract

“…By Theorem 2.8,

Bim (I^{⊥})

contains a non‐zero compact operator. Thus, again by [17, Corollary 1.5 ],

I

is an ideal of multiplicity.

□

…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 94%

“…A closed set

E \subseteq H

is called an

M

‐set (respectively, an

M_{1}

‐set) if the ideal

J_{H} false(E false)

(respectively,

I_{H} false(E false)

) is an ideal of multiplicity. Corollaries 2.7(i) and 2.9 together with [17, Corollary 3.6] imply the following: Corollary If

E \subseteq H

is a closed set such that

θ^{- 1} false(E false)

is an

M

‐set (respectively, an

M_{1}

‐set), then

E

is an

M

‐set (respectively, an

M_{1}

‐set).…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 97%

“…For the following theorem, we recall from [17] that a closed ideal

J \subseteq A (G)

is an ideal of multiplicity if

J^{⊥} \cap C_{r}^{*} false(G false) \neq false{0 false}

, where

C_{r}^{*} false(G false)

is the reduced

C^{*}

‐algebra of

G

. Theorem Let

I

be a closed ideal of

A (H)

.…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 99%

“…Proof By [17, Corollary 1.5 ], if

J

is an ideal of multiplicity, then

Bim (J^{⊥})

contains a non‐zero compact operator. By Theorem 2.8,

Bim (I^{⊥})

contains a non‐zero compact operator.…”

Section: Synthetic and Transference Properties Of Group Homomorphismsmentioning

confidence: 99%

“…In [13], Shulman, Todorov and Turowska proved that if

G

is a locally compact second countable group and

E \subseteq G

, then

E

is an

M

‐set (respectively,

M_{1}

‐set) if and only if

E^{*}

is an

M

‐set (respectively,

M_{1}

‐set). Subsequently, Todorov and Turowska in [17] proved that an ideal

J \subseteq A (G)

is an ideal of multiplicity if and only if

Bim (J^{⊥})

is an operator space of multiplicity.…”

Section: Introductionmentioning

confidence: 99%

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On synthetic and transference properties of group homomorphisms

Eleftherakis

2020

Bull. London Math. Soc.

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Synthetic properties of locally compact groups: preservation and transference

2022

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Reduced spectral synthesis and compact operator synthesis

Shulman

Todorov

Turowska

2020

Advances in Mathematics

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We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory, studied previously. We show that a closed subset E of a second countable locally compact group G satisfies reduced local spectral synthesis if and only if the subset E * = {(s, t) : ts −1 ∈ E} of G×G satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten p-classes.

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Transference and preservation of uniqueness

Cited by 7 publications

References 20 publications

On synthetic and transference properties of group homomorphisms

On synthetic and transference properties of group homomorphisms

Synthetic properties of locally compact groups: preservation and transference

Reduced spectral synthesis and compact operator synthesis

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