Harmonic operators: the dual perspective

Neufang, Matthias; Runde, Volker

doi:10.1007/s00209-006-0039-6

Cited by 14 publications

(30 citation statements)

References 26 publications

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“…The lemma below shows that we may regard each of (T (L 2 (G), ) and (T (L 2 (G), )) as a lifting of L 1 (G). This phenomenon has been observed in [39][40][41] for the case where G is commutative or co-commutative.…”

Section: Proposition 51 Let G Be a Locally Compact Quantum Group Tsupporting

confidence: 54%

“…We remark that in the setting of locally compact groups G, the above type of multiplicative structure on T (L 2 (G)) has been studied in [1,[39][40][41][42]. As defined in Section 3, the *-antiautomorphism R : B(L 2 (G)) → B(L 2 (G)) extends the unitary antipode R of G, mapping L ∞ (G) and L ∞ (Ĝ) onto L ∞ (G) and L ∞ (Ĝ ), respectively.…”

Section: Luc(g) and Ruc(g)mentioning

confidence: 99%

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Completely bounded multipliers over locally compact quantum groups

Neufang

Ruan

2011

Proceedings of the London Mathematical Society

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In this paper, we consider several interesting multiplier algebras associated with a locally compact quantum group G. Firstly, we study the completely bounded right multiplier algebra M r cb (L1(G)). We show that M r cb (L1(G)) is a dual Banach algebra with a natural operator predual Q r cb (L1(G)), and the completely isometric representation of M r cb (L1(G)) on B(L2(G)), studied recently by Junge, Neufang and Ruan, is actually weak*-weak* continuous if the quantum group G has the right co-approximation property. Secondly, we study the space LUC(G) of left uniformly continuous functionals on L1(G) and its Banach algebra dual LUC(G) * . We prove that LUC(G) is a unital C*-subalgebra of L∞(G) if the quantum group G is semi-regular. We show the connection between LUC(G) * and the quantum measure algebra M (G), as well as their representations on L∞ (G) and B(L2(G)). Finally, we study the right uniformly continuous complete quotient space UCQ r (G) and its Banach algebra dual UCQ r (G) * . For quantum groups G with the right coapproximation property, we establish a completely contractive injection Q r cb (L1(G)) → UCQ r (G) which is compatible with the relation C0(G) ⊆ LUC(G). For co-amenable quantum groups G, we obtain the weak*-weak* homeomorphic and completely isometric algebra isomorphism M r cb (L1(G)) ∼ = M (G) and the completely isometric isomorphism UCQ r (G) ∼ = LUC(G).

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Section: Proposition 51 Let G Be a Locally Compact Quantum Group Tsupporting

confidence: 54%

Section: Luc(g) and Ruc(g)mentioning

confidence: 99%

Completely bounded multipliers over locally compact quantum groups

Neufang

Ruan

2011

Proceedings of the London Mathematical Society

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“…We next extend the notions of σ-harmonic functionals [4] and operators [16] to jointly harmonic functionals and operators: Definition 2.11. Let Σ ⊆ M cb A(G).…”

Section: Proof By Theorem 28ñ(j) = Bim(n(j)) and Thus By Lemma 2mentioning

confidence: 99%

“…Notice that supp G T coincides with the zero set of the ideal I T (see also [16,Proposition 3.3]). More generally, let us define the G-support of a subset A of B(L 2 (G)) by supp G (A) = Z(I A ).…”

Section: It Is Easy To Verify That I a Is A Closed Ideal Of A(g)mentioning

confidence: 99%

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Ideals of the Fourier algebra, supports and harmonic operators

Anoussis¹,

Katavolos²,

Todorov³

2016

Math. Proc. Camb. Phil. Soc.

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Abstract. We examine the common null spaces of families of HerzSchur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in [1] can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.

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