Completely positive definite functions and Bochner's theorem for locally compact quantum groups

被引:13
作者
Daws, Matthew [1 ]
Salmi, Pekka [2 ]
机构
[1] Univ Leeds, Sch Math, Leeds LS2 9JT, W Yorkshire, England
[2] Univ Oulu, Dept Math Sci, FI-90014 Oulu, Finland
基金
英国工程与自然科学研究理事会;
关键词
Quantum group; Positive definite function; Bochner's theorem; UNIFORM CONTINUITY; PROPERTY T; ALGEBRAS; MULTIPLIERS;
D O I
10.1016/j.jfa.2013.01.017
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We prove two versions of Bochner's theorem for locally compact quantum groups. First, every completely positive definite "function" on a locally compact quantum group G arises as a transform of a positive functional on the universal C*-algebra C-0(ll) ((G) over cap) of the dual quantum group. Second, when G is coamenable, complete positive definiteness may be replaced with the weaker notion of positive definiteness, which models the classical notion. A counterexample is given to show that the latter result is not true in general. To prove these results, we show two auxiliary results of independent interest: products are linearly dense in L-#(1)(G) and when G is coamenable, the Banach *-algebra L-#(1)(G) has a contractive bounded approximate identity. (c) 2013 Elsevier Inc. All rights reserved.
引用
收藏
页码:1525 / 1546
页数:22
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共 47 条
[41]   Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1, 1) [J].
Martijn Caspers .
Communications in Mathematical Physics, 2014, 331 :1041-1069
[42]   Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1,1) [J].
Caspers, Martijn .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2014, 331 (03) :1041-1069
[43]   Stinespring's theorem for unbounded operator valued local completely positive maps and its applications [J].
Bhat, B. V. Rajarama ;
Ghatak, Anindya ;
Pamula, Santhosh Kumar .
INDAGATIONES MATHEMATICAE-NEW SERIES, 2021, 32 (02) :547-578
[44]   Twisting and Rieffel’s Deformation of Locally Compact Quantum GroupsDeformation of the Haar Measure [J].
Pierre Fima ;
Leonid Vainerman .
Communications in Mathematical Physics, 2009, 286 :1011-1050
[45]   Actions of locally compact (quantum) groups on ternary rings of operators, their crossed products, and generalized Poisson boundaries [J].
Salmi, Pekka ;
Skalski, Adam .
KYOTO JOURNAL OF MATHEMATICS, 2017, 57 (03) :667-691
[46]   Another proof of Joseph and Letzter's separation of variables theorem for quantum groups [J].
P. Baumann .
Transformation Groups, 2000, 5 :3-20
[47]   Krieger's type for ergodic non-singular Poisson actions of non-(T) locally compact groups [J].
Danilenko, Alexandre, I .
ERGODIC THEORY AND DYNAMICAL SYSTEMS, 2023, 43 (07) :2317-2353