Descriptive set-theoretic aspects of closed sets of uniqueness in the non-abelian setting. (English) Zbl 1501.43005
Summary: We study the family of closed sets of (extended) uniqueness of a locally compact group \(G\) which is not necessarily abelian. We prove some preservation properties concerning this family of sets as well as their operator-theoretic counterpart, locate their descriptive complexity and establish sufficient conditions for the non-existence of a Borel basis.
MSC:
| 43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |
| 03E15 | Descriptive set theory |
References:
| [1] | W. B. Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433-532. · Zbl 0334.46070 |
| [2] | N. Bari, Sur l’unicité du développement trigonométrique, Fund. Math. 9 (1927), 62-115. · JFM 53.0261.01 |
| [3] | B. Bekka and P. Harpe, Unitary Representations of Groups, Duals, and Characters, Math. Surveys Monogr. 250, Amer. Math. Soc., 2020. · Zbl 1475.22011 |
| [4] | M. Blümlinger, Characterization of measures in the group C * -algebra of a locally compact group, Math. Ann. 289 (1991), 393-402. · Zbl 0702.43003 |
| [5] | M. Bożejko, Sets of uniqueness on noncommutative locally compact groups, Proc. Amer. Soc. 64 (1977), 93-96. · Zbl 0364.43007 |
| [6] | G. Debs et J. Saint-Raymond, Ensembles boréliens d’unicité et d’unicité au sens large, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 3, 217-239. · Zbl 0618.42004 |
| [7] | C. Dunkl and D. Ramirez, Translation in measure algebras and the correspondence to Fourier transforms vanishing at infinity, Michigan Math. J. 17 (1970), 311-319. · Zbl 0188.20601 |
| [8] | G. Eleftherakis, Bilattices and Morita equivalence of masa bimodules, Proc. Edin-burgh Math. Soc. 59 (2016), 605-621. · Zbl 06709051 |
| [9] | P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. · Zbl 0169.46403 |
| [10] | E. Kaniuth and A. Lau, Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups, Math. Surveys Monogr. 231, Amer. Math. Soc., 2018. · Zbl 1402.43001 |
| [11] | R. Kaufman, M-sets and distributions, Astérisque 5 (1973), 225-230. · Zbl 0281.43006 |
| [12] | A. Kechris, Classical Descriptive Set Theory, Grad. Texts in Math. 156, Springer, 1995. · Zbl 0819.04002 |
| [13] | A. Kechris, Set theory and uniqueness for trigonometric series, preprint, Caltech, 2007. |
| [14] | A. Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, 1987. · Zbl 0642.42014 |
| [15] | É. Matheron, Sigma-idéaux polaires et ensembles d’unicité dans les groupes abéliens localement compacts, Ann. Inst. Fourier (Grenoble) 46 (1996), 493-533. · Zbl 0854.43006 |
| [16] | É. Matheron and M. Zelený, Descriptive set theory of families of small sets, Bull. Symbolic Logic 13 (2007), 482-537. · Zbl 1152.03042 |
| [17] | S. Saeki, Helson sets which disobey spectral synthesis, Proc. Amer. Math. Soc. 47 (1975), 371-377. · Zbl 0297.43007 |
| [18] | V. Shulman, I. Todorov and L. Turowska, Operator synthesis I. Synthetic sets, bilat-tices and tensor algebras, J. Funct. Anal. 209 (2004), 293-331. · Zbl 1071.47066 |
| [19] | V. Shulman, I. Todorov and L. Turowska, Sets of multiplicity and closable multipliers on group algebras, J. Funct. Anal. 268 (2015), 1454-1508. · Zbl 1330.46055 |
| [20] | V. Shulman, I. Todorov and L. Turowska, Reduced spectral synthesis and compact operator synthesis, Adv. Math. 367 (2020), art. 107109, 49 pp. · Zbl 1450.43003 |
| [21] | V. Tardivel, Fermés d’unicité dans les groupes abéliens localement compacts, Studia Math. 91 (1988), 1-15. · Zbl 0666.43002 |
| [22] | A. Ülger, Relatively weak * -closed ideals of A(G), sets of synthesis and sets of unique-ness, Colloq. Math. 136 (2014), 271-296. · Zbl 1306.43005 |
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