Multipliers for Gelfand pairs. (English) Zbl 1482.43002
Let \(G\) be a locally compact group and \(K\) be a compact subgroup of \(G\). A function \(f: G\rightarrow\mathbb{C}\) is said to be \(K\)-bi-invariant if \( f (k_1xk_2)= f(x)\) for all \(x\in G\) and \(k_1, k_2\in K\). Let \(L^1(G/\!/K)\) be the set of integrable \(K\)-bi-invariant functions. The space \(L^1(G/\!/K)\) is a Banach algebra with the convolution product and \(\|.\|_1\).
In this paper, the authors study the multipliers and the double multipliers for the convolution Banach algebra \(L^1(G/\!/K)\) and give a characterization of the multipliers on the commutative Banach algebra \(L^1(G/\!/K)\).
In this paper, the authors study the multipliers and the double multipliers for the convolution Banach algebra \(L^1(G/\!/K)\) and give a characterization of the multipliers on the commutative Banach algebra \(L^1(G/\!/K)\).
Reviewer: M. J. Mehdipour (Shiraz)
MSC:
| 43A22 | Homomorphisms and multipliers of function spaces on groups, semigroups, etc. |
| 43A20 | \(L^1\)-algebras on groups, semigroups, etc. |
References:
| [1] | F. T. Birtel, Isomorphisms and isometric multipliers,Proc. Amer. Math. Soc., 13(2), (1962) 204-210. · Zbl 0117.09701 |
| [2] | R. Bloom, H. Heyer,Harmonic analysis of probability measures on hypergroups, de Gruyter Stud. Math., 20, Walter de Gruyter, Berlin and Hawthorne, (1995). · Zbl 0828.43005 |
| [3] | S. N. Degenfeld-Schonburg,Multipliers for hypergroups:concrete examples, application to time series, PhD thesis, Technische Universität München, München (2012). Available at https://dnb.info/1031513728/34 |
| [4] | A. Deitmar and S. Echterhoff,Principles of harmonic analysis, Springer International Publishing, 2nd Ed. (2014). · Zbl 1300.43001 |
| [5] | J. Dieudonné, Gelfand pairs and spherical functions,Internat. J. Math. and Math. Sci., 2(2), (1979) 155162. · Zbl 0441.33015 |
| [6] | G. van Dijk,Introduction to harmonic analysis and generalized Gelfand pairs,De Gruyter Studies in Mathematics 36, Berlin, Germany (2009). · Zbl 1184.43001 |
| [7] | R. E. Edwards, Operators commuting with translations,Pacific J. Math., 15, (1965) 85-95. · Zbl 0141.12504 |
| [8] | J. Faraut,Analyse harmonique sur les paires de Gelfand et les espaces hyperboliques, Les Cours du CIMPA, (1980). |
| [9] | G. B. Folland,A course in abstract harmonic analysis, CRC Press, (1995). · Zbl 0857.43001 |
| [10] | S. Helgason,Multipliers of Banach algebras, Annals of Mathematics, Second Series, 64, (1956) 240-256. · Zbl 0072.32303 |
| [11] | I. N. Herstein,Rings with involution, The University of Chicago Press, Chicago (1976). · Zbl 0343.16011 |
| [12] | K. Kangni,Analyse de Fourier non-commutative, Edilivre (2018). |
| [13] | E. Kaniuth,A course in commutative Banach algebras, Springer-Verlag, (2009). · Zbl 1190.46001 |
| [14] | V. Kumar, R. Sarma, N. S. Kumar, Characterisation of multipliers on hypergroups, to appear inActa Math. Vietnamica. · Zbl 1441.43008 |
| [15] | N. P. Landsman,C∗-algebras, HilbertC∗-modules and Quantum mechanics, arXiv:math-ph/9807030v1 24 Jul (1998). |
| [16] | R. Larsen,An introduction to the theory of multipliers, Springer-Verlag, (1971). · Zbl 0213.13301 |
| [17] | E. Michael, Topologies on spaces of subsets,Trans. Amer. Math. Soc., 71, (1951), 152-182. · Zbl 0043.37902 |
| [18] | A. Riazi, and M. Adib,ϕ-Multipliers on Banach algebras without order,Int. Journal of Math. Analysis, 3(3), (2009) 121-132. · Zbl 1180.47026 |
| [19] | R. Spector, Mesures invariantes sur les hypergroupes,Trans. Amer. Math. Soc., 239 (1978) 147-165. · Zbl 0428.43001 |
| [20] | J. Wang , Multipliers of commutative Banach algebras,Pacific J. Math., 11, (1961) 1131-1149. · Zbl 0127.33302 |
| [21] | J. G. Wendel, On isometric isomorphism of group algebras,Pacific J. Math., 1, (1951) 305-311. · Zbl 0043.03102 |
| [22] | J. G. Wendel, Left centralizers and isomorphisms of group algebras,Pacific J. Math., 2, (1952) 251-261. · Zbl 0049.35702 |
| [23] | H. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
