Amini, Massoud Module Connes amenability of hypergroup measure algebras. (English) Zbl 1348.43002 Open Math. 13, 737-756 (2015). In this paper, the author firstly gives general facts on dual Banach algebras and Banach modules and studies module diagonals. Let \(K_{0}\) be a closed subhypergroup of a locally compact measured hypergroup \(K\). He shows that \(M(K)\) is a dual Banach algebra and a dual Banach \(M(K_{0})\)-bimodule with compatible actions for each hypergroup \(K\). He finds formulas for the corresponding multiplication operator and its first and second conjugates. Also, he gives several examples of hypergroups. As a main theorem, he shows that if \(M(K)\) has a normal module virtual diagonal, then \(K\) is amenable. Reviewer: Ömer Gök (Istanbul) MSC: 43A10 Measure algebras on groups, semigroups, etc. 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 43A62 Harmonic analysis on hypergroups Keywords:hypergroup; normal module virtual diagonal; separate continuity; dual algebra PDFBibTeX XMLCite \textit{M. Amini}, Open Math. 13, 737--756 (2015; Zbl 1348.43002) Full Text: DOI References: [1] [1] Runde, V., Amenability for dual Banach algebras, Studia Math., 2001, 148, 47-66.; · Zbl 1003.46028 [2] Runde, V., Connes-amenability and normal, virtual diagonals for measure algebras, J. London Math. Soc., 2003, 67, 643-656.; · Zbl 1040.22002 [3] Runde, V., Connes-amenability and normal, virtual diagonals for measure algebras II, Bull. Austral. Math. Soc., 2003, 68, 325-328.; · Zbl 1042.22001 [4] Runde, V., Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand., 2004, 95, 124-144.; · Zbl 1087.46035 [5] Johnson, B.E., Kadison, R.V., Ringrose, J., Cohomology of operator algebras III, Bull. Soc. Math. France, 1972, 100, 73-79.; · Zbl 0234.46066 [6] Jewett, R.I., Spaces with an abstract convolution of measures, Advances in Math., 1975, 18, 1-110.; · Zbl 0325.42017 [7] Bloom, W.R., Heyer, H., Harmonic Analysis of Probability Measures on Hypergroups, Walter de Gruyter, Berlin, 1995.; · Zbl 0828.43005 [8] Amini, M., Module amenability for semigroup algebras, Semigroup Forum, 2004, 69, 243-254.; · Zbl 1059.43001 [9] M. A. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Func. Anal., 1967, 1, 443-491.; · Zbl 0181.41303 [10] Daws, M., Dual Banach algebras: representations and injectivity, Studia Math., 2007, 178(3), 231-275.; · Zbl 1115.46038 [11] Ryan, R., Introduction to Tensor Products of Banach Spaces, Springer-Verlag, London, 2002.; · Zbl 1090.46001 [12] Corach, G., Galé, J. E., Averaging with virtual diagonals and geometry of representations. In: Banach algebras ’97, Walter de Grutyer, Berlin, 87-100, 1998.; · Zbl 0920.46038 [13] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 1951, 71, 152-182.; · Zbl 0043.37902 [14] T. H. Koornwinder, Alan L. Schwartz, Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle, Constr. Approx., 1997, 13, 537-567.; · Zbl 0937.33009 [15] Grothendieck, A., Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math., 1953, 74, 168-186.; · Zbl 0046.11702 [16] M. Skantharajah, Amenable hypergroups, Illinois J. Math., 1992, 36(1), 15-46.; · Zbl 0755.43003 [17] Johnson, B.E., Separate continuity and measurability, Proc. Amer. Math. Soc., 1969, 20, 420-422.; · Zbl 0181.14502 [18] Lasser, R., Amenability and weak amenability of ‘1-algebras of polynomial hypergroups, Studia Math., 2007, 182, 183-196.; · Zbl 1126.43003 [19] Lasser, R., Various amenability properties of the L1-algebra of polynomial hypergroups and applications, J. Comput. Appl. Math., 2009, 233, 786-792.; · Zbl 1182.43008 [20] Amini, M., Bodaghi, A., Ebrahimi Bagha, D., Module amenability of the second dual and module topological center of semigroup algebras, Semigroup Forum, 2010, 80, 302-312.; · Zbl 1200.43001 [21] Runde V., Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer-Verlag, Berlin, 2002.; · Zbl 0999.46022 [22] Doran, R.S., Wichman, J., Approximate Identities and Factorization in Banach Modules, Lecture Notes in Mathematics 768, Springer-Verlag, Berlin, 1979.; [23] Lasser, R., Orthogonal polynomials and hypergroups, Rend. Mat., 1983, 3, 185-209.; · Zbl 0538.33010 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.