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$ C\sp{\ast} $-algebras of multivariable Wiener-Hopf operators


Authors: Paul S. Muhly and Jean N. Renault
Journal: Trans. Amer. Math. Soc. 274 (1982), 1-44
MSC: Primary 46L05; Secondary 45E10, 47B35
DOI: https://doi.org/10.1090/S0002-9947-1982-0670916-3
MathSciNet review: 670916
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Abstract: The $ {C^ \ast }$-algebra $ \mathfrak{W}$ generated by the Wiener-Hopf operators defined over a subsemigroup of a locally compact group is shown to be the image of a groupoid $ {C^ \ast }$-algebra under a suitable representation. When the subsemigroup is either a polyhedral cone or a homogeneous, self-dual cone in an Euclidean space, this representation may be used to show that $ \mathfrak{W}$ is postliminal and to find a composition series with very explicit subquotients. This yields a concrete parameterization of the spectrum of $ \mathfrak{W}$ and exhibits the topology on it.


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  • [1] E. Alfsen, F. Shultz and E. Størmer, A Gelfand-Naimark theorem for Jordan algebras, Adv. in Math. 28 (1978), 11-56. MR 0482210 (58:2292)
  • [2] J. M. Ash, ed., Studies in harmonic analysis, Studies in Math., vol. 13, Math. Assoc. Amer., Washington, D. C., 1976. MR 0442565 (56:946)
  • [3] C. Berger and L. Coburn, Wiener-Hopf operators on $ {U_2}$, Integral Equations Operator Theory 2 (1979), 139-173. MR 543881 (81c:47031)
  • [4] C. Berger, L. Coburn and A. Koranyi, Opérateurs de Wiener-Hopf sur les spheres de Lie, preprint. MR 584284 (81g:22018)
  • [5] A. Berman, Cones, matrices and mathematical programming, Lecture Notes in Econ. and Math. Systems, vol. 79, Springer-Verlag, New York, 1973. MR 0363463 (50:15901)
  • [6] H. Braun and M. Koecher, Jordan-algebren, Springer-Verlag, New York, 1966. MR 0204470 (34:4310)
  • [7] L. Coburn, The $ {C^ \ast }$-algebra generated by an isometry. II, Trans. Amer. Math. Soc. 137 (1969), 211-217. MR 0236720 (38:5015)
  • [8] L. Coburn and R. G. Douglas, On $ {C^ \ast }$-algebras of operators on a half-space. I, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 59-67. MR 0358417 (50:10883)
  • [9] L. Coburn, R. G. Douglas, D. Schaeffer and I. Singer, On $ {C^ \ast }$-algebras of operators on a half-space. II. Index theory, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 69-79. MR 0358418 (50:10884)
  • [10] J. Dixmier, Algèbres quasi-unitaires, Comment. Math. Helv. 26 (1952), 275-322. MR 0052697 (14:660b)
  • [11] -, $ {C^ \ast }$-algebras, North-Holland, Amsterdam and New York, 1977.
  • [12] R. G. Douglas, On the $ {C^ \ast }$-algebra of a one-parameter semi-group of isometries, Acta Math. 128 (1972), 143-151. MR 0394296 (52:15099)
  • [13] R. G. Douglas and R. Howe, On the $ {C^ \ast }$-algebra of Toeplitz operators on the quarter plane, Trans. Amer. Math. Soc. 158 (1971), 203-217. MR 0288591 (44:5787)
  • [14] A. Dynin, Inversion problem for singular integral operators: $ {C^ \ast }$-approach, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 4668-4670. MR 507929 (80c:46066)
  • [15] E. Effros and F. Hahn, Locally compact transformation groups and $ {C^ \ast }$-algebras, Mem. Amer. Math. Soc. No. 75 (1967). MR 0227310 (37:2895)
  • [16] K. Friedrichs, Pseudo-differential operators, Courant Institute of Mathematical Sciences, New York University, New York, 1970.
  • [17] L. Fuchs, Partially ordered algebraic systems, Addison-Wesley, Reading, Mass., 1963. MR 0171864 (30:2090)
  • [18] J. Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885-911. MR 0146297 (26:3819)
  • [19] L. Gol'denstein and I. Gohberg, On a multi-dimensional integral equation on a half-space whose kernel is a function of the difference of the arguments and a discrete analogue of this equation, Dokl. Akad. Nauk SSSR 131 (1960), 9-12 = Soviet Math. Dokl. 1 (1960), 173-176. MR 0117519 (22:8298)
  • [20] I. Gohberg and I. Feldman, Convolution equations and projection methods for their solution, Transl. Math. Monographs, No. 41, Amer. Math. Soc., Providence, R. I., 1974. MR 0355675 (50:8149)
  • [21] A. Guichardet, Tensor products of $ {C^ \ast }$-algebras. I, II, Aarhus Univ., 1969.
  • [22] P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math. 35 (1934), 29-64. MR 1503141
  • [23] V. Klee, Some characterizations of convex polyhedra, Acta Math. 102 (1959), 79-107. MR 0105651 (21:4390)
  • [24] M. Koecher, Jordan algebras and their applications, mimeographed notes, University of Minnesota, 1962.
  • [25] J. Renault, A groupoid approach to $ {C^ \ast }$-algebras, Lecture Notes in Math., vol. 793, Springer-Verlag, New York, 1980. MR 584266 (82h:46075)
  • [26] I. Simonenko, Operators of convolution type in cones, Mat. Sb. (N.S.) 74 (116) (1967), 298-313 = Math. USSR-Sb. 3 (1967), 279-294. MR 0222723 (36:5773)
  • [27] H. Takai, On a duality for crossed products of $ {C^ \ast }$-algebras, J. Funct. Anal. 19 (1975), 25-39. MR 0365160 (51:1413)
  • [28] M. Takesaki, Covariant representations of $ {C^ \ast }$-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273-303. MR 0225179 (37:774)
  • [29] D. Topping, An isomorphism invariant for spin factors, J. Math. Mech. 15 (1966), 1055-1064. MR 0198271 (33:6430)

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DOI: https://doi.org/10.1090/S0002-9947-1982-0670916-3
Article copyright: © Copyright 1982 American Mathematical Society

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