Convergence of Fourier series in discrete crossed products of von Neumann algebras

Mercer, Richard

doi:10.1090/S0002-9939-1985-0784174-0

Convergence of Fourier series in discrete crossed products of von Neumann algebras
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by Richard Mercer

Proc. Amer. Math. Soc. 94 (1985), 254-258

DOI: https://doi.org/10.1090/S0002-9939-1985-0784174-0

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Abstract:

The convergence of the generalized Fourier series $\Sigma \pi (x(g))u(g)$ is considered in the crossed product of a von Neumann algebra by a discrete group. An example from classical theory shows that this series does not converge in any of the usual topologies. It is proven that this series does converge in a topology introduced by Bures which is well suited to a crossed product situation. As an elementary application, we answer the question: In what topology is an infinite matrix (representing a bounded operator) the sum of its diagonals?

References

Donald Bures, Abelian subalgebras of von Neumann algebras, Memoirs of the American Mathematical Society, No. 110, American Mathematical Society, Providence, R.I., 1971. MR 0296706
Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
Robert R. Kallman, A generalization of free action, Duke Math. J. 36 (1969), 781–789. MR 256181
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210528
Şerban Strătilă, Modular theory in operator algebras, Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells, 1981. Translated from the Romanian by the author. MR 696172
Noboru Suzuki, Crossed products of rings of operators, Tohoku Math. J. (2) 11 (1959), 113–124. MR 105624, DOI 10.2748/tmj/1178244632
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728