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

HTML articles powered by AMS MathViewer

- by Richard Mercer
- Proc. Amer. Math. Soc.
**94**(1985), 254-258 - DOI: https://doi.org/10.1090/S0002-9939-1985-0784174-0
- PDF | Request permission

## 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**

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**94**(1985), 254-258 - MSC: Primary 46L10; Secondary 47C15
- DOI: https://doi.org/10.1090/S0002-9939-1985-0784174-0
- MathSciNet review: 784174