Feichtinger's algebra and the Stone-von Neumann Theorem
given at strobl11 (18.06.11 11:50)
Since I have first heard Hans Feichtinger lecture on the minimal strongly character invariant Segal algebra $S_0$, I have been deeply impressed by the beauty of the Feichtinger algebra and the various approaches Hans Feichtinger has developed to this important function spaces. It is with great admiration of the advisor, researcher and teacher Hans Feichtinger that I am going to discuss the usefulness of Feichtinger's algebra for the theory of operator algebras in this talk.
My intention is to present some further evidence of the connection between operator algebras
and time-frequency analysis, in particular the relevance of Morita-Rieffel equivalence of operator algebras.
About four decades ago Rieffel interpreted the Stone-von Neumann Theorem on the irreducible representations of the canonical commutation relations in terms of operator algebras, which is mostly unknown to researchers outside of the community of operator algebras. In modern terminology the Stone-von Neumann Theorem is equivalent to the Morita-Rieffel equivalence of the compact operators and the complex numbers.
In this talk I want to present this perspective on the Stone-von Neumann theorem, since it
allows one to interpret well-known facts of time-frequency analysis in a wider context. Another reason for this choice of topic is the relevance of Feichtinger's algebra $S_0$ in this approach to Stone-von Neumann's Theorem. More precisely, $S_0$ turns out to be a natural equivalence bimodule between the $C^*$-algebra of compact operators and the complex numbers.
Therefore this approach leads to an extension of Rieffel's original argument, which demonstrates once more Feichtinger's paradigm: $S_0$ is in most cases a good substitute for the Schwartz class of test functions.
Let me conclude with my sincere gratitude to Hans Feichtinger for his constant encouragement, inspiration and support since the day, when I was fortunate to walk into the Seminarraum at Währinger Straße in Vienna to attend his course on functional analysis.