Abstract
Using the operator valued Fourier transform, the \(C^{*}\)-algebras of connected real two-step nilpotent Lie groups are characterized as algebras of operator fields defined over their spectra. In particular, it is shown by explicit computations, that the Fourier transform of such \(C^{*}\)-algebras fulfills the norm controlled dual limit property.
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This work is supported by the Fonds National de la Recherche, Luxembourg (Project Code 3964572).
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Appendix
Appendix
Lemma 1
Let V be a finite-dimensional euclidean vector space and S an invertible, skew-symmetric endomorphism. Then V can be decomposed into an orthogonal direct sum of two-dimensional S-invariant subspaces.
Proof
S extends to a complex endomorphism \( S_{\mathbb {C}}\) on the complexification \(V_{\mathbb {C}}\) of V, which has purely imaginary eigenvalues.
If \( i\lambda \in i{\mathbb {R}}\) is an eigenvalue, then also \( -i\lambda \) is a spectral element. Denote by \( E_{i\lambda } \) the corresponding eigenspace. These eigenspaces are orthogonal to each other with respect to the Hilbert space structure of \( V_{\mathbb {C}}\) coming from the euclidean scalar product \( \langle \cdot ,\cdot \rangle \) on V.
Let for \( i\lambda \) in the spectrum of \( S_{\mathbb {C}}\)
If \( \lambda \ne 0 \), \( dim (V^{\lambda } ) \) is even and \(V^{\lambda } \) is S -invariant and orthogonal to \(V^{\lambda '} \), whenever \( |\lambda |\ne |\lambda '| \):
Indeed, one then has for \( x\in V^{\lambda }, x'\in V^{\lambda '}\) that
Therefore,
Thus, one has
Suppose that \(dim (V^{\lambda } )>2\), choose a vector \( x\in V^{\lambda } \) of length 1 and let \(y=S(x)\). Since \(S_{\mathbb {C}}^2=-\lambda ^2 \text {Id}\), both on \(E_{i\lambda }\) and on \(E_{-i\lambda }\),
This shows that \( W_1^{\lambda }:= \text {span} \{x,y\} \) is an S-invariant subspace of \( V^{\lambda } \). If \( V_1^{\lambda } \) denotes the orthogonal complement of \( W_1^{\lambda } \) in \( V^{\lambda } \), then \(V_1^{\lambda } \) is S -invariant, since \(S^t=-S\).
In this way one can find a decomposition of \( V^{\lambda } \) into an orthogonal direct sum of two-dimensional S -invariant subspaces \(W_j^{\lambda } \) and by summing up over the eigenvalues, one obtains the required decomposition of V.
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Günther, JK., Ludwig, J. The \(C^{*}\)-algebras of connected real two-step nilpotent Lie groups. Rev Mat Complut 29, 13–57 (2016). https://doi.org/10.1007/s13163-015-0177-7
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DOI: https://doi.org/10.1007/s13163-015-0177-7