The \(C^*\)-algebras of the Heisenberg group and of thread-like Lie groups. (English) Zbl 1230.22004
The authors study the \(C^*\)-algebras \(C^*(H_n)\) associated to Heisenberg groups \(H_n\) for every \(n\geq 1\), and the \(C^*\)-algebras of thread-like Lie groups \(G_N\) for \(N\geq 3\) in terms of algebras of operator fields.
The Heisenberg group \(H_n\) is the Lie group with underlying space \(\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}\) and operation \((x,y,t)\cdot(x',y',t')=(x+x', y+y', t+t' +1/2(x\cdot y'-x'\cdot y))\), where \(z\cdot w\) denotes the scalar product on \(\mathbb{R}^n\). The authors construct a \(C^*\)-algebra \(\mathcal{F}_n\) associated to a family of operator fields with fibers in the compact operators on \(L^2(\mathbb{R}^n)\) for all \(\lambda\in \mathbb{R}\setminus \{0\}\) and with value at \(0\) in \(C^*(\mathbb{R}^{2n})\). They then construct a linear map \(\nu\) from \(C^*(\mathbb{R}^{2n})\) to \(\mathcal{F}_n\) which turns out to be a cross section for the quotient map from \(C^*(H_n)\) corresponding to the ideal \(C_0(\mathbb{R}\setminus \{0\}, \mathcal{K})\). The image \(D_\nu(H_n)\) in \(\mathcal{F}_n\) obtained from this map is then shown to be isomorphic to \(C^*(H_n)\).
The thread-like group \(G_N\) for \(N\geq 3\) is the group \(\exp(\mathfrak{g}_N)\) associated to the \(N\)-dimensional real nilpotent Lie algebra \(\mathfrak{g}_N\). The unitary dual of \(G_N\) is described by means of methods due to R. J. Archbold et al. [Adv. Math. 158, No.1, 26–65 (2001; Zbl 0978.22005)] and R. J. Archbold, J. Ludwig and G. Schlichting [ Math. Z. 255, No. 2, 245–282 (2007; Zbl 1194.22001)]. The main application is a realisation of \(C^*(G_N)\) as a \(C^*\)-algebra of operator fields.
The Heisenberg group \(H_n\) is the Lie group with underlying space \(\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}\) and operation \((x,y,t)\cdot(x',y',t')=(x+x', y+y', t+t' +1/2(x\cdot y'-x'\cdot y))\), where \(z\cdot w\) denotes the scalar product on \(\mathbb{R}^n\). The authors construct a \(C^*\)-algebra \(\mathcal{F}_n\) associated to a family of operator fields with fibers in the compact operators on \(L^2(\mathbb{R}^n)\) for all \(\lambda\in \mathbb{R}\setminus \{0\}\) and with value at \(0\) in \(C^*(\mathbb{R}^{2n})\). They then construct a linear map \(\nu\) from \(C^*(\mathbb{R}^{2n})\) to \(\mathcal{F}_n\) which turns out to be a cross section for the quotient map from \(C^*(H_n)\) corresponding to the ideal \(C_0(\mathbb{R}\setminus \{0\}, \mathcal{K})\). The image \(D_\nu(H_n)\) in \(\mathcal{F}_n\) obtained from this map is then shown to be isomorphic to \(C^*(H_n)\).
The thread-like group \(G_N\) for \(N\geq 3\) is the group \(\exp(\mathfrak{g}_N)\) associated to the \(N\)-dimensional real nilpotent Lie algebra \(\mathfrak{g}_N\). The unitary dual of \(G_N\) is described by means of methods due to R. J. Archbold et al. [Adv. Math. 158, No.1, 26–65 (2001; Zbl 0978.22005)] and R. J. Archbold, J. Ludwig and G. Schlichting [ Math. Z. 255, No. 2, 245–282 (2007; Zbl 1194.22001)]. The main application is a realisation of \(C^*(G_N)\) as a \(C^*\)-algebra of operator fields.
Reviewer: Nadia Larsen (Oslo)
MSC:
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 22E27 | Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) |
| 46L05 | General theory of \(C^*\)-algebras |
References:
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| [2] | Archbold R.J., Ludwig J., Schlichting G.: Limit sets and strengths of convergence for sequences in the duals of thread-like Lie groups. Math. Z. 255(2), 245–282 (2007) · Zbl 1194.22001 · doi:10.1007/s00209-006-0023-1 |
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