Multiplications of Symbols on the 2-Sphere

Pedro de M. Rios and Eldar Straume
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Pedro de M. Rios: Universidade de São Paulo, Departamento de Matemática, ICMC
Eldar Straume: Norwegian University of Science and Technology, Department of Mathematical Sciences

Chapter Chapter 7 in Symbol Correspondences for Spin Systems, 2014, pp 105-133 from Springer

Abstract: Abstract Given any symbol correspondence $$W^{j} = W_{\vec{c}}^{j}$$ , the algebra of operators in $$\mathcal{B}(\mathcal{H}_{j}) \simeq M_{\mathbb{C}}(n + 1)$$ can be imported to the space of symbols $$W_{\vec{c}}^{j}(\mathcal{B}(\mathcal{H}_{j})) \simeq \mathit{Poly}_{\mathbb{C}}(S^{2})_{\leq n} \subset C_{\mathbb{C}}^{\infty }(S^{2})$$ . The 2-sphere, with such an algebra on a subset of its function space, has become known as the “fuzzy sphere” [47]. However, there is no single “fuzzy sphere”, as each symbol correspondence defined by characteristic numbers $$\vec{c} = (c_{1},\ldots,c_{n})$$ gives rise to a distinct (although isomorphic) algebra on the space of symbols $$\mathit{Poly}_{\mathbb{C}}(S^{2})_{\leq n}$$ , as we shall investigate in some detail, in this chapter.

Keywords: Spherical Harmonic; Invariant Function; Characteristic Number; Spherical Geometry; Transition Kernel (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-08198-4_7

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