Abstract
We define and study a continuous functional calculus for a commutative family of normal elements of a \(C^{*}\)-algebra. We obtain that this functional calculus is unique, continuous and satisfies the spectral mapping theorem. We also provide two applications. The first one concerns the existence of a particular orthonormal basis on a locally Hilbert space. The second for multi-dimensional continuous versions of N. Wiener and P. Lé vy theorems.
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Mazighi, M., Kinani, A.E. Some applications of simultaneous continuous functional calculus. Rend. Circ. Mat. Palermo, II. Ser 72, 1629–1638 (2023). https://doi.org/10.1007/s12215-022-00752-9
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DOI: https://doi.org/10.1007/s12215-022-00752-9
Keywords
- \(C^{*}\)-algebra
- Normal element
- Continuous function
- Simultaneous continuous functional calculus
- Weighted algebra
- Wiener theorem
- Lévy theorem