# Spectra of Abelian *C*^{∗}-Subalgebra Sums

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## Abstract

Let *C*_{b}(*X*) be the *C*^{∗}-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space *X*. Moreover, let \(\mathfrak {A}_{0}\) be some ideal and \(\mathfrak {A}_{1}\) be some unital *C*^{∗}-subalgebra of *C*_{b}(*X*). For \(\mathfrak {A}_{0}\) and \(\mathfrak {A}_{1}\) having trivial intersection, we show that the spectrum of their vector space sum equals the disjoint union of their individual spectra, whereas their topologies are nontrivially interwoven. Indeed, they form a so-called twisted-sum topology which we will investigate beforehand. Within the whole framework, e.g., the one-point compactification of *X* and the spectrum of the algebra of asymptotically almost periodic functions can be described.

## Keywords

Spectra of abelian*C*

^{∗}-algebras Twisted-sum topology Asymptotically almost periodic functions Mathematical foundations of loop quantum cosmology

## Mathematics Subject Classification (2010)

46J40 (Primary) 46L05 54A10 54C50 (Secondary)## Notes

### Acknowledgements

The author thanks Maximilian Hanusch for numerous discussions and many helpful comments on a draft of the present article. Moreover, the author gratefully acknowledges discussions with Thomas Tonev concerning asymptotically almost periodic functions. The author has been supported by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft under grant FL 622/1-1.

### Funding Information

The author has been supported by the Emmy-Noether-Programm of the Deutsche Forschungsgemeinschaft under grant FL 622/1-1.

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