tailieunhanh - Spectra of abelian C-subalgebra sums

Let Cb(X) be the C∗-algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space X. Moreover, let A0 be some ideal and A1 be some unital C-subalgebra of Cb(X). For A0 and A1 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. | Vietnam Journal of Mathematics (2019) 47:195–208 Spectra of Abelian C ∗ -Subalgebra Sums Christian Fleischhack1 Received: 15 January 2018 / Accepted: 4 November 2018 / Published online: 18 January 2019 © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019 Abstract Let Cb (X) be the C ∗ -algebra of bounded continuous functions on some non-compact, but locally compact Hausdorff space X. Moreover, let A0 be some ideal and A1 be some unital C ∗ -subalgebra of Cb (X). For A0 and A1 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, ., 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) 1 Introduction It is well-known that the spectrum of a direct sum of two abelian C ∗ -algebras equals the topological direct sum of the respective individual spectra. Sometimes, however, one is given only a vector space direct sum of two C ∗ -algebras. This applies, most prominently, to A + C1 when one adjoins a unit to the non-unital C ∗ -algebra A. Another example is the algebra C0 (X) + CAP (X) of asymptotically almost periodic functions [6] on a non-compact locally compact abelian group X, being our main motivation [5]. It is now natural to ask whether there are still general arguments on how to determine the spectrum in these cases. Or, to put it into a more abstract form: how does the spectrum of a sum of any two .