Unitary groups as a complete invariant

The general linear group as a complete invariant for $C^*$-algebras

“…However, it turns out that in many cases θ ϕ is essentially an orthoisomorphism. More precisely, generalising the notion of oddly decomposability given in [1] (see Definition 2.1), we show in Theorem 2.9 that there exist a partitioning of the non-trivial elements of IpAq into two set I o , Iō, such that the mapθ ϕ : IpAq Ñ IpBq defined byθ…”

“…In this subsection, inspired by [1], we show that an isomorphism between the general linear groups of simple, unital, purely infinite C˚-algebras induces an isomorphism between their K 0 -groups. Recall that if A is a purely infinite simple C˚-algebra, then every nonzero projection in A is infinite, and K 0 pAq " trps : p P PpAq, p ‰ 0u (see p. 73-85 in [19]).…”

“…The following additional properties of the map θ can be easily checked by adapting the arguments in [1] and [8] to the present situation. Proposition 1.1.…”

“…, n are orthogonal projections adding to q. We can select projections r [1]. 1 AND ADAM SIERAKOWSKI 2 Notation 2.3.…”

On the extension of unitary group isomorphisms of unital UHF-algebras

“…In [1], Al-Rawashdeh, Booth and Giordano proved that if the unitary groups of two simple unital AH-algebras of slow dimension growth and of real rank zero are isomorphic as abstract groups, then their K 0 -ordered groups are isomorphic. Also, using Elliott-Gong and Dadarlat's classification theorem, they proved that such C * -algebras are isomorphic if and only if their unitary groups are isomorphic as topological groups.…”