A classification of inductive limit C∗$C^{*}$‐algebras with ideal property

Abstract

Let A$A$ be an AH algebra A=limn→∞(An=⨁i=1tnPn,iM[n,i]false(C(Xn,i)false)Pn,i,ϕn,m)$A=\lim \nolimits _{n\rightarrow \infty }(A_{n}=\bigoplus \nolimits _{i=1} ^{t_{n}}P_{n,i} M_{[n,i]}(C(X_{n,i}))P_{n,i}, \phi _{n,m})$, where Xn,i$X_{n,i}$ are compact metric spaces, tn$t_{n}$ and false[n,ifalse]$[n,i]$ are positive integers, Pn,i∈Mfalse[n,ifalse](Cfalse(Xn,ifalse))$P_{n,i}\in M_{[n,i]} (C(X_{n,i}))$ are projections, and ϕn,m:An→Am$\phi _{n,m}: A_n\rightarrow A_m$ (for m>n$m&gt;n$) are homomorphisms satisfying ϕn,m=ϕm−1,m∘ϕm−2,m−1∘⋯∘ϕn+1,n+2∘ϕn,n+1$\phi _{n,m}=\phi _{m-1,m} \circ\; \phi _{m-2,m-1}\;\circ\; \cdots \;\circ\; \phi _{n+1,n+2}\;\circ\; \phi _{n, n+1}$. Suppose that A$A$ has the ideal property: each closed two‐sided ideal of A$A$ is generated by the projections inside the ideal, as a closed two‐sided ideal (see Pacnicn, Pacific J. Math. 192 (2000), 159–183). In this article, we will classify all AH algebras with the ideal property (of no dimension growth — that is, supn,idim(Xn,i)<+∞$sup_{n,i}dim(X_{n,i})&lt;+\infty$). This result generalizes and unifies the classification of AH algebras of real rank zero in Dadarlat and Gong (Geom. Funct. Anal. 7 (1997), 646–711), Elliott and Gong (Ann. of Math. (2) 144 (1996), 497–610) and the classification of simple AH algebras in Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320), and Gong (Doc. Math. 7 (2002), 255–461). This completes one of two important possible generalizations of Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320) suggested in the introduction of Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320). The invariants for the classification include the scaled ordered total K$K$‐group false(K̲(A),K̲false(Afalse)+,normalΣAfalse)$(\underline{K}(A), \underline{K}(A)_{+},\Sigma A)$ (as already used in the real rank zero case in Dadarlat and Gong, Geom. Funct. Anal. 7 (1997) 646–711), for each false[pfalse]∈normalΣA$[p]\in \Sigma A$, the tracial state space Tfalse(pApfalse)$T(pAp)$ of the cut down algebra pAp$pAp$ with a certain compatibility, (which is used by Steven (Field Inst. Commun. 20 (1998), 105–148), and Ji and Jang (Canad. J. Math. 63 (2011) no. 2, 381–412) for AI algebras with the ideal property), and a new ingredient, the invariant U(pAp)/DUfalse(pApfalse)¯$U(pAp)/\overline{DU(pAp)}$ with a certain compatibility condition, where DU(pAp)¯$\overline{DU(pAp)}$ is the closure of commutator subgroup DUfalse(pApfalse)$DU(pAp)$ of the unitary group Ufalse(pApfalse)$U(pAp)$ of the cut down algebra pAp$pAp$. In Gong, Jiang and Li (Ann. K‐Theory 5 (2020), no.1, 43–78), a counterexample is presented to show that this new ingredient must be included in the invariant. The discovery of this new invariant is analogous to that of the order structure on the total K‐theory when one advances from the classification of simple real rank zero C∗$C^*$‐algebras to that of non‐simple real rank zero C∗$C^*$‐algebras in Dadarlat and Gong (Geom. Funct. Anal. 7 (1997), 646–711), Dadarlat and Loring (Duke Math. J. 84 (1996), ...

The unitary Cuntz semigroup on the classification of non-simple C⁎-algebras

“…See e.g. [13]. However, whenever dealing with non-simple C * -algebras, one has to take into account KT u of all the ideals (which can be plenty) together with many more compatibility conditions.…”

On classification of non-unital amenable simple C*-algebras, II

“…As mentioned earlier, there are known simple AH-algebras which are not Z-stable. There is also a question whether non-simple Z-stable C * -algebras could be classified, and if so, what would be the right formulation of the invariant (see [47], [37], [38], [48], [10], [49], [35], [36], and [76]). Therefore Theorem 5.13 is not even the beginning of the end.…”

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