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

Abstract

Let A $A$ be an AH algebra A = lim n → ∞ ( A n = ⨁ i = 1 t n P n , i M [ n , i ] ( C ( X n , i ) ) P n , i , ϕ n , 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 X n , i $X_{n,i}$ are compact metric spaces, t n $t_{n}$ and [ n , i ] $[n,i]$ are positive integers, P n , i ∈ M [ n , i ] ( C ( X n , i ) ) $P_{n,i}\in M_{[n,i]} (C(X_{n,i}))$ are projections, and ϕ n , m : A n → A m $\phi _{n,m}: A_n\rightarrow A_m$ (for m > n $m>n$ ) are homomorphisms satisfying ϕ n , m = ϕ m − 1 , m ∘ ϕ m − 2 , m − 1 ∘ ⋯ ∘ ϕ n + 1 , n + 2 ∘ ϕ n , 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, s u p n , i d i m ( X n , i ) < + ∞ $sup_{n,i}dim(X_{n,i})<+\infty$ ). 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 ( K ̲ ( A ) , K ̲ ( A ) + , Σ A ) $(\underline{K}(A), \underline{K}(A)_{+},\Sigma A)$ (as already used in the real rank zero case in Dadarlat and Gong, Geom. Funct. Anal. 7 (1997) 646–711), for each [ p ] ∈ Σ A $[p]\in \Sigma A$ , the tracial state space T ( p A p ) $T(pAp)$ of the cut down algebra p A p $pAp$ 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 ( p A p ) / D U ( p A p ) ¯ $U(pAp)/\overline{DU(pAp)}$ with a certain compatibility condition, where D U ( p A p ) ¯ $\overline{DU(pAp)}$ is the closure of commutator subgroup D U ( p A p ) $DU(pAp)$ of the unitary group U ( p A p ) $U(pAp)$ of the cut down algebra p A p $pAp$ . 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), 355–377), Eilers (J. Funct. Anal. 139 (1996), 325–348), and Gong (J. Funct. Anal. 152 (1998), 281–329) (see Introduction below). Let us point out that the Hausdorffified algebraic K 1 $K_1$ -group U ( A ) / D U ( A ) ¯ $U(A)/\overline{DU(A)}$ , was first used by Nielsen and Thomsen to classify homomorphisms up to approximately unitary equivalent. This classification serves as the uniqueness theorem for the classification of simple A T $\mathbb {T}$ algebras in Nielsen and Thomsen (Expo. Math. 14 (1996), 17–56).