The ideal property in crossed products

Pasnicu, Cornel

doi:10.1090/s0002-9939-03-07032-1

Cited by 9 publications

(4 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even if we assume, in the above class of examples, that 𝐴 is the commutative 𝐶 * -algebra 𝐶(𝑋) with 𝑑𝑖𝑚(𝑋) = 0 and 𝐺 = ℤ, it is not known whether 𝐴 ⋊ 𝛼 𝐺 is of real rank zero (it is known that 𝐴 ⋊ 𝛼 𝐺 is not simple if 𝛼 is not minimal). However, it follows from the above result or from [55] that 𝐶(𝑋) ⋊ 𝛼 ℤ has the ideal property. Hence, it is important and natural to extend the classification of simple 𝐶 * -algebras and the classification of real rank zero 𝐶 * -algebras to 𝐶 * -algebras with the ideal property.…”

Section: Introductionmentioning

confidence: 93%

“…Even if we assume, in the above class of examples, that

A

is the commutative

C^{*}

‐algebra

C(X)

with

dim(X)=0

and

G=\mathbb {Z}

, it is not known whether

A\rtimes _{\alpha }G

is of real rank zero (it is known that

A\rtimes _{\alpha }G

is not simple if

\alpha

is not minimal). However, it follows from the above result or from [55] that

C(X)\rtimes _{\alpha }\mathbb {Z}

has the ideal property. Hence, it is important and natural to extend the classification of simple

C^{*}

‐algebras and the classification of real rank zero

C^{*}

‐algebras to

C^{*}

‐algebras with the ideal property.…”

Section: Introductionmentioning

confidence: 96%

See 1 more Smart Citation

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

Gong

Jiang

2022

Trans London Maths Soc

View full text Add to dashboard Cite

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$m>n$) 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)<+∞$sup_{n,i}dim(X_{n,i})<+\infty$). 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), ...

show abstract

Section: Introductionmentioning

confidence: 93%

“…Even if we assume, in the above class of examples, that

A

is the commutative

C^{*}

‐algebra

C(X)

with

dim(X)=0

and

G=\mathbb {Z}

, it is not known whether

A\rtimes _{\alpha }G

is of real rank zero (it is known that

A\rtimes _{\alpha }G

is not simple if

\alpha

is not minimal). However, it follows from the above result or from [55] that

C(X)\rtimes _{\alpha }\mathbb {Z}

has the ideal property. Hence, it is important and natural to extend the classification of simple

C^{*}

‐algebras and the classification of real rank zero

C^{*}

‐algebras to

C^{*}

‐algebras with the ideal property.…”

Section: Introductionmentioning

confidence: 96%

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

Gong

Jiang

2022

Trans London Maths Soc

View full text Add to dashboard Cite

show abstract

“…Even if we assume, in the above class of examples, that A is the commutative C * -algebra C(X) with dim(x) = 0 and G = Z, it is not known whether A ⋊ α G is of real rank zero (it is known that A ⋊ α G is not simple if α is not minimal). However it follows from the above result or from [Pa2] that C(X) ⋊ α Z has ideal property. Hence it is important and natural to extend the classification of simple C * -algebras and the classification of real rank zero C * -algebras to C * -algebras with ideal property.…”

mentioning

confidence: 90%

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

Gong,

Jiang,

2016

Preprint

View full text Add to dashboard Cite

Let A be an AH algebra A = lim n→∞ (A n = tn i=1P n,i M [n,i] (C(X n,i ))P n,i , φ n,m ), where X n,i are compact metric spaces, t n and [n, i] are positive integers, and P n,i ∈ M [n,i] (C(X n,i )) are projections. Suppose that A has the ideal property: each closed two sided ideal of A is generated by the projections inside the ideal, as closed two sided ideal. In this article, we will classify all AH algebras with ideal property of no dimension growth-that is Sup n,i dim(X n,i ) < +∞. This result generalizes and unifies the classification of AH algebras of real rank zero in [EG] and [DG] and the classification of simple AH algebras in [G5] and [EGL1]. By this paper, we have completed one of two so considered most important possible generalizations of [EGL1](see the introduction of [EGL1]). The invariants for the classification including scale ordered total K −group (K(A), K(A) + , ΣA)(as already used in real rank zero case in [DG]), for each [p] ∈ ΣA, T (pAp)-the tracial state space of cut down algebra pAp with certain compatibility, (which is used by [Stev] and [Ji-Jiang] for AI algebras with ideal property), and a new ingredient, the invariant U(pAp)/DU(pAp), with certain compatibility condition, where DU(pAp) is the closure of commutator subgroup DU(pAp) of the unitary group U(pAp) of the cut down algebra pAp. We will also present a counterexample if this new ingredient is missed in the invariant. The discovery of this new invariant is an analogy to that of order structure on total K-theory when one advanced from the classification of simple real rank zero C * -algebras to that of non simple real rank zero C * -algebras in [G2], [Ei], [DL] and [DG] (see introduction below). We will also prove that the so called AT AI algebras in [Jiang1] (and AT AF algebra in [Fa])-the inductive limits of direct sums of simple T AI (and T AF ) algebras with UCT -, are in our class. Here the concept of simple T AI (and T AF ) algebras was introduced by Lin to axiomatize the decomposition theorem of [G5] (and of [EG2] respectively) and is partially inspired by Popa's work [Popa].

show abstract

“…It follows from [30] that if h is an arbitrary aperiodic homeomorphism of the Cantor set X, then C * (Z, X, h) has the ideal property, that is, every ideal is generated as an ideal in the algebra by its projections. This certainly suggests that the reduced C*-algebra of an almost AF Cantor groupoid should have the ideal property.…”

Section: Examples and Open Problemsmentioning

confidence: 99%