Positive combinations and sums of projections in purely infinite simple 𝐶*-algebras and their multiplier algebras

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang

doi:10.1090/s0002-9939-2011-10995-x

Cited by 13 publications

(22 citation statements)

References 16 publications

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“…In [11,Proposition 2.7] we proved that if a C*-algebra A has such universal constants and if furthermore A + is the closure of PCPs in A then every positive invertible element of A is a PCP.…”

Section: Positive Linear Combinations Of Projectionsmentioning

confidence: 99%

Strict Comparison of Positive Elements in Multiplier Algebras

Kaftal¹,

Ng²,

Zhang³

2017

Can. j. math.

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Abstract.Main result: If a C*-algebra A is simple, σ-unital, has nitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra M(A) also has strict comparison of positive elements by traces.e same results holds if " nitely many extremal traces" is replaced by "quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary σ-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If A is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

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Section: Positive Linear Combinations Of Projectionsmentioning

confidence: 99%

Strict Comparison of Positive Elements in Multiplier Algebras

Kaftal¹,

Ng²,

Zhang³

2017

Can. j. math.

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“…We obtained in [12, Proposition 2.7] extending a B(H) result by Fong and Murphy [9], that if B is a unital C*-algebra that is the span of its projections with control on the coefficients and if PCPs are norm dense in B + , then every positive invertible element of B is a PCP. As in [12] and [17], the key tool for constructing PCP decompositions in our setting is given by the following result, which is an immediate consequence of Corollary 6.1 and [12, Lemma 2.9]: The next step is to prove that if M(A ⊗ K) has real rank zero, every positive element in a corner P M(A ⊗ K) P that has sufficiently large range with respect to P is also PCP (see statement below). The proof is modeled on one of [17, Lemma 6.4] but with some substantial differences, so for clarity and completeness sake, we present a proof here.…”

Section: Positive Combination Of Projections In M(a ⊗ K)mentioning

confidence: 99%

“…As a consequence of this result we then obtain in Theorem 5.1 that every element P (M(A ⊗ K))P is a linear combination of projections in P (M(A ⊗ K))P with control on the coefficients (see Section 5). The control on the coefficients permits us to prove ( [12] and [17]) that every positive locally invertible element in P (M(A ⊗ K))P is PCP (Corollary 6.1).…”

Section: Introductionmentioning

confidence: 99%

Strict comparison of projections and positive combinations of projections in certain multiplier algebras

Kaftal¹,

Ng²,

Zhang³

2015

J. Operator Theory

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In this paper we investigate whether positive elements in the multiplier algebras of certain finite C*-algebras can be written as finite linear combinations of projections with positive coefficients (PCP). Our focus is on the category of underlying C*-algebras that are separable, simple, with real rank zero, stable rank one, finitely many extreme traces, and strict comparison of projections by the traces. We prove that the strict comparison of projections holds also in the multiplier algebra M(A ⊗ K). Based on this result and under the additional hypothesis that M(A ⊗ K) has real rank zero, we characterize which positive elements of M(A ⊗ K) are of PCP.

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“…We will present the proof through the chain of the following lemmas. Our first result extends to the present setting the main tool that we used in [25] and [26]. Proof.…”

Section: Positive Combinations Of Projectionsmentioning

confidence: 56%

“…Such an estimate is stated in [30, Theorems 3.1, 3.3, 3.4]. A discussion of that estimate is also given in our previous paper [25].…”

Section: Linear Combination Of Projectionsmentioning

confidence: 81%

Commutators and linear spans of projections in certain finite C*-algebras

Kaftal

Zhang

2014

Journal of Functional Analysis

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Assume that A is a unital separable simple C*-algebra with real rank zero, stable rank one, strict comparison of projections, and that its tracial simplex T(A) has a finite number of extremal points. We prove that every self-adjoint element a in A with τ (a) = 0 for all τ ∈ T(A) is the sum of two commutators in A and that that every positive element of A is a linear combination of projections with positive coefficients. Assume that A is as above but σ−unital. Then an element (resp. a positive element) a of A is a linear combination (resp. a linear combination with positive coefficients) of projections if and only ifτ (Ra) < ∞ for every τ ∈ T(A), and if and only if , whereτ denotes the extension of τ to a tracial weight on A * * and Ra ∈ A * * denotes the range projection of a. Assume that A is unital and as above but T(A) has infinitely many extremal points. Then A is not the linear span of its projections. This result settles two open problems of Marcoux in [30].

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Positive combinations and sums of projections in purely infinite simple 𝐶*-algebras and their multiplier algebras

Cited by 13 publications

References 16 publications

Strict Comparison of Positive Elements in Multiplier Algebras

Strict Comparison of Positive Elements in Multiplier Algebras

Strict comparison of projections and positive combinations of projections in certain multiplier algebras

Commutators and linear spans of projections in certain finite C*-algebras

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