2017
DOI: 10.4153/cjm-2016-015-3
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Strict Comparison of Positive Elements in Multiplier Algebras

Abstract: 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 applic… Show more

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Cited by 7 publications
(27 citation statements)
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“…However a decomposition into a tridiagonal series plus remainder was obtained and used in [36]. A refinement of that construction, but with fewer hypotheses on A, was obtained in [14] where we proved that if A is σ-unital, then every positive element T ∈ M(A) + can be decomposed into the sum of a selfadjoint element in A of arbitrarily small norm and a bidiagonal series. A bidiagonal series D := ∞ k=1 d k is a strictly converging series with summands [14] can be modified to show that the bidiagonal series can be chosen in K o ({e n }).…”
Section: The Minimal Idealmentioning
confidence: 98%
See 2 more Smart Citations
“…However a decomposition into a tridiagonal series plus remainder was obtained and used in [36]. A refinement of that construction, but with fewer hypotheses on A, was obtained in [14] where we proved that if A is σ-unital, then every positive element T ∈ M(A) + can be decomposed into the sum of a selfadjoint element in A of arbitrarily small norm and a bidiagonal series. A bidiagonal series D := ∞ k=1 d k is a strictly converging series with summands [14] can be modified to show that the bidiagonal series can be chosen in K o ({e n }).…”
Section: The Minimal Idealmentioning
confidence: 98%
“…A trace τ on A is naturally extended to the trace τ ⊗ Tr on A ⊗ K, and so we can identify T (A ⊗ K) with T (A). For more details, see [32], [11] and also [14] and [12].…”
Section: Preliminariesmentioning
confidence: 99%
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“…If T ∈ M(B) + , then by [11], Theorem 4.2, for any ǫ > 0, there exist a bidiagonal series k≥1 t k and a self-adjoint element a ǫ ∈ B with a ǫ < ǫ and T = k≥1 t k + a ǫ .…”
Section: Pedersen's Questionmentioning
confidence: 99%
“…Proof. For 0 < δ < 1, fixed, we have b − (b − δ) + ≤ δ and as B is simple purely infinite, a (b − δ) + .Then for any ǫ > 0, there exists by[11], Lemma 2.2, x ∈ B with x 2 ≤ 1 δ and (a − ǫ) + = xbx * . Then a − xbx * ≤ a − (a − ǫ) + ≤ ǫ.…”
mentioning
confidence: 97%