The general Stone-Weierstrass problem

Akemann, Charles A.

doi:10.1016/0022-1236(69)90015-9

Cited by 116 publications

(169 citation statements)

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“…We now give an Urysohn lemma for TROs, based on Akemann's Urysohn lemma for C * -algebras [1,2,3,5]. See also [23] for a related result, with a different proof strategy (which relies on results of the second author [33]).…”

Section: Proof (I)⇒(ii)mentioning

confidence: 99%

Open partial isometries and positivity in operator spaces

Blecher

Neal

2007

Studia Math.

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Abstract. We first study positivity in C * -modules using tripotents (= partial isometries) which are what we call open. This is then used to study ordered operator spaces via an "ordered noncommutative Shilov boundary" which we introduce. This boundary satisfies the usual universal diagram/property of the noncommutative Shilov boundary, but with all the arrows completely positive. Because of their independent interest, we also systematically study open tripotents and their properties.1. Introduction. We are interested here in cones of positive operators X + = {x ∈ X : x ≥ 0}, for a space X of bounded linear operators on a Hilbert space, where ≥ denotes the usual ordering of such operators. Besides the intrinsic interest of such objects (for example, operator positivity plays a central role in many areas of mathematical physics today), our work is a sequel to [13], which was a first step in a new approach to positivity in an operator space X, namely studying it in terms of the "noncommutative Shilov boundary" of X (see [6,26,12]). The latter object is a Hilbert C * -module, or, equivalently, a ternary ring of operators (or TRO for short), by which we will mean a closed subspace Z of a C * -algebra A such that ZZ * Z ⊂ Z. If X contains positive operators, then so will any containing TRO. The starting point of the present investigation and [13] was the question of whether, in this case, all morphisms in the universal property of the noncommutative Shilov boundary can also be chosen to be positive (allowing this boundary to be used as a new tool in the study of ordered operator spaces). To answer this, one is led immediately to study positivity in TROs, and we

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Section: Proof (I)⇒(ii)mentioning

confidence: 99%

Open partial isometries and positivity in operator spaces

Blecher

Neal

2007

Studia Math.

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“…Suppose S ç B(H). Then (1) appr(S)" = [T: {V~XTVR} is convergent whenever {V") is an invertibly bounded sequence such that { V~XSV) is convergent for every S in § }, (2) // {V"} is an invertibly bounded sequence such that {V~XSV"} is convergent for every SinS, then m(A) = lim V~XAV" defines a homeomorphic isomorphism from appr(S)" onto appr(7r( §))".…”

Section: Corollary 23 If S Isa Separable Subset Ofb(h) Then #"( §)mentioning

confidence: 99%

“…Statement (5) follows from (2)-(4) and the fact [44, Proposition 2.1] that every operator is approximately equivalent to a direct sum of irreducible operators. Statement (1) follows from the proofs of XV.6.1 and XV.6.2 in [14].…”

Section: Suppose T G B(h) Then (L)ß(t) < A(t) < ß(T)\ (2) If T = Pmentioning

confidence: 99%

An asymptotic double commutant theorem for 𝐶*-algebras

Hadwin¹

1978

Trans. Amer. Math. Soc.

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Abstract. An asymptotic version of von Neumann's double commutant theorem is proved in which C*-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and «■ is a representation of the C*-algebra generated by I, then ir(T) is similar to a normal (subnormal) operator.1. Introduction. One of the reasons for the success of the theory of von Neumann algebras is J. von Neumann's double commutant theorem [46], which gives an alternate description of the weak closure of a selfadjoint algebra of operators. It is the purpose of this paper to prove an asymptotic version of the double commutant theorem that gives an alternate description of the norm closure of a selfadjoint algebra of operators. This asymptotic double commutant theorem helps to unify asymptotic versions of various operator-theoretic concepts (e.g., similarity, reflexivity, reductivity). Applications are made to the study of operators that are similar to normal (or subnormal) operators. Also a proof is given that a strongly reductive, nonseparable, commutative, norm closed algebra of operators is selfadjoint.Throughout, H denotes a separable, infinite-dimensional complex Hubert space, B(H) denotes the set of operators (bounded linear transformations) on H, and %(H) denotes the set of compact operators on H. Also § denotes a separable, nonempty subset of B(H). However, in §8 the separability

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“…By part b of Lemma 0.2, if {r α } is an increasing approximate unit for K f = {a ∈ N : f (a * a + aa * ) = 0} with r α ↑ r ∈ N * * , then r α → 1 in the weak* topology of N . Then r is an open projection, hence regular in N by [1], Prop. II.14 since N is a von Neumann algebra.…”

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confidence: 92%