Semisimple Triangular AF Algebras

Donsig, Allan P.

doi:10.1006/jfan.1993.1016

Cited by 25 publications

(49 citation statements)

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“…A Banach algebra is called semisimple if its Jacobson radical is zero. Donsig characterized semisimplicity for TAF algebras in terms of mixing embeddings [8,Corollary 6]. If ϕ : A n → A n+1 is an embedding and p and q are projections in D pn such that p q, then ϕ mixes p and q if there are restrictions r of ϕ(p) and s of ϕ(q) such that s r. If ϕ mixes p and q for every such p and q in D pn , then the embedding ϕ is called mixing.…”

Section: Lemma 4 If a Is A Taf Algebra Then The Convex Hull (No Clomentioning

confidence: 99%

“…Since A is semisimple, by [8,Corollary 6], we can choose a presentation lim −→ (A n ; ϕ n ) for A, where each ϕ n : A n → A n+1 is mixing and A n is isometrically isomorphic to T pn for some n. Choose a system of matrix units for A so that for each n, {e…”

Section: Lemma 4 If a Is A Taf Algebra Then The Convex Hull (No Clomentioning

confidence: 99%

“…Recall from [8] (A n ; ϕ n ) for A. We claim there exists some positive integer m such that none of the maps ϕ n,m , for any n ≥ m, are mixing.…”

Section: Lemma 6 Let ϕ : T N → T M Be An Embedding Which Is Not Miximentioning

confidence: 99%

“…We claim there exists some positive integer m such that none of the maps ϕ n,m , for any n ≥ m, are mixing. Indeed, since A is not semisimple, then by [8,Theorem 4] there is an off-diagonal matrix unit g in A with no links. Suppose g belongs to A m for some m, and let n ≥ m. Let e be the range projection of g and f the domain projection; hence e ≺ f .…”

Section: Lemma 6 Let ϕ : T N → T M Be An Embedding Which Is Not Miximentioning

confidence: 99%

“…Their theory has grown rapidly, cf. [8,9,18,19,20]. We focus on the extreme point structure, and our results have a different flavor than those for nest algebras.…”

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confidence: 99%

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Extreme points in triangular UHF algebras

Hudson

Katsoulis

Larson

1997

Trans. Amer. Math. Soc.

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Abstract. We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points.

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Section: Lemma 4 If a Is A Taf Algebra Then The Convex Hull (No Clomentioning

confidence: 99%

Section: Lemma 4 If a Is A Taf Algebra Then The Convex Hull (No Clomentioning

confidence: 99%

“…Recall from [8] (A n ; ϕ n ) for A. We claim there exists some positive integer m such that none of the maps ϕ n,m , for any n ≥ m, are mixing.…”

Section: Lemma 6 Let ϕ : T N → T M Be An Embedding Which Is Not Miximentioning

confidence: 99%

Section: Lemma 6 Let ϕ : T N → T M Be An Embedding Which Is Not Miximentioning

confidence: 99%

“…Their theory has grown rapidly, cf. [8,9,18,19,20]. We focus on the extreme point structure, and our results have a different flavor than those for nest algebras.…”

mentioning

confidence: 99%

See 3 more Smart Citations