Representations of triangular subalgebras of groupoid <i>C</i>*-algebras

Muhly, Paul S.; Solel, Baruch

doi:10.1017/s1446788700000392

Cited by 7 publications

(24 citation statements)

References 20 publications

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“…The automorphism group α is determined by a cocycle c on G and so, therefore, A is determined by c. The invariant subspace lattice of π(A), lat(π(A)), turns out to be intimately tied to the asymptotic behavior of c. Our analysis of this relation answers some questions raised in [21]; it gives new proofs and extensions of some results of Orr and Peters [22]; and it provides the new perspective on the results of Kadison and Singer [14] and Arveson [1] alluded to above.…”

Section: Theorem 12 If a Is A Strongly Dirichlet Subalgebra Of B Anmentioning

confidence: 63%

“…In this section we complement the results of [21] and show how the analysis in [14] and [1] relates to our perspective. For this purpose, we follow the notation of [21] and [19].…”

Section: Coordinatized Operator Algebrasmentioning

confidence: 86%

“…More precisely, if B is such a von Neumann algebra, with ideal of relatively compact operators K, and if A = B ∩ alg(N ), where N is a nest whose projections lie in B, then A ∩ K is ultraweakly dense in A. In Section 4 we use Theorems 1.1 and 1.2 and the analysis that enters into their proofs to complement the results of [21]. In the setting of [21],…”

Section: Theorem 12 If a Is A Strongly Dirichlet Subalgebra Of B Anmentioning

confidence: 99%

“…In Section 4 we use Theorems 1.1 and 1.2 and the analysis that enters into their proofs to complement the results of [21]. In the setting of [21],…”

Section: Theorem 12 If a Is A Strongly Dirichlet Subalgebra Of B Anmentioning

confidence: 99%

“…For this purpose, we follow the notation of [21] and [19]. We fix a second countable, locally compact, Hausdorff, r-discrete, principal groupoid G. We view G as an equivalence on G (0) × G (0) .…”

Section: Coordinatized Operator Algebrasmentioning

confidence: 99%

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Automorphism Groups and Invariant Subspace Lattices

Muhly

Solel

1997

Trans. Amer. Math. Soc.

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Abstract. Let (B, R, α) be a C * -dynamical system and let A = B α ([0, ∞)) be the analytic subalgebra of B. We extend the work of Loebl and the first author that relates the invariant subspace structure of π(A), for a C * -representation π on a Hilbert space Hπ, to the possibility of implementing α on Hπ. We show that if π is irreducible and if lat π(A) is trivial, then π(A) is ultraweakly dense in L(H π ). We show, too, that if A satisfies what we call the strong Dirichlet condition, then the ultraweak closure of π(A) is a nest algebra for each irreducible representation π. Our methods give a new proof of a "density" theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid C * -algebras.

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Section: Theorem 12 If a Is A Strongly Dirichlet Subalgebra Of B Anmentioning

confidence: 63%

“…In this section we complement the results of [21] and show how the analysis in [14] and [1] relates to our perspective. For this purpose, we follow the notation of [21] and [19].…”

Section: Coordinatized Operator Algebrasmentioning

confidence: 86%

Section: Theorem 12 If a Is A Strongly Dirichlet Subalgebra Of B Anmentioning

confidence: 99%

“…In Section 4 we use Theorems 1.1 and 1.2 and the analysis that enters into their proofs to complement the results of [21]. In the setting of [21],…”

Section: Theorem 12 If a Is A Strongly Dirichlet Subalgebra Of B Anmentioning

confidence: 99%

Section: Coordinatized Operator Algebrasmentioning

confidence: 99%

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Automorphism Groups and Invariant Subspace Lattices

Muhly

Solel

1997

Trans. Amer. Math. Soc.

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The Lattice of Ideals of a Triangular AF Algebra

Donsig

Hudson

1996

Journal of Functional Analysis

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We study triangular AF (TAF) algebras in terms of their lattices of closed twosided ideals. Not (isometrically) isomorphic TAF algebras can have isomorphic lattices of ideals; indeed, there is an uncountable family of pairwise non-isomorphic algebras, all with isomorphic lattices of ideals. In the positive direction, if A and B are strongly maximal TAF algebras with isomorphic lattices of ideals, then there is a bijective isometry between the subalgebras of A and B generated by their order preserving normalizers. This bijective isometry is the sum of an algebra isomorphism and an anti-isomorphism. Using this, we show that if the TAF algebras are generated by their order preserving normalizers and are triangular subalgebras of primitive C*-algebras, then the lattices of ideals are isomorphic if and only if the algebras are either (isometrically) isomorphic or anti-isomorphic. Finally, we use complete distributivity to show that there are TAF algebras whose lattices of ideals can not arise from TAF algebras generated by their order preserving normalizers. Our techniques rely on constructing a topological binary relation based on the lattice of ideals; this relation is closely connected to the spectrum or fundamental relation (also a topological binary relation) of the TAF algebra.1996 Academic Press, Inc.

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