Primitive Triangular UHF Algebras

Hudson, Timothy D.; Katsoulis, Elias G.

doi:10.1006/jfan.1998.3336

Cited by 7 publications

(14 citation statements)

References 21 publications

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“…By Lemma 2.1 in [25], each of the representations τ a , a ∈ A, is algebraically irreducible. Since τ = a∈A τ a is faithful for A we conclude that the intersection of all kernels of algebraically irreducible representations for A equals zero, i.e., an operator semisimple algebra is indeed semisimple.…”

Section: Representation Theorems For Operator Algebrasmentioning

confidence: 96%

Geometry of the unit ball and representation theory for operator algebras

Katsoulis¹

2004

Pacific J. Math.

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Section: Representation Theorems For Operator Algebrasmentioning

confidence: 96%

Geometry of the unit ball and representation theory for operator algebras

Katsoulis¹

2004

Pacific J. Math.

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“…We can describe many of the epimorphisms for these two classes of algebras, by using the results of the second two authors on primitivity [9].…”

Section: Epimorphismsmentioning

confidence: 99%

“…This shows that the spectrum, or fundamental relation [20], a topological binary relation which provides coordinates for limit algebras and is a useful tool in classifications, is a complete algebraic isomorphism invariant for this class (Corollary 2.6). In recent work, the second two authors studied primitivity for limit algebras [9], showing that a variety of limit algebras are primitive. These results, together with automatic continuity, give descriptions of epimorphisms between various classes of limit algebras, namely lexicographic algebras (Theorem 3.2) and Z-analytic algebras (Theorem 3.3).…”

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confidence: 99%

Algebraic isomorphisms of limit algebras

Donsig

Hudson

Katsoulis

2000

Trans. Amer. Math. Soc.

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Abstract. We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.In this paper, we study automatic continuity for limit algebras. Automatic continuity involves algebraic conditions on a linear operator from one Banach algebra into another that guarantee the norm continuity of the operator. This is a generalization, via the open mapping theorem, of the uniqueness of norms problem. Recall that a Banach algebra A is said to have a unique (Banach algebra) topology if any two complete algebra norms on A are equivalent, so that the norm topology determined by a Banach algebra is unique. Uniqueness of norms, automatic continuity, and related questions have played an important and long-standing role in the theory of Banach algebras [6,29,28,10, 11,2].Limit algebras, whose theory has grown rapidly in recent years, are the nonselfadjoint analogues of UHF and AF C-algebras. We first prove that algebraic isomorphisms between limit algebras are automatically continuous (Theorem 1.4). This proof uses the ideal theory of limit algebras as well as key results from the theory of automatic continuity for Banach algebras. Combining this with [23, Theorem 8.3] verifies Power's conjecture that the C-envelope of a limit algebra is an invariant for purely algebraic isomorphisms, for limits of finite dimensional nest algebras, and in particular, for all triangular limit algebras (Corollary 1.6). In [5], the first two authors studied triangular limit algebras in terms of their lattices of ideals. By combining automatic continuity with this work, we show that within the class of algebras generated by their order preserving normalizers (see below for definitions), algebraically isomorphic algebras are isometrically isomorphic (Theorem 2.5). This shows that the spectrum, or fundamental relation [20], a topological binary relation which provides coordinates for limit algebras and is a useful tool in classifications, is a complete algebraic isomorphism invariant for this class (Corollary 2.6). In recent work, the second two authors studied primitivity for limit algebras [9], showing that a variety of limit algebras are primitive. These results, together with automatic continuity, give descriptions of epimorphisms between various classes of limit algebras, namely lexicographic algebras (Theorem 3.2) and Z-analytic algebras (Theorem 3.3).

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“…Let M ax(A) be the space of maximal ideals of A with the hull-kernel topology. Then each M ∈ M ax(A) is the kernel of a character on A, and the map M → M ∩D defines a homeomorphism from M ax(A) onto X, the maximal ideal space of D [6]. We shall identify these two spaces.…”

Section: Introductionmentioning

confidence: 99%

A Dauns-Hofmann theorem for TAF-algebras

Somerset

1999

Proc. Amer. Math. Soc.

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Abstract. Let A be a TAF-algebra, Z(A) the centre of A, Id(A) the ideal lattice of A, and Mir(A) the space of meet-irreducible elements of Id(A), equipped with the hull-kernel topology. It is shown that Mir(A) is a compact, locally compact, second countable, T 0 -space, that Id(A) is an algebraic lattice isomorphic to the lattice of open subsets of Mir(A), and that Z(A) is isomorphic to the algebra of continuous, complex functions on Mir(A). If A is semisimple, then Z(A) is isomorphic to the algebra of continuous, complex functions on P rim(A), the primitive ideal space of A. If A is strongly maximal, then the sum of two closed ideals of A is closed.

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Primitive Triangular UHF Algebras

Cited by 7 publications

References 21 publications

Geometry of the unit ball and representation theory for operator algebras

Geometry of the unit ball and representation theory for operator algebras

Algebraic isomorphisms of limit algebras

A Dauns-Hofmann theorem for TAF-algebras

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