Classification of limits of triangular matrix algebras

Hopenwasser, Alan; Power, S. C.

doi:10.1017/s0013091500005927

Cited by 21 publications

(31 citation statements)

References 10 publications

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“…easy to see that two points x,y e X = Π {0, l} n are in the same orbit iff they have the same tails, and in that case the ordering is given by x -< y if either x = y or else 3 iV G N with XJV < yâ nd x n -y n , n > N. Thus the ordering on each orbit ("reverse lexicographic order") has the property that each element except for ϊ = (1,1,1 For Example 1.4, view X as described in [4]: X = Π {0, l} n , n=-oo where x, y G X belong to the same orbit if they have the same tails (both to the left and to the right), and x -< y if for some JV, %N < 2/iv, and x n = y n for n < N. Let X_ n C X be the set of points x = (rrfc)£L-oo with x k = 0, fe < -n. Define a measure μ on U X-n by by A(x 0 ) = Σ X{2~~1. This is well defined as xt -0 for -£ =-oo sufficiently large.…”

Section: Let 21 Be a Primitive Af Algebra T C 21 A Strongly Maximamentioning

confidence: 99%

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Some representations of TAF algebras

Orr¹,

Peters²

1995

Pacific J. Math.

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Section: Let 21 Be a Primitive Af Algebra T C 21 A Strongly Maximamentioning

confidence: 99%

“…Quite recently there has been an interest in triangular subalgebras of AF C*-algebras (cf. [8], [4], [9], [10], [15], [16], to name a few). While analogues between the two theories have been noticed -indeed, they are hard to ignore -no direct connection has been established between them.…”

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confidence: 99%

Some representations of TAF algebras

Orr¹,

Peters²

1995

Pacific J. Math.

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“…This family includes a range of fundamental examples, such as the standard, refinement, and alternation limit algebras [1,7,15,17], the lexicographic algebras [14,24,25], and the Z-analytic algebras [16,18], as well as non-analytic algebras [8,31]. We include the necessary definitions below, but refer the reader to [3,5] for further details on order preservation and TAF algebras generated by their order preserving normalizers.…”

Section: Algebraic Isomorphisms Between Order Preserving Taf Algebrasmentioning

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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“…0055 years and become an important source of examples of nonselfadjoint operator algebras. Although the main emphasis of the research has dealt with classifying these algebras up to isometric isomorphism [2,5,8,16,18,20,23,24], analytic algebras [17,21,29,30,33,34], ideals [4,10,22], reflexivity [19], representation theory [15], and other topics [7,9,26,28] have also received attention. A closely related family of algebras are subalgebras of groupoid C*-algebras, as developed in [12 14].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, this isometry is a sum (but not necessarily a direct sum) of an algebra isomorphism and an algebra anti-isomorphism. This class includes many algebras studied in the literature, such as the standard limit algebras [2,16,23], the refinement limit algebras [16,22,23], the alternation limit algebras classified (independently) in [8] and [20], Z-analytic TAF algebras [17,21], and the strongly maximal non-analytic TAF algebra constructed (independently) in [10] and [30] and proved non-analytic in [30].…”

Section: Introductionmentioning

confidence: 99%

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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Classification of limits of triangular matrix algebras

Cited by 21 publications

References 10 publications

Some representations of TAF algebras

Some representations of TAF algebras

Algebraic isomorphisms of limit algebras

The Lattice of Ideals of a Triangular AF Algebra

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