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1
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Finite generation in C*-algebras and Hilbert C*-modules

100%
Blecher D., Kania T.
Studia Mathematica
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2014
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tom 224
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nr 2
143-151
EN
We characterize C*-algebras and C*-modules such that every maximal right ideal (resp. right submodule) is algebraically finitely generated. In particular, C*-algebras satisfy the Dales-Żelazko conjecture.
2
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Order theory and interpolation in operator algebras

100%
Blecher D., Read C.
Studia Mathematica
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2014
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tom 225
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nr 1
61-95
EN
In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator algebra and the C*-algebra it generates. In much of this it is not necessary that the algebra have an approximate identity. Many of our results apply immediately to function algebras, but we will not take the time to point these out, although most of these applications seem new.
3
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Open projections in operator algebras II: Compact projections

100%
Blecher D., Neal M.
Studia Mathematica
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2012
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tom 209
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nr 3
203-224
EN
We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of 'peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is weak* closed.
4
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Outers for noncommutative $H^{p}$ revisited

100%
Blecher D., Labuschagne L.
Studia Mathematica
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2013
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tom 217
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nr 3
265-287
EN
We continue our study of outer elements of the noncommutative $H^{p}$ spaces associated with Arveson's subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in $H^{p}$ actually satisfy the stronger condition that there exist aₙ ∈ A with haₙ ∈ Ball(A) and haₙ → 1 in p-norm.
5
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Open projections in operator algebras I: Comparison theory

100%
Blecher D., Neal M.
Studia Mathematica
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2012
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tom 208
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nr 2
117-150
EN
We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rørdam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the authors and Hay), and the open partial isometries (tripotents) introduced by the authors.
6
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Open partial isometries and positivity in operator spaces

100%
Blecher D., Neal M.
Studia Mathematica
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2007
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tom 182
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nr 3
227-262
EN
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.
7
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Noncommutative function theory and unique extensions

100%
Blecher D., Labuschagne L.
Studia Mathematica
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2007
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tom 178
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nr 2
177-195
EN
We generalize, to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő $L^{p}$-distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of $H^{∞}$ from the 1960's. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary condition for every completely contractive homomorphism on a unital subalgebra of a C*-algebra to have a unique completely positive extension.
8
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On a generalization of W*-modules

100%
Blecher D., Kraus J.
Banach Center Publications
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2010
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tom 91
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nr 1
77-86
EN
a recent paper of the first author and Kashyap, a new class of Banach modules over dual operator algebras is introduced. These generalize the W*-modules (that is, Hilbert C*-modules over a von Neumann algebra which satisfy an analogue of the Riesz representation theorem for Hilbert spaces), which in turn generalize Hilbert spaces. In the present paper, we describe these modules, giving some motivation, and we prove several new results about them.
9
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Ideals and hereditary subalgebras in operator algebras

81%
Almus M., Blecher D., Read C.
Studia Mathematica
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2012
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tom 212
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nr 1
65-93
EN
This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach-algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (or HSA's, which are in some sense a generalization of ideals). Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which are 'weakly compact'. We also give several examples answering natural questions that arise in such an investigation.
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