The structure of free smigroup algebras

Davidson, Kenneth R.; Katsoulis, Elias G.; Pitts, David R.

doi:10.1515/crll.2001.028

Cited by 60 publications

(124 citation statements)

References 21 publications

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“…, E k−1,k−1 provide one-dimensional 'anchor' subspaces which generate the irreducible subspaces associated with the irreducible subrepresentations. Further, some thought shows that the free semigroup algebra S (the wot (weak operator topology)-closed algebra generated by the isometries from the dilation [9][10][11]) associated with each subrepresentation is unitarily equivalent to the tractable 'one-dimensional atomic' free semigroup algebra arising in the literature [9]. Thus the free semigroup algebra of the full representation is unitarily equivalent to the direct sum of k − 1 copies of this algebra.…”

Section: Example 33mentioning

confidence: 99%

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Kribs¹

2003

Proceedings of the Edinburgh Mathematical Society

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We show that the representations of the Cuntz C * -algebras On which arise in wavelet analysis and dilation theory can be classified through a simple analysis of completely positive maps on finitedimensional space. Based on this analysis, we find an application in quantum information theory; namely, a structure theorem for the fixed-point set of a unital quantum channel. We also include some open problems motivated by this work. There has been considerable recent interest in the analysis of completely positive maps on finite-dimensional space. There are a number of reasons for this including connections with wavelet analysis [3,4,15], dilation theory [11,16], representation theory of the Cuntz C * -algebras O n [3,4,10,11], and quantum information theory [1,17,21,22]. The results obtained in the current paper have implications for each of these areas. In presenting this work, another goal we have is to push further the connections between the various areas mentioned above.In § 1 we establish a result for completely positive maps. While we focus on the finitedimensional setting, this is not necessary in the proof. A structure theorem for the fixedpoint set of a unital quantum channel is contained in § 2. In particular, we prove that the fixed-point set is a C * -algebra which is equal to the commutant of the algebra generated by any choice of row contraction which determines the channel. We discuss the twodimensional channels [17,21], and use the theorem to classify them by their fixed-point sets. The representation theory for O n is considered in § 3. We focus on a subclass of representations arising in dilation theory and wavelet analysis [3, 4, 10, 11, 15]. Each of these representations determines a completely positive map on finite-dimensional space. We ask if these representations can be classified just by examining the map. An affirmative answer is provided by the result on completely positive maps from the first section. Finally, in § 4 we pose some open questions which are motivated by this work.

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Section: Example 33mentioning

confidence: 99%

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Kribs¹

2003

Proceedings of the Edinburgh Mathematical Society

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“…(ii) The noncommutative analytic Toeplitz algebras L n , n ≥ 2 [2,9,10,11,20,25,26], the fundamental examples of free semigroup algebras, arise from the graphs with a single vertex and n distinct loop edges. For instance, in the case n = 2 with loop edges e = xex = f = xf x, the Hilbert space is identified with unrestricted 2-variable Fock space H 2 .…”

Section: Examples 12 (I)mentioning

confidence: 99%

Partly Free Algebras From Directed Graphs

Kribs

Power

2004

Current Trends in Operator Theory and Its Applications

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We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras. These are the weak operator topology closed algebras generated by the left regular representations of semigroupoids associated with finite or countable directed graphs. We expand our analysis of partly free algebras from previous work and obtain a graph-theoretic characterization of when a free semigroupoid algebra with countable graph is partly free. This analysis carries over to norm closed quiver algebras. We also discuss new examples for the countable graph case.Every finite or countable directed graph G recursively generates a Fock space Hilbert space and a family of partial isometries. These operators are of Cuntz-Krieger-Toeplitz type and also arise through the left regular representations of free semigroupoids determined by directed graphs. This was initially discovered by Muhly [23], and, in the case of finite graphs, more recent work with Solel [24] considered the norm closed algebras generated by these representations, which they called quiver algebras. In [19], we developed a structure theory for the weak operator topology closed algebras L G generated by the left regular representations coming from both finite and countable directed graphs; we called these algebras free semigroupoid algebras. In doing so, we found a unifying framework for a number of classes of algebras which appear in the literature, including; noncommutative analytic Toeplitz algebras L n [2,10,11,20,25,26] (the prototypical free semigroup algebras), the classical analytic Toeplitz algebra H ∞ [14,16], and certain finite dimensional digraph algebras [19]. But this approach gives rise to a diverse collection of new examples which include finite dimensional algebras, algebras with free behaviour, algebras which can be represented as matrix function algebras, and examples which mix these possibilities. The general theme of our work in [19] was a marriage of simple graph-theoretic properties with properties of the operator algebra. Furthermore, our technical analyses were chiefly spatial in nature; for instance, we proved the graph is a complete unitary invariant of both the free semigroupoid algebra and the quiver algebra.In the next section we give a short introduction to free semigroupoid algebras and discuss a number of examples. The second section contains an expanded analysis of a first author partially supported by a Canadian NSERC Post-doctoral Fellowship.

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“…Hence ltsr(L n ) = ∞. But by the Structure Theorem for free semigroup algebras mentioned above [7], either there is a homomorphism of S n onto L n or S n is a von Neumann algebra containing two isometries with orthogonal ranges. Either way, rtsr(S n ) = ltsr(S n ) = ∞.…”

Section: Non-commutative Operator Algebras Generated By Isometriesmentioning

confidence: 95%

“…A theorem of Davidson, Katsoulis, and Pitts [7] shows that if S n is a free semigroup algebra, then there exists a projection P ∈ S n such that S = MP ⊕ P ⊥ MP ⊥ , where M is the von Neumann algebra generated by S n , and SP ⊥ = P ⊥ SP ⊥ is completely isometrically isomorphic to L n .…”

Section: Non-commutative Operator Algebras Generated By Isometriesmentioning

confidence: 99%

On the topological stable rank of non-selfadjoint operator algebras

et al. 2008

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Abstract. We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.

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The structure of free smigroup algebras

Cited by 60 publications

References 21 publications

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

QUANTUM CHANNELS, WAVELETS, DILATIONS AND REPRESENTATIONS OF $\mathcal{O}_{n}$

Partly Free Algebras From Directed Graphs

On the topological stable rank of non-selfadjoint operator algebras

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