Subproduct Systems and Cartesian Systems: New Results on Factorial Languages and their Relations with Other Areas

Gerhold, Malte; Skeide, Michael

doi:10.31390/josa.1.4.05

Cited by 4 publications

(3 citation statements)

References 38 publications

(56 reference statements)

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“…As in Remark 2.1(a), one can use Arveson subproduct systems of finite dimensional Hilbert spaces to construct Tsirelson subproduct systems of finite dimensional Hilbert spaces, thus producing C * -subproduct systems of matrix algebras from finite-dimensional Arveson subproduct systems. We note that a complete classification of Arveson subproduct systems of 2dimensional Hilbert spaces was obtained by B. Tsirelson in [21,22] (see also [9] for related results and applications).…”

Section: Background: Definitions and Examplesmentioning

confidence: 88%

$C^*$-subproduct and product systems

Floricel¹,

Ketelboeter²

2022

Preprint

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We introduce and study two-parameter subproduct and product systems of C * -algebras as the operator-algebraic analogues of, and in relation to, Tsirelson's two-parameter product systems of Hilbert spaces. Using inductive limit techniques and related arguments, we show that (i) any C * -subproduct system can be embedded into a C *product system, called the inductive dilation of the C * -subproduct system; and (ii) any unital C * -subproduct system can be assembled into a quasi-local C * -bialgebra, whose co-multiplication is constructed from the co-multiplication of the C * -subproduct system, and from its inductive dilation. We also discuss co-multiplicative families of states of C *subproduct systems and show that they correspond to idempotent states of the associated quasi-local C * -bialgebras. We then use the GNS construction to obtain Tsirelson subproduct systems of Hilbert spaces from such co-multiplicative families of states, and describe the relationship between the inductive dilation of a unital C * -suproduct system and the Bhat dilation of the Tsirelson subproduct system of Hilbert spaces associated with a co-multiplicative family of states. All these results are illustrated concretely at the level of C * -subproduct systems of commutative C * -algebras.

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Section: Background: Definitions and Examplesmentioning

confidence: 88%

$C^*$-subproduct and product systems

Floricel¹,

Ketelboeter²

2022

Preprint

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“…So it follows from the previous paragraph that (F(H n )) n∈N 0 yields a standard graded algebra if (H n ) n∈N 0 is a subproduct system. Similar considerations play an important role in [19] where dimension sequences of finite-dimensional subproduct systems are investigated.…”

Section: (Co)monoidal Systems In Nonprobabilistic Categoriesmentioning

confidence: 92%

Categorial Independence and Lévy Processes

Gerhold¹,

Lachs²,

Schürmann³

2022

SIGMA

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We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.

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“…[21, Corollary 37]). In particular, our sequences are an example of cardinality sequences of word systems: due to [18, Proposition 3.2], for every finite‐dimensional subproduct systems of Hilbert spaces

{false{H_{m} false}}_{m \in N_{0}}

, there exists a word system

{false{X_{m} false}}_{m \in N_{0}}

such that

dim false(H_{m} false) = false| X_{m} false|

for all

m \in {double-struckN}_{0}

(see also [3, Lemma 1.1] for a noncommutative algebraic version of this claim). However, the subproduct system associated to the word system described above is, in general, not isomorphic to the original one.…”

Section: Fusion Rules For An Su(2)‐equivariant Subproduct Systemmentioning

confidence: 99%

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Arici

Kaad

2021

Trans London Maths Soc

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Motivated by the study of symmetries of C∗‐algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)‐equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz–Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K‐theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK‐equivalent to the algebra of complex numbers. In this way, we obtain a six‐term exact sequence of K‐groups containing a noncommutative analogue of the Euler class.

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Subproduct Systems and Cartesian Systems: New Results on Factorial Languages and their Relations with Other Areas

Cited by 4 publications

References 38 publications

$C^*$-subproduct and product systems

$C^*$-subproduct and product systems

Categorial Independence and Lévy Processes

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

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