scholarly journals QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450019 ◽  
Author(s):  
RALF MEYER ◽  
SUTANU ROY ◽  
STANISŁAW LECH WORONOWICZ
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Products ◽  
Group Representations ◽  
Basic Properties ◽  
C Algebras ◽  
Twisted Tensor Product

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.

scholarly journals The maximal quantum group-twisted tensor product of C*-algebras

2018 ◽  
Vol 12 (1) ◽  
pp. 279-330 ◽  
Author(s):  
Sutanu Roy ◽  
Thomas Timmermann
Keyword(s):  
Tensor Product ◽  
Quantum Group ◽  
C Algebras ◽  
Twisted Tensor Product

Cartan's constructions and the twisted Eilenberg–Zilber theorem

2010 ◽  
Vol 17 (1) ◽  
pp. 13-23
Author(s):  
Víctor Álvarez ◽  
José Andrés Armario ◽  
María Dolores Frau ◽  
Pedro Real
Keyword(s):  
Perturbation Theory ◽  
Tensor Product ◽  
Cartesian Product ◽  
Tensor Products ◽  
Standard Product ◽  
Homological Perturbation Theory ◽  
Twisted Tensor Product ◽  
Homological Perturbation ◽  
Twisted Cartesian Product

Abstract Let 𝐺 × τ 𝐺′ be the principal twisted Cartesian product with fibre 𝐺, base 𝐺 and twisting function where 𝐺 and 𝐺′ are simplicial groups as well as 𝐺 × τ 𝐺′; and 𝐶𝑁(𝐺) ⊗𝑡 𝐶𝑁 (𝐺′) be the twisted tensor product associated to 𝐶𝑁 (𝐺 × τ 𝐺′) by the twisted Eilenberg–Zilber theorem. Here we prove that the pair 𝐶𝑁(𝐺) ⊗𝑡 𝐶𝑁(𝐺′), μ) is a multiplicative Cartan's construction where μ is the standard product on 𝐶𝑁(𝐺) ⊗ 𝐶𝑁(𝐺′). Furthermore, assuming that a contraction from 𝐶𝑁(𝐺′) to 𝐻𝐺′ exists and using the techniques from homological perturbation theory, we extend the former result to other “twisted” tensor products of the form 𝐶𝑁(𝐺) ⊗ 𝐻𝐺′.

scholarly journals P-Tensor Product for Group C*-Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 627
Author(s):  
Yufang Li ◽  
Zhe Dong
Keyword(s):  
Tensor Product ◽  
Free Group ◽  
Discrete Group ◽  
Tensor Products ◽  
Haagerup Property ◽  
C Algebras ◽  
P Tensor

In this paper, we introduce new tensor products ⊗ p ( 1 ≤ p ≤ + ∞ ) on C ℓ p * ( Γ ) ⊗ C ℓ p * ( Γ ) and ⊗ c 0 on C c 0 * ( Γ ) ⊗ C c 0 * ( Γ ) for any discrete group Γ . We obtain that for 1 ≤ p < + ∞ C ℓ p * ( Γ ) ⊗ m a x C ℓ p * ( Γ ) = C ℓ p * ( Γ ) ⊗ p C ℓ p * ( Γ ) if and only if Γ is amenable; C c 0 * ( Γ ) ⊗ m a x C c 0 * ( Γ ) = C c 0 * ( Γ ) ⊗ c 0 C c 0 * ( Γ ) if and only if Γ has Haagerup property. In particular, for the free group with two generators F 2 we show that C ℓ p * ( F 2 ) ⊗ p C ℓ p * ( F 2 ) ≇ C ℓ q * ( F 2 ) ⊗ q C ℓ q * ( F 2 ) for 2 ≤ q < p ≤ + ∞ .

Tensor products which do not preserve operator algebras

1990 ◽  
Vol 108 (2) ◽  
pp. 395-403 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Operator Algebras ◽  
Tensor Products ◽  
Preserve Operator ◽  
Infinite Dimensional ◽  
Hilbert Space Operators ◽  
Algebra Of Operators

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).

Property T for locally compact quantum groups

2015 ◽  
Vol 26 (03) ◽  
pp. 1550024 ◽  
Author(s):  
Xiao Chen ◽  
Chi-Keung Ng
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Short Paper ◽  
Locally Compact ◽  
C Algebras ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Compact Quantum Groups ◽  
Dimensional Irreducible Representation ◽  
Property T

In this short paper, we obtained some equivalent formulations of property T for a general locally compact quantum group 𝔾, in terms of the full quantum group C*-algebras [Formula: see text] and the *-representation of [Formula: see text] associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if 𝔾 is of Kac type, we show that 𝔾 has property T if and only if every finite-dimensional irreducible *-representation of [Formula: see text] is an isolated point in the spectrum of [Formula: see text] (this also generalizes the corresponding locally compact group result). In addition, we give a way to construct property T discrete quantum groups using bicrossed products.

Finite Lattices of Projections in Factors and Approximately Finite C*-Algebras

1992 ◽  
Vol 35 (1) ◽  
pp. 116-125
Author(s):  
S. C. Power
Keyword(s):  
Tensor Product ◽  
General Result ◽  
Finite Lattice ◽  
Tensor Products ◽  
C Algebras ◽  
Finite Lattices

AbstractA unique factorisation theorem is obtained for tensor products of finite lattices of commuting projections in a factor. This leads to unique tensor product factorisations for reflexive subalgebras of the hyperfinite II1 factor which have irreducible finite commutative invariant projection lattices. It is shown that the finite refinement property fails for simple approximately finite C*-algebras, and this implies that there is no analogous general result for finite lattice subalgebras in this context.

scholarly journals CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

2020 ◽  
pp. 1-12
Author(s):  
BHARAT TALWAR ◽  
RANJANA JAIN
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hausdorff Space ◽  
Separable Hilbert Space ◽  
Locally Compact ◽  
C Algebras ◽  
Finite Sum ◽  
Locally Compact Hausdorff Space ◽  
Quotient Property

Abstract For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

scholarly journals The similarity problem for tensor products of certain C*-algebras

2004 ◽  
Vol 70 (3) ◽  
pp. 385-389 ◽  
Author(s):  
Florin Pop
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
C Algebras ◽  
Similarity Problem ◽  
Bounded Representation

We prove that every bounded representation of the tensor product of two C*-algebras, one of which is nuclear and contains matrices of any order, is similar to a *-representation.

scholarly journals The “maximal” tensor product of operator spaces

1999 ◽  
Vol 42 (2) ◽  
pp. 267-284 ◽  
Author(s):  
Timur Oikhberg ◽  
Gilles Pisier
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Tensor Products ◽  
Operator Space ◽  
Operator Spaces ◽  
Two Factors

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

scholarly journals ON ITERATED TWISTED TENSOR PRODUCTS OF ALGEBRAS

2008 ◽  
Vol 19 (09) ◽  
pp. 1053-1101 ◽  
Author(s):  
PASCUAL JARA MARTÍNEZ ◽  
JAVIER LÓPEZ PEÑA ◽  
FLORIN PANAITE ◽  
FREDDY VAN OYSTAEYEN
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Noncommutative Geometry ◽  
Differential Forms ◽  
Tensor Products ◽  
Number Of Factors ◽  
Algebra Theory ◽  
Twisted Tensor Product

We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We find conditions for constructing an iterated product of three factors and prove that they are enough for building an iterated product of any number of factors. As an example of the geometrical aspects of our construction, we show how to construct differential forms and involutions on iterated products starting from the corresponding structures on the factors and give some examples of algebras that can be described within our theory. We prove a certain result (called "invariance under twisting") for a twisted tensor product of two algebras, stating that the twisted tensor product does not change when we apply certain kind of deformation. Under certain conditions, this invariance can be iterated, containing as particular cases a number of independent and previously unrelated results from Hopf algebra theory.

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