scholarly journals The structure of the observable algebra determined by a Hopf *-subalgebra in Hopf spin models

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 485-500
Author(s):  
Xiaomin Wei ◽  
Lining Jiang ◽  
Qiaoling Xin
Keyword(s):  
Tensor Product ◽  
Lattice Site ◽  
Inductive Limit ◽  
Spin Models ◽  
Hopf Subalgebra ◽  
C Algebra ◽  
Finite Algebras ◽  
Finite Dimensional ◽  
Observable Algebra ◽  
Twisted Tensor Product

Let H be a finite dimensional Hopf C*-algebra, H1 a Hopf*-subalgebra of H. This paper focuses on the observable algebra AH1 determined by H1 in nonequilibrium Hopf spin models, in which there is a copy of H1 on each lattice site, and a copy of ? on each link, where ? denotes the dual of H. Furthermore, using the iterated twisted tensor product of finite +*-algebras, one can prove that the observable algebraAH1 is *-isomorphic to the C*-inductive limit ... o H1 o ? o H1 o ? o H1 o ... .

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

Some C*-Algebras with Outer Derivations, II

1974 ◽  
Vol 26 (1) ◽  
pp. 185-189 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Tensor Product ◽  
Finite Number ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Direct Sum ◽  
Finite Dimensional

In this paper we shall consider the class of C*-algebras which are inductive limits of sequences of finite-dimensional C*-algebras. We shall give a complete description of those C*-algebras in this class every derivation of which is inner.Theorem. Let A be a C*-algebra. Suppose that A is the inductive limit of a sequence of finite-dimensional C*-algebras. Then the following statements are equivalent:(i) every derivation of A is inner;(ii) A is the direct sum of a finite number of algebras each of which is either commutative, the tensor product of a finite-dimensional and a commutative with unit, or simple with unit.

scholarly journals $K$-Continuity Is Equivalent To $K$-Exactness

2016 ◽  
Vol 118 (1) ◽  
pp. 95
Author(s):  
Otgonbayar Uuye
Keyword(s):  
Tensor Product ◽  
Approximation Theory ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation

Let $A$ be a $C^{*}$-algebra. It is well known that the functor $B \mapsto A \otimes B$ of taking the minimal tensor product with $A$ preserves inductive limits if and only if it is exact. $C^{*}$-algebras with this property play an important role in the structure and finite-dimensional approximation theory of $C^{*}$-algebras. We consider a $K$-theoretic analogue of this result and show that the functor $B \mapsto K_{0}(A \otimes B)$ preserves inductive limits if and only if it is half-exact.

scholarly journals Jones Type Basic Construction on Hopf Spin Models

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1547
Author(s):  
Cao Tianqing ◽  
Xin Qiaoling ◽  
Wei Xiaomin ◽  
Jiang Lining
Keyword(s):  
Hopf Algebra ◽  
Crossed Product ◽  
Spin Models ◽  
Drinfeld Double ◽  
Field Algebra ◽  
Basic Construction ◽  
Finite Dimensional ◽  
Observable Algebra

Let H be a finite dimensional C∗-Hopf algebra and A the observable algebra of Hopf spin models. For some coaction of the Drinfeld double D(H) on A, the crossed product A⋊D(H)^ can define the field algebra F of Hopf spin models. In the paper, we study C∗-basic construction for the inclusion A⊆F on Hopf spin models. To achieve this, we define the action α:D(H)×F→F, and then construct the resulting crossed product F⋊D(H), which is isomorphic A⊗End(D(H)^). Furthermore, we prove that the C∗-basic construction for A⊆F is consistent to F⋊D(H), which yields that the C∗-basic constructions for the inclusion A⊆F is independent of the choice of the coaction of D(H) on A.

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

2020 ◽  
pp. 1-24
Author(s):  
KONRAD AGUILAR
Keyword(s):  
Inductive Limit ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Ideal Space ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Af Algebras ◽  
The Ideal

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

scholarly journals Classification of certain inductive limit actions of compact groups on af algebras

2020 ◽  
pp. 2050128
Author(s):  
Qingyun Wang
Keyword(s):  
Compact Group ◽  
Inductive Limit ◽  
Complete Classification ◽  
Compact Groups ◽  
C Algebra ◽  
Finite Dimensional ◽  
K Theory ◽  
Af Algebras

Let [Formula: see text] be an AF algebra, [Formula: see text] be a compact group. We consider inductive limit actions of the form [Formula: see text], where [Formula: see text] is an action on the finite-dimensional C*-algebra [Formula: see text] which fixes each matrix summand. We give a complete classification up to conjugacy of such actions using twisted equivariant K-theory.

scholarly journals DIAGONALS IN TENSOR PRODUCTS OF OPERATOR ALGEBRAS

2002 ◽  
Vol 45 (3) ◽  
pp. 647-652 ◽  
Author(s):  
Vern I. Paulsen ◽  
Roger R. Smith
Keyword(s):  
Tensor Product ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Operator Algebras ◽  
Direct Proof ◽  
Secondary 46L05 ◽  
C Algebra ◽  
Finite Dimensional ◽  
Hochschild Cohomology Group ◽  
Primary 46L06

AbstractIn this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra $A$ possesses a diagonal in the Haagerup tensor product of $A$ with itself, then $A$ must be isomorphic to a finite-dimensional $C^*$-algebra. Consequently, for operator algebras, the first Hochschild cohomology group $H^1(A,X)=0$ for every bounded, Banach $A$-bimodule $X$, if and only if $A$ is isomorphic to a finite-dimensional $C^*$-algebra.AMS 2000 Mathematics subject classification: Primary 46L06. Secondary 46L05

scholarly journals DIRECTLY FINITE ALGEBRAS OF PSEUDOFUNCTIONS ON LOCALLY COMPACT GROUPS

2014 ◽  
Vol 57 (3) ◽  
pp. 693-707
Author(s):  
YEMON CHOI
Keyword(s):  
Invertible Element ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Complex Line ◽  
Locally Compact ◽  
The Real ◽  
C Algebra ◽  
Finite Algebras ◽  
Unimodular Group ◽  
Reduced Group

AbstractAn algebraAis said to be directly finite if each left-invertible element in the (conditional) unitization ofAis right invertible. We show that the reduced group C*-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras ofp-pseudofunctions, showing that these algebras are directly finite ifGis amenable and unimodular, or unimodular with the Kunze–Stein property. An exposition is also given of how existing results from the literature imply thatL1(G) is not directly finite whenGis the affine group of either the real or complex line.

scholarly journals FINITE-DIMENSIONAL ORDERED VECTOR SPACES WITH RIESZ INTERPOLATION AND EFFROS–SHEN’S UNIMODULARITY CONJECTURE

2016 ◽  
Vol 101 (2) ◽  
pp. 277-287
Author(s):  
AARON TIKUISIS
Keyword(s):  
Vector Space ◽  
Inductive Limit ◽  
Space Structure ◽  
Vector Spaces ◽  
Ordered Vector Space ◽  
Finite Dimensional ◽  
Ordered Vector Spaces

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$, any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$.

scholarly journals Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1737
Author(s):  
Mariia Myronova ◽  
Jiří Patera ◽  
Marzena Szajewska
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Geometrical Structures ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Branching Rules ◽  
Definition Of ◽  
Representation Orbit

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified.

Sign in / Sign up

Export Citation Format

Share Document