scholarly journals C⁎-Basic Construction from the Conditional Expectation on the Drinfeld Double

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Qiaoling Xin ◽  
Lining Jiang ◽  
Tianqing Cao
Keyword(s):  
Finite Group ◽  
Conditional Expectation ◽  
Crossed Product ◽  
Drinfeld Double ◽  
Basic Construction ◽  
A Finite Group

Let D(G) be the Drinfeld double of a finite group G and D(G;H) be the crossed product of C(G) and CH, where H is a subgroup of G. Then the sets D(G) and D(G;H) can be made C⁎-algebras naturally. Considering the C⁎-basic construction C⁎〈D(G),e〉 from the conditional expectation E of D(G) onto D(G;H), one can construct a crossed product C⁎-algebra C(G/H×G)⋊CG, such that the C⁎-basic construction C⁎〈D(G),e〉 is C⁎-algebra isomorphic to C(G/H×G)⋊CG.

scholarly journals A CHARACTERIZATION OF SATURATED C*-ALGEBRAIC BUNDLES OVER FINITE GROUPS

2010 ◽  
Vol 88 (3) ◽  
pp. 363-383
Author(s):  
KAZUNORI KODAKA ◽  
TAMOTSU TERUYA
Keyword(s):  
Finite Group ◽  
Finite Type ◽  
Finite Groups ◽  
Conditional Expectation ◽  
Crossed Product ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet A be a unital C*-algebra. Let (B,E) be a pair consisting of a unital C*-algebra B containing A as a C*-subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B1 be the C*-basic construction induced by (B,E). In this paper, we shall show that any such pair (B,E) satisfying the conditions that A′∩B=ℂ1 and that A′∩B1 is commutative is constructed by a saturated C*-algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A′∩B1 is commutative.

scholarly journals The Linear Span of Projections in AH Algebras and for Inclusions ofC*-Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Dinh Trung Hoa ◽  
Toan Minh Ho ◽  
Hiroyuki Osaka
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Conditional Expectation ◽  
Linear Span ◽  
Crossed Product ◽  
Car Algebra ◽  
Crossed Product Algebra ◽  
Fixed Point Algebra ◽  
A Finite Group ◽  
Product Algebra

In the first part of this paper, we show that an AH algebraA=lim→(Ai,ϕi)has the LP property if and only if every element of the centre ofAibelongs to the closure of the linear span of projections inA. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unitalC*-algebrasP⊂Awith a finite Watatani index, if a faithful conditional expectationE:A→Phas the Rokhlin property in the sense of Kodaka et al., thenPhas the LP property under the condition thatAhas the LP property. As an application, letAbe a simple unitalC*-algebra with the LP property,αan action of a finite groupGontoAut(A). Ifαhas the Rokhlin property in the sense of Izumi, then the fixed point algebraAGand the crossed product algebraA ⋊α Ghave the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

scholarly journals CONJUGATE PAIRS OF SUBFACTORS AND ENTROPY FOR AUTOMORPHISMS

2011 ◽  
Vol 22 (04) ◽  
pp. 577-592 ◽  
Author(s):  
MARIE CHODA
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
View Point ◽  
Jones Index ◽  
Outer Automorphisms ◽  
Outer Action ◽  
A Finite Group

Based on the fact that, for a subfactor N of a II1factor M, the first nontrivial Jones index is two and then M is decomposed as the crossed product of N by an outer action of ℤ2, we study pairs {N, uNu*} from the view-point of entropy for two subalgebras of M in connection with the entropy for automorphisms, where the inclusion of II1factors N ⊂ M is given with M being the crossed product of N by a finite group of outer automorphisms and u is a unitary in M.

Certain properties for crossed products by automorphisms with a certain non-simple tracial Rokhlin property

2012 ◽  
Vol 33 (5) ◽  
pp. 1391-1400 ◽  
Author(s):  
XIAOCHUN FANG ◽  
QINGZHAI FAN
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Infinite Dimensional ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra ◽  
Tracial Rokhlin Property

AbstractLet $\Omega $ be a class of unital $C^*$-algebras. Then any simple unital $C^*$-algebra $A\in \mathrm {TA}(\mathrm {TA}\Omega )$ is a $\mathrm {TA}\Omega $ algebra. Let $A\in \mathrm {TA}\Omega $ be an infinite-dimensional $\alpha $-simple unital $C^*$-algebra with the property SP. Suppose that $\alpha :G\to \mathrm {Aut}(A)$ is an action of a finite group $G$ on $A$ which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra $C^*(G,A,\alpha )$ belongs to $\mathrm {TA}\Omega $.

scholarly journals THE DUAL STRUCTURE OF CROSSED PRODUCT -ALGEBRAS WITH FINITE GROUPS

2013 ◽  
Vol 88 (2) ◽  
pp. 243-249 ◽  
Author(s):  
FIRUZ KAMALOV
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Orbit Space ◽  
Crossed Product ◽  
Irreducible Representations ◽  
Natural Action ◽  
Dual Structure ◽  
Projective Representations ◽  
A Finite Group ◽  
Sigma G

AbstractWe study the space of irreducible representations of a crossed product ${C}^{\ast } $-algebra ${\mathop{A\rtimes }\nolimits}_{\sigma } G$, where $G$ is a finite group. We construct a space $\widetilde {\Gamma } $ which consists of pairs of irreducible representations of $A$ and irreducible projective representations of subgroups of $G$. We show that there is a natural action of $G$ on $\widetilde {\Gamma } $ and that the orbit space $G\setminus \widetilde {\Gamma } $ corresponds bijectively to the dual of ${\mathop{A\rtimes }\nolimits}_{\sigma } G$.

scholarly journals The Jiang–Su Absorption for Inclusions of Unital C*-algebras

2018 ◽  
Vol 70 (2) ◽  
pp. 400-425 ◽  
Author(s):  
Hiroyuki Osaka ◽  
Tamotsu Teruya
Keyword(s):  
Finite Group ◽  
Conditional Expectation ◽  
C Algebra ◽  
C Algebras ◽  
Crossed Product Algebra ◽  
Fixed Point Algebra ◽  
Stably Finite ◽  
A Finite Group ◽  
Product Algebra ◽  
Tracial Rokhlin Property

AbstractWe introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital C*-algebras P ⊂ A with index finite, and show that an action α from a finite group G on a simple unital C*- algebra A has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation E: A → AG has the tracial Rokhlin property. Let be a class of infinite dimensional stably finite separable unital C*-algebras that is closed under the following conditions:(1) If A ∊ and B ≅ A, then B ∊ .(2) If A ∊ and n ∊ ℕ, then Mn(A) ∊ .(3) If A ∊ and p ∊ A is a nonzero projection, then pAp ∊ .Suppose that any C*-algebra in is weakly semiprojective. We prove that if A is a local tracial -algebra in the sense of Fan and Fang and a conditional expectation E: A → P is of index-finite type with the tracial Rokhlin property, then P is a unital local tracial -algebra.The main result is that if A is simple, separable, unital nuclear, Jiang–Su absorbing and E: A → P has the tracial Rokhlin property, then P is Jiang–Su absorbing. As an application, when an action α from a finite group G on a simple unital C*-algebra A has the tracial Rokhlin property, then for any subgroup H of G the fixed point algebra AH and the crossed product algebra H is Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup W(A) is hereditary to W(P) if A is simple, separable, exact, unital, and E: A → P has the tracial Rokhlin property.

TILTED ALGEBRAS AND CROSSED PRODUCTS

2015 ◽  
Vol 58 (3) ◽  
pp. 559-571
Author(s):  
YANAN LIN ◽  
ZHENQIANG ZHOU
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Artin Algebra ◽  
Tilted Algebra ◽  
Tilted Algebras ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra

AbstractWe consider an artin algebra A and its crossed product algebra Aα#σG, where G is a finite group with its order invertible in A. Then, we prove that A is a tilted algebra if and only if so is Aα#σG.

Sign in / Sign up

Export Citation Format

Share Document