scholarly journals TOPOLOGICAL STABLE RANK OF INCLUSIONS OF UNITAL C*-ALGEBRAS

2006 ◽  
Vol 17 (01) ◽  
pp. 19-34 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
TAMOTSU TERUYA
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Finite Type ◽  
Stable Rank ◽  
C Algebras ◽  
Rank One ◽  
Topological Stable Rank ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr (B)) when A has topological stable rank one. We show that tsr (B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr (D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊α G has topological stable rank less than or equal to two. These results are affirmative data to a generalization of a question by Blackadar in 1988.

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.

scholarly journals STABLE RANK FOR INCLUSIONS OF C*-ALGEBRAS

2008 ◽  
Vol 19 (09) ◽  
pp. 1011-1020 ◽  
Author(s):  
HIROYUKI OSAKA
Keyword(s):  
Finite Groups ◽  
Stable Rank ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Common Unit ◽  
Topological Stable Rank

When a unital C*-algebra A has topological stable rank one (write tsr (A) = 1), we know that tsr (pAp) = 1 for a non-zero projection p ∈ A. When, however, tsr (A) ≥ 2, it is generally false. We prove that if a unital C*-algebra A has a simple unital C*-subalgebra D of A with common unit such that D has Property (SP) and sup p ∈ P(D) tsr (pAp) < ∞, then tsr (A) ≤ 2. As an application let A be a simple unital C*-algebra with tsr (A) = 1 and Property (SP), [Formula: see text] finite groups, αk actions from Gk to Aut ((⋯((A × α1 G1) ×α2 G2)⋯) ×αk-1 Gk-1). (G0 = {1}). Then [Formula: see text]

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 $.

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.

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.

Elementary operators and subhomogeneous C*-algebras

2010 ◽  
Vol 54 (1) ◽  
pp. 99-111 ◽  
Author(s):  
Ilja Gogić
Keyword(s):  
Tensor Product ◽  
Finite Type ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
C Algebras ◽  
Direct Sum ◽  
Elementary Operators ◽  
Uniformly Bounded

AbstractLet A be a C*-algebra and let ΘA be the canonical contraction form the Haagerup tensor product of M(A) with itself to the space of completely bounded maps on A. In this paper we consider the following conditions on A: (a) A is a finitely generated module over the centre of M(A); (b) the image of ΘA is equal to the set E(A) of all elementary operators on A; and (c) the lengths of elementary operators on A are uniformly bounded. We show that A satisfies (a) if and only if it is a finite direct sum of unital homogeneous C*-algebras. We also show that if a separable A satisfies (b) or (c), then A is necessarily subhomogeneous and the C*-bundles corresponding to the homogeneous subquotients of A must be of finite type.

scholarly journals PERIODIC ORBITS OF A DYNAMICAL SYSTEM RELATED TO A KNOT

2011 ◽  
Vol 20 (03) ◽  
pp. 411-426 ◽  
Author(s):  
LILYA LYUBICH
Keyword(s):  
Dynamical System ◽  
Finite Group ◽  
Finite Type ◽  
Periodic Orbits ◽  
Weak Topology ◽  
Commutator Subgroup ◽  
Alexander Polynomial ◽  
Shift Of Finite Type ◽  
A Finite Group

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom (K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σx of Hom (K, Σ) onto itself: σxρ(a) = ρ(xax-1) ∀ a ∈ K, ρ ∈ Hom (K, Σ). As proven in [5], the dynamical system ( Hom (K, Σ), σx) is a shift of finite type. In the case when Σ is abelian, Hom (K, Σ) is finite. In this paper we calculate the periods of orbits of ( Hom (K, ℤ/p), σx), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed

1997 ◽  
Vol 17 (6) ◽  
pp. 1359-1370
Author(s):  
CARLOS E. DURÁN
Keyword(s):  
Finite Group ◽  
Symmetric Space ◽  
Geodesic Flow ◽  
Complete Integrability ◽  
Integrals Of Motion ◽  
Completely Integrable ◽  
Rank One ◽  
A Finite Group

We show that the geodesic flow of a metric all of whose geodesics are closed is completely integrable, with tame integrals of motion. Applications to classical examples are given; in particular, it is shown that the geodesic flow of any quotient $M/\Gamma$ of a compact, rank one symmetric space $M$ by a finite group acting freely by isometries is completely integrable by tame integrals.

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