scholarly journals The block Schur product is a Hadamard product

2020 ◽  
Vol 126 (3) ◽  
pp. 603-616
Author(s):  
Erik Christensen
Keyword(s):  
Fourier Coefficients ◽  
Hadamard Product ◽  
Natural Generalization ◽  
Continuous Functions ◽  
Crossed Product ◽  
Convolution Product ◽  
C Algebra ◽  
Schur Product ◽  
Crossed Product Algebra ◽  
Product Algebra

Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product. We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

scholarly journals Semigroups of local homeomorphisms and interaction groups

2007 ◽  
Vol 27 (6) ◽  
pp. 1737-1771 ◽  
Author(s):  
R. EXEL ◽  
J. RENAULT
Keyword(s):  
Compact Space ◽  
Crossed Product ◽  
C Algebra ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractGiven a semigroup of surjective local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. We also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

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 $.

FREE-PRODUCT GROUPS, CUNTZ-KRIEGER ALGEBRAS, AND COVARIANT MAPS

1991 ◽  
Vol 02 (04) ◽  
pp. 457-476 ◽  
Author(s):  
JOHN SPIELBERG
Keyword(s):  
Free Product ◽  
Compact Convex ◽  
Negative Answer ◽  
Crossed Product ◽  
Crossed Products ◽  
Explicit Computation ◽  
Completely Positive ◽  
Cyclic Groups ◽  
Crossed Product Algebra ◽  
Product Algebra

A construction is given relating a finitely generated free-product of cyclic groups with a certain Cuntz-Krieger algebra, generalizing the relation between the Choi algebra and 02. It is shown that a certain boundary action of such a group yields a Cuntz-Krieger algebra by the crossed-product construction. Certain compact convex spaces of completely positive mappings associated to a crossed-product algebra are introduced. These are used to generalize a problem of J. Anderson regarding the representation theory of the Choi algebra. An explicit computation of these spaces for the crossed products under study yields a negative answer to this problem.

scholarly journals DENSE m-CONVEX FRÉCHET SUBALGEBRAS OF OPERATOR ALGEBRA CROSSED PRODUCTS BY LIE GROUPS

1993 ◽  
Vol 04 (04) ◽  
pp. 601-673 ◽  
Author(s):  
LARRY B. SCHWEITZER
Keyword(s):  
Continuous Functions ◽  
Crossed Product ◽  
C Algebra ◽  
Schwartz Functions ◽  
Differentiable Functions ◽  
Dense Subalgebra ◽  
Fréchet Algebra ◽  
Locally Convex ◽  
Locally Convex Algebra ◽  
Projective Completion

Let A be a dense Fréchet *-subalgebra of a C*-algebra B. (We do not require Fréchet algebras to be m-convex.) Let G be a Lie group, not necessarily connected, which acts on both A and B by *-automorphisms, and let σ be a sub-polynomial function from G to the nonnegative real numbers. If σ and the action of G on A satisfy certain simple properties, we define a dense Fréchet *-subalgebra G ⋊σ A of the crossed product L1 (G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in σ. We give conditions on σ and the action of G on A which imply the m-convexity of the dense subalgebra G ⋊σ A. A locally convex algebra is said to be m-convex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Fréchet algebra, and is useful in modern operator theory. If G acts as a transformation group on a locally compact space M, we develop a class of dense subalgebras for the crossed product L1 (G, C0 (M)), where C0 (M) denotes the continuous functions on M vanishing at infinity with the sup norm topology. We define Schwartz functions S (M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense Fréchet *-subalgebra G ⋊σ S (M) of rapidly vanishing, G-differentiable functions from G to S (M). If the reciprocal of σ is in Lp (G) for some p, we prove that our group algebras Sσ (G) are nuclear Fréchet spaces, and that G ⋊σ A is the projective completion [Formula: see text].

CROSSED PRODUCTS OF CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

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.

Isomorphism Classes of Solenoidal Algebras I

1993 ◽  
Vol 36 (4) ◽  
pp. 414-418 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Dynamical System ◽  
Growth Rate ◽  
Unit Circle ◽  
Compact Space ◽  
Periodic Points ◽  
Crossed Product ◽  
Roots Of Unity ◽  
Isomorphism Classes ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractEach g ∊ ℤ[x] defines a homeomorphism of a compact space We investigate the isomorphism classes of the C*-crossed product algebra Bg associated with the dynamical system An isomorphism invariant Eg of the algebra Bg is shown to determine the algebra Bg up to * or * anti-isomorphism if degree g ≤ 1 and 1 is not a root of g or if degree g = 2 and g is irreducible. It is also observed that the entropy of the dynamical system is equal to the growth rate of the periodic points if g has no roots of unity as zeros. This slightly extends the previously known equality of these two quantities under the assumption that g has no zeros on the unit circle.

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.

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