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AF flows and continuous symmetries
, 2008
"... We consider AF flows, i.e., oneparameter automorphism groups of a unital simple AF C ∗algebra which leave invariant the dense union of an increasing sequence of finitedimensional *subalgebras, and derive two properties for these; an absence of continuous symmetry breaking and a kind of real rank ..."
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We consider AF flows, i.e., oneparameter automorphism groups of a unital simple AF C ∗algebra which leave invariant the dense union of an increasing sequence of finitedimensional *subalgebras, and derive two properties for these; an absence of continuous symmetry breaking and a kind of real rank zero property for the almost fixed points. 1
Approximate AF flows
, 2005
"... Dedicated to George A. Elliott on the occasion of his sixtieth birthday When α is a flow on a unital AF algebra A such that there is an increasing sequence (An) of finitedimensional αinvariant C ∗subalgebras of A with dense union, we call α an AF flow. We show that an approximate AF flow is a coc ..."
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Dedicated to George A. Elliott on the occasion of his sixtieth birthday When α is a flow on a unital AF algebra A such that there is an increasing sequence (An) of finitedimensional αinvariant C ∗subalgebras of A with dense union, we call α an AF flow. We show that an approximate AF flow is a cocycle perturbation of an AF flow. 1
UHF flows and the flip automorphism
, 2000
"... By a flow on a unital C ∗algebra A we mean a strongly continuous oneparameter automorphism group. The infinitesimal generator δα of a flow α is a closed derivation in A, by which we mean that δα is a closed linear operator which is defined on a dense *algebra ..."
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By a flow on a unital C ∗algebra A we mean a strongly continuous oneparameter automorphism group. The infinitesimal generator δα of a flow α is a closed derivation in A, by which we mean that δα is a closed linear operator which is defined on a dense *algebra
Rohlin flows on the Cuntz algebra O∞
, 2005
"... It is shown that certain quasifree flows on the Cuntz algebra O ∞ have the Rohlin property and therefore are cocycleconjugate with each other. This, in particular, shows that any unital separable nuclear purely infinite simple C ∗algebra has a Rohlin flow. 1 ..."
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It is shown that certain quasifree flows on the Cuntz algebra O ∞ have the Rohlin property and therefore are cocycleconjugate with each other. This, in particular, shows that any unital separable nuclear purely infinite simple C ∗algebra has a Rohlin flow. 1
Multiplier cocycles of a flow on a C∗algebra
"... We show that a multiplier cocycle of a flow on a nonunital C∗algebra can be approximated by a normcontinuous cocycle in the strict topology. As an application among others we show that a flow is approximately inner if and only if the restriction of the flow to a full invariant hereditary C∗subal ..."
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We show that a multiplier cocycle of a flow on a nonunital C∗algebra can be approximated by a normcontinuous cocycle in the strict topology. As an application among others we show that a flow is approximately inner if and only if the restriction of the flow to a full invariant hereditary C∗subalgebra is approximately inner. 1
Quasidiagonal flows
, 2008
"... We introduce two notions for flows on quasidiagonal C ∗algebras, quasidiagonal and pseudodiagonal flows; the former being apparently stronger than the latter. We derive basic facts about these flows and give various examples. In addition we extend results of Voiculescu from quasidiagonal C ∗al ..."
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We introduce two notions for flows on quasidiagonal C ∗algebras, quasidiagonal and pseudodiagonal flows; the former being apparently stronger than the latter. We derive basic facts about these flows and give various examples. In addition we extend results of Voiculescu from quasidiagonal C ∗algebras to these flows.
Flows on a separable C∗algebra
"... By a flow α on a C∗algebra A we mean a homomorphism α: R→Aut(A) such that t 7 → αt(x) is continuous for each x ∈ A, where Aut(A) is the automorphism group of A. When α is a flow, we denote by δα the generator of α, which is a closed derivation in A, i.e., δα is a closed linear map defined on a dens ..."
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By a flow α on a C∗algebra A we mean a homomorphism α: R→Aut(A) such that t 7 → αt(x) is continuous for each x ∈ A, where Aut(A) is the automorphism group of A. When α is a flow, we denote by δα the generator of α, which is a closed derivation in A, i.e., δα is a closed linear map defined on a dense ∗subalgebra D(δα) of A into A such that δα(x) ∗ = δα(x∗) and δα(xy) = δα(x)y + xδα(y) for x, y ∈ D(δα). See [3, 4, 1, 23] for characterizations of generators and more. Given h ∈ Asa, δα+ad ih is again a generator. We denote by α(h) the flow generated by δα+ad ih. We call α(h) an inner perturbation of α. More generally, if u is an αcocycle, i.e., u: R→U(A) is continuous such that usαs(ut) = us+t, s, t ∈ R, then t 7 → Adutαt is a flow, called a cocycle perturbation of α. Note that an inner perturbation is a cocycle perturbation; α(h) is obtained as Aduα, where u is the (differentiable) αcocycle defined by dut/dt = utαt(ih). In general a cocycle perturbation of α is given as t 7 → Ad vα(h)t Ad v ∗ for some v ∈ U(A) and h ∈ Asa. A (nondegenerate) representation pi of the system (A,α) is called covariant if there is a unitary flow U on the Hilbert space Hpi such that AdUtpi = piαt, t ∈ R. In general we do not seem to know a good characterization for existence of covariant irreducible representations, but in the following discussions