7,642 research outputs found

    UHF flows and the flip automorphism

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    A UHF flow is an infinite tensor product type action of the reals on a UHF algebra AA and the flip automorphism is an automorphism of AAA\otimes A sending xyx\otimes y into yxy\otimes x. If α\alpha is an inner perturbation of a UHF flow on AA, there is a sequence (un)(u_n) of unitaries in AAA\otimes A such that αtαt(un)un\alpha_t\otimes \alpha_t(u_n)-u_n converges to zero and the flip is the limit of \Ad u_n. We consider here whether the converse holds or not and solve it with an additional assumption: If AAAA\otimes A\cong A and α\alpha absorbs any UHF flow β\beta (i.e., αβ\alpha\otimes\beta is cocycle conjugate to α\alpha), then the converse holds; in this case α\alpha is what we call a universal UHF flow.Comment: 18 page

    Trace-scaling automorphisms of certain stable AF algebras

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    Trace scaling automorphisms of stable AF algebras with dimension group totally ordered are outer conjugate if the scaling factors are the same (not equal to one). This is an adaptation of a similar result for the AFD type II_infty factor by Connes and extends the previous result for stable UHF algebras.Comment: 12 pages, late

    Exploring Vacuum Structure around Identity-Based Solutions

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    We explore the vacuum structure in bosonic open string field theory expanded around an identity-based solution parameterized by a(>=1/2)a(>=-1/2). Analyzing the expanded theory using level truncation approximation up to level 20, we find that the theory has the tachyon vacuum solution for a>1/2a>-1/2. We also find that, at a=1/2a=-1/2, there exists an unstable vacuum solution in the expanded theory and the solution is expected to be the perturbative open string vacuum. These results reasonably support the expectation that the identity-based solution is a trivial pure gauge configuration for a>1/2a>-1/2, but it can be regarded as the tachyon vacuum solution at a=1/2a=-1/2.Comment: 12 pages, 5 figures; new numerical data up to level (20,60) included; Contribution to the proceedings of "Second International Conference on String Field Theory and Related Aspects" (Steklov Mathematical Institute, Moscow, Russia, April 12-19, 2009

    Continuous and discrete flows on operator algebras

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    Let (N,R,θ)(N,\R,\theta) be a centrally ergodic W* dynamical system. When NN is not a factor, we show that, for each t0t\not=0, the crossed product induced by the time tt automorphism θt\theta_t is not a factor if and only if there exist a rational number rr and an eigenvalue ss of the restriction of θ\theta to the center of NN, such that rst=2πrst=2\pi. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if (A,R,α)(A,\R,\alpha) is a minimal unital C* dynamical system and AA is either prime or commutative but not simple, then, for each t0t\not=0, the crossed product induced by the time tt automorphism αt\alpha_t is not simple if and only if there exist a rational number rr and an eigenvalue ss of the restriction of α\alpha to the center of AA, such that rst=2πrst=2\pi.Comment: 7 page

    Computationally efficient algorithms for the two-dimensional Kolmogorov-Smirnov test

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    Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2^d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n^2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Omega(n^2.lg(n)), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test
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