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13
Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces
"... A metric space X has Markov type 2, if for any reversible finitestate Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is d ..."
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Cited by 41 (24 self)
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A metric space X has Markov type 2, if for any reversible finitestate Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its biLipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type 2 (in particular, Lp for p> 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp. 1
A Constructive Proof of Gleason’s Theorem
 J. Func. Anal
, 1999
"... Gleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace class. In this paper we give a constructive proof of that theorem. A measure on the projections of a real or c ..."
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Cited by 12 (2 self)
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Gleason's theorem states that any totally additive measure on the closed subspaces, or projections, of a Hilbert space of dimension greater than two is given by a positive operator of trace class. In this paper we give a constructive proof of that theorem. A measure on the projections of a real or complex Hilbert space assigns to
Contents
, 2004
"... 1.1 Statement of the problem...................... 1 1.2 A motivation for studying extension maps............. 2 ..."
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1.1 Statement of the problem...................... 1 1.2 A motivation for studying extension maps............. 2
A NOTE ON LENGTH AND ANGLE
"... Our slogan is that defining length comparable regardless of direction is the same as defining angle. We temper this idealism with only one proviso, to wit the parallelogram law. An abstract vector space does not come equipped with notions of length and angle other than that one may compare vectors ‘ ..."
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Our slogan is that defining length comparable regardless of direction is the same as defining angle. We temper this idealism with only one proviso, to wit the parallelogram law. An abstract vector space does not come equipped with notions of length and angle other than that one may compare vectors ‘in the same direction’: It always seems to make sense to say that the vector αv has α  times the length of the vector v, where α is a scalar and   is some absolute value on the field of scalars. No doubt, we also think of a nonzero vector as having nonzero length, but that plays only a concluding rôle below. Throughout, V is a vector space over a field K. We intend to restrict ourselves to subfields of R, and eventually of C. Nevertheless, to maintain the generality of our remarks, we do our best not to use properties peculiar to such fields until absolutely necessary. The field K will come accompanied with an absolute value  , that is, a positive definite map K − → R preserving multiplication in K and obeying the triangle inequality. In mildly technical language, Definition. Length is the composition of a map V−→K respecting multiplication by scalars, and an absolute value on K. In different words, length is a map to the ordered semigroup R≥0: v ↦ → ‖v ‖ with the homogeneity property (1) ‖αv ‖ = α‖v ‖ , α ∈ K. Comparability of lengths arises from the ordering on R. In principle, we could assume that the absolute value  , and hence length ‖ ‖, takes its values in some more general ordered ring.
SECOND DERIVATIVE TEST FOR ISOMETRIC EMBEDDINGS IN Lp
, 1997
"... Abstract. An old problem of P. Levy is to characterize those Banach spaces which embed isometrically in Lp. We show a new criterion in terms of the second derivative of the norm. As a consequence we show that, if M is a twice differentiable Orlicz function with M ′(0) = M ′′(0) = 0, then the ndim ..."
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Abstract. An old problem of P. Levy is to characterize those Banach spaces which embed isometrically in Lp. We show a new criterion in terms of the second derivative of the norm. As a consequence we show that, if M is a twice differentiable Orlicz function with M ′(0) = M ′′(0) = 0, then the ndimensional Orlicz space ℓn M, n ≥ 3 does not embed isometrically in Lp with 0 < p ≤ 2. These results generalize and clear up the recent solution to the 1938 Schoenberg’s problem on positive definite functions. 1.
FUZZY STABILITY OF ADDITIVE–QUADRATIC FUNCTIONAL EQUATIONS
, 903
"... Abstract. In this paper we investigate the generalized Hyers Ulam stability of the functional equation in fuzzy Banach spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) 1. Introduction and ..."
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Abstract. In this paper we investigate the generalized Hyers Ulam stability of the functional equation in fuzzy Banach spaces. f(2x + y) + f(2x − y) = f(x + y) + f(x − y) + 2f(2x) − 2f(x) 1. Introduction and
unknown title
, 812
"... Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasiBanach spaces ..."
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Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasiBanach spaces
Aronszajn’s Criterion for Euclidean Space
, 2009
"... We give a simple proof of a characterization of euclidean space due to Aronszajn and derive a wellknown characterization due to Jordan & von Neumann as a corollary. A norm     on a vector space V is euclidean if there is an inner product 〈 , 〉 on V such that v  = √ 〈v,v〉. Characterizatio ..."
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We give a simple proof of a characterization of euclidean space due to Aronszajn and derive a wellknown characterization due to Jordan & von Neumann as a corollary. A norm     on a vector space V is euclidean if there is an inner product 〈 , 〉 on V such that v  = √ 〈v,v〉. Characterizations of euclidean normed spaces abound. Amir [1] surveys some 350 characterizations, starting with a wellknown classic of Jordan & von Neumann [3]: a norm is euclidean iff it satisfies the parallelogram identity: v + w  = 2v  2 + 2w  2 − v − w  2 Aronszajn proved that the algebraic details of this identity are mostly irrelevant: if the norms of two sides and one diagonal of a parallelogram determine the norm of the other diagonal then the norm is euclidean. Formally, Aronszajn’s criterion is the following property, as illustrated in figure 1(a). ∀v1 w1 v2 w2 · v1  = v2  ∧ w1  = w2  ∧ v1 − w1  = v2 − w2 ⇒ v1 + w1  = v2 + w2. Aronszajn’s announcement of this characterization [2] does not give a proof. Amir’s proof forms part of a long chain of interrelated results. In this note we give a short, selfcontained proof of the theorem and derive the Jordanvon Neumann theorem as a corollary. We begin with a lemma showing that the Aronszajn criterion ensures a useful supply of isometries. Figure 1(b) illustrates the parallelograms that feature in the proof. This note was inspired by joint work with Robert M. Solovay and John Harrison on decidability for logical theories of normed spaces. I am grateful to Bob and John for ther comments. 1 ap + (b+1)q ap + bq 0 v2 w1 v1+ w1 v 1 v 1 w2 v2 v2 − w1 w2 w2
ON EXTREMAL POSITIVE MAPS ACTING BETWEEN TYPE I FACTORS
, 812
"... Abstract. The paper is devoted to the problem of classification of extremal positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. It is shown that every positive map with the property that rank φ(P) ≤ 1 for any onedimensional projection P is a rank 1 preserver. It allows to c ..."
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Abstract. The paper is devoted to the problem of classification of extremal positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. It is shown that every positive map with the property that rank φ(P) ≤ 1 for any onedimensional projection P is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rank φ(P) ≤ 1 for some onedimensional projection P and satisfy the condition of local complete positivity. It allows us to give a negative answer for Robertson’s problem in some special cases. 1.