Continuous Reinhardt domains from a Jordan viewpoint

Stachó, László L.

doi:10.4064/sm185-2-6

Cited by 3 publications

(6 citation statements)

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“…In particular, the standard Banach-Alaoglu theorem is sufficient. It should be noted that Stachó writes at the end of the introduction of [22] that [22,Theorem 7.1] is contained implicitly in a result of Villanueva [23], while Fernando Bombal, in private communication, points to the earlier paper [2].…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…The proof of [22,Theorem 7.1], which appears in an appendix to that paper, is completely independent of the rest of [22] and accessible, though there are a few misprints: φ should be Φ in the first line of the statement of the theorem and on line 5 of page 21, and the equality sign on line 4 of page 21 should be less than or equal to. A key tool is the Alaoglu-Bourbaki theorem.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…Of course, there is already much information in this direction in the literature. In fact, we use a result of Stachó [22,Theorem 7.1], stated in Proposition 2.2, as a springboard for our work. Stachó's theorem provides an integral representation for a continuous positive multilinear functional on C(X) n , where X is a locally compact Hausdorff space.…”

Section: Introductionmentioning

confidence: 99%

“…Stachó's theorem provides an integral representation for a continuous positive multilinear functional on C(X) n , where X is a locally compact Hausdorff space. We are grateful to Laszlo Stachó for supplying Proposition 3.4 and to him and Fernando Bombal for helpful correspondence concerning results such as [22,Theorem 7.1], which arise in the representation of polymeasures.…”

Section: Introductionmentioning

confidence: 99%

“…This is done by a standard disjointification argument, again using the positive homogeneity of F . The rotation invariance of F is not needed for the latter step, but may now be invoked, together with (20) and (21), to yield that there is a c ≥ 0 such that (22) F (C 1 , . .…”

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confidence: 99%

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Characterizing the dual mixed volume via additive functionals

Dulio¹,

Gardner²,

Peri³

2016

Indiana Univ. Math. J.

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Integral representations are obtained of positive additive functionals on finite products of the space of continuous functions (or of bounded Borel functions) on a compact Hausdorff space. These are shown to yield characterizations of the dual mixed volume, the fundamental concept in the dual Brunn-Minkowski theory. The characterizations are shown to be best possible in the sense that none of the assumptions can be omitted. The results obtained are in the spirit of a similar characterization of the mixed volume in the classical Brunn-Minkowski theory, obtained recently by Milman and Schneider, but the methods employed are completely different.

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Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Definitions and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Characterizing the dual mixed volume via additive functionals

Dulio¹,

Gardner²,

Peri³

2016

Indiana Univ. Math. J.

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On linear isometries of Banach lattices in C 0(Ω)-spaces

Isidro

2009

Proc Math Sci

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On linear isometries of Banach lattices inConsider the space C 0 ( ) endowed with a Banach lattice-norm · that is not assumed to be the usual spectral norm · ∞ of the supremum over . A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry U of the Banach lattice (C 0 ( ), · ) induces a partition of into a family of finite subsets S ⊂ along with a bijection T : → which preserves cardinality, and a family [u(S): S ∈ ] of surjective linear maps u(S):Here we endow the space of finite sets S ⊂ with a topology for which the bijection T and the map u are continuous, thus completing the analogy with the classical result.

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Norm of Sums of Peirce Projections

Zalar

2009

Asian-European J. Math.

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SlovenijaPeirce projections P 1 , P 1/2 and P 0 are one of the fundamental technical tools in the theory of JB*-triples. It is well known that all three of them are contractive. We show that the sum of two Peirce projections need not be contractive. We also give the upper estimate for the norm of such a sum, valid in all JC*-triples, and present a conjecture of the exact value of this norm, based on some numerical experiments. 521 Asian-European J. Math. 2009.02:521-531. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/21/15. For personal use only. 522 B. ZalarIt is possible to develop a theory of JBW*-factors, which is somewhat richer than the theory of von Neumann algebras. There exist 6 families of type I factors. Four of them are infinite. Factors of type B(H, K) are called rectangular. Factors of type {x ∈ B(H) : x T = ±x} are called Hermitian and symplectic respectively. Every Hilbert space with dimension ≥ 3 gives rise to a construction of a spin factor, but we do not need to go into details here. JB*-triples for which all type I representations are of one of the above four types, are called JC*-triples. Every JC*-triple can be imbedded into B(H) for a suitable Hilbert space H. There exist also two exceptional type I factors, constructed from octonion matrices. They are both finite dimensional, their dimensions being 16 and 27, and are therefore more interesting from algebraic viewpoint than a functional-analytic. It is known that neither of them is embeddable into B(H).An element u ∈ W is called a tripotent if {uuu} = u. These elements play a similar role in JB*-theory as orthogonal projections play in C*-theory. In the case of the triple B(H, K) it is not difficult to see that tripotents are precisely partial isometries from H into K. If we consider the operator T (x) = {uux}, it can be shown that the Jordan identity implies T (2T − 1)(T − 1) = 0 which shows that W decomposes asOf course this decomposition, which is called Peirce decomposition, depends on the choice of u. The corresponding Peirce projections P i : W → W , whose ranges are W i , can all be given with an explicit algebraic formula P 1 (x) = {u{uxu}u},Peirce projections are one of the most widely used technical tools in research about JB*-triples. It is known that all three Peirce projections are contractive. Moreover P 1 + P 0 is also contractive. The purpose of this paper is to show that the remaining combinations P 1 + P 1/2 and P 1/2 + P 0 are in general not contractive and to give a reasonable estimate for their norms in JC* case.The reader can find proofs of the above explained facts in classical papers [3,9,10,11,12,13,16] and surveys [7,18,19,23]. For those interested in various modern trends in JB*-theory, a starting sample is [1,2,4,5,6,8,14,17,20,21,22]. Example of a lower estimateIn this section we consider the special situation when W = B(H, K) is a JB*-triple of bounded linear operators between two complex Hilbert spaces H and K. We use a rather standard notation β ⊗ α, where α ∈ H and β ∈ K, f...

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Continuous Reinhardt domains from a Jordan viewpoint

Cited by 3 publications

References 11 publications

Characterizing the dual mixed volume via additive functionals

Characterizing the dual mixed volume via additive functionals

On linear isometries of Banach lattices in C 0(Ω)-spaces

Norm of Sums of Peirce Projections

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