Characterizations of tripotents in JB*-triples

Hügli, Remo V.

doi:10.7146/math.scand.a-15005

Cited by 5 publications

(3 citation statements)

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“…Since x 3 = {xxx} holds in a JB * -triple and for orthogonal elements x and y we have {x + y, x + y, x + y} = {xxx} + {yyy}, it follows that x + y ≤ 2 1/3 max( x , y ) and by iteration that x + y ≤ 2 3 −n max( x , y ), so that x + y = max( x , y ) for orthogonal elements x, y. The converse is false in general, but is true in case one of x, y is a tripotent (as pointed out to us independently by R. Hügli and A. Peralta, [25,Th. 4.1]).…”

Section: Preliminariesmentioning

confidence: 82%

Existence of contractive projections on preduals of JBW*-triples

Neal

Russo

2011

Isr. J. Math.

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In 1965, Ron Douglas proved that if X is a closed subspace of an L 1 -space and X is isometric to another L 1 -space, then X is the range of a contractive projection on the containing L 1 -space. In 1977 Arazy-Friedman showed that if a subspace X of C 1 is isometric to another C 1 -space (possibly finite dimensional), then there is a contractive projection of C 1 onto X. In 1993 Kirchberg proved that if a subspace X of the predual of a von Neumann algebra M is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of M onto X.We widen significantly the scope of these results by showing that if a subspace X of the predual of a JBW * -triple A is isometric to the predual of another JBW * -triple B, then there is a contractive projection on the predual of A with range X, as long as B does not have a direct summand which is isometric to a space of the form L ∞ (Ω, H), where H is a Hilbert space of dimension at least two. The result is false without this restriction on B.

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Section: Preliminariesmentioning

confidence: 82%

Existence of contractive projections on preduals of JBW*-triples

Neal

Russo

2011

Isr. J. Math.

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“…We need some further information about the relationships of inner automorphisms and L ∞ -orthogonality, which is equivalent with algebraic orthogonality for tripotent elements [20]. A crucial step, providing the transitivity of Inn(A) on the set of frames F(A), is given by the following lemma, stated for finite-rank triples, which obviously include the finite-dimensional cases.…”

Section: Lemma 33mentioning

confidence: 99%

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

Hügli

2011

Integr. Equ. Oper. Theory

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It is shown that if P is a weak * -continuous projection on a JBW * -triple A with predual A * , such that the range P A of P is an atomic subtriple with finite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of P A, then P is contractive if and only if it commutes with all inner derivations of P A. This provides characterizations of 1-complemented elements in a large class of subspaces of A * in terms of commutation relations.Mathematics Subject Classification (2010). Primary 17C65; Secondary 47D27.

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“…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].…”

Section: Slovenijamentioning

confidence: 99%

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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Characterizations of tripotents in JB*-triples

Cited by 5 publications

References 9 publications

Existence of contractive projections on preduals of JBW*-triples

Existence of contractive projections on preduals of JBW*-triples

Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

Norm of Sums of Peirce Projections

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