Characterizing projections among positive operators in the unit sphere

Let E and P be subsets of a Banach space X, and let us define the unit sphere around E in P as the set Sph(E; P) := {x is an element of P : parallel to x - b parallel to = 1 for all b is an element of E}. Given a C*-algebra A and a subset E subset of A; we shall write Sph(+) (E) or Sph(A)(+) (E) for the set Sph(E; S(A(+))); where S(A(+)) denotes the unit sphere of A(+). We prove that, for every complex Hilbert space H, the following statements are equivalent for every positive element a in the unit sphere of B(H): (a) a is a projection; (b) Sph(B)((H))(+) (Sph(B)((H))(+) ({a})) = {a}. We also prove that the equivalence remains true when B(H) is replaced with an atomic von Neumann algebra or with K(H-2), where H-2 is an infinite-dimensional and separable complex Hilbert space

Preservers of Triple Transition Pseudo-Probabilities in Connection with Orthogonality Preservers and Surjective Isometries

We prove that every bijection preserving triple transition pseudoprobabilities between the sets of minimal tripotents of two atomic JBW ∗ - triples automatically preserves orthogonality in both directions. Consequently, each bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW ∗ -triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW ∗ -triples. This result can be regarded as triple version of the celebrated Wigner theorem for Wigner symmetries on the posets of minimal projections in B(H). We also present a Tingley type theorem by proving that every surjective isometry between the sets of minimal tripotents in two atomic JBW ∗ -triples admits an extension to a real linear surjective isometry between these two JBW ∗ -triples. We also show that the class of surjective isometries between the sets of minimal tripotents in two atomic JBW ∗ -triples is, in general, strictly wider than the set of bijections preserving triple transition pseudo-probabilities.Universidad de Granada/CBUAERDF/Ministry of Science and Innovation -State Research Agency PID2021-122126NB-C31Junta de Andalucia FQM375 PY20 00255IMAG-Maria de Maeztu Grant CEX2020-001105-M/AE

The alternative Dunford-Pettis property on projective tensor products

A Banach space X has the Dunford–Pettis property (DPP) if and only if whenever (xn) and (pn) are weakly null sequences in X and X*, respectively, we have pn(xn)→ 0. Freedman introduced a stricly weaker version of the DPP called the alternative Dunford–Pettis property (DP1). A Banach space X has the DP1 if whenever xn ! x weakly in X, with kxnk = kxk, and (xn) is weakly null in X*, we have that xn(xn)→ 0. The authors study the DP1 on projective tensor products of C*-algebras and JB*-triples. Their main result, Theorem 3.5, states that if X and Y are Banach spaces such that X contains an isometric copy of c0 and Y contains an isometric copy of C[0, 1], then Xˆ_Y , the projective tensor product of X and Y , does not have the DP1. As a corollary, they get that if X and Y are JB*-triples such that X is not reflexive and Y contains `1, then Xˆ_Y does not have the DP1. Furthermore, if A and B are infinite-dimensional C*-algebras, then Aˆ_B has the DPP if and only if Aˆ_B has the DP1 if and only if both A and B have the DPP and do not contain `1

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

Second author was partially supported by MCIN / AEI / 10. 13039 / 501100011033 / FEDER “Una manera de hacer Europa” project no. PGC2018-093332-B-I00, Junta de Andalucía grants FQM375, A-FQM-242-UGR18 and PY20 00255, and by the IMAG–María de Maeztu grant CEX2020- 001105-M / AEI / 10.13039 / 501100011033. We deeply appreciate the useful and detailed suggestions and comments received from the anonymous referees of this paper. Special thanks are given to one of the reviewers for pointing out a mistake in one of the statements of an earlier version.There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states P—i.e., the set of one-dimensional projections on a complex Hilbert space H– and the orthomodular lattice L of closed subspaces of H. These six groups are isomorphic when the dimension of H is ≥3. The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on L identifies with the bijections on P preserving transition probabilities only if dim(H)≥3. Despite of the difficulties caused by M2(C), rank two algebras are used for quantum mechanics description of the spin state of spin-12 particles. However, there is a counterexample for Uhlhorn’s version of Wigner’s theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, let M and N be two atomic JBW∗-triples not containing rank–one Cartan factors, and let U(M) and U(N) denote the set of all tripotents in M and N, respectively. We show that each bijection Φ:U(M)→U(N), preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Molnár to the wider setting of atomic JBW∗-triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.MCIN / AEI / 10. 13039 / 501100011033 / FEDER “Una manera de hacer Europa” project no. PGC2018-093332-B-I00Junta de Andalucía grants FQM375, A-FQM-242-UGR18 and PY20 00255MAG–María de Maeztu grant CEX2020- 001105-M / AEI / 10.13039 / 50110001103

Linear orthogonality preservers between function spaces associated with commutative JB*-triples

It is known, by Gelfand theory, that every commutative JB*-triple admits a representation as a space of continuous functions of the form C-0(T) (L) = {alpha epsilon C-0(L) : alpha(lambda t) =lambda alpha(t), A lambda epsilon T, t epsilon L}, where L is a principal T-bundle and T denotes the unit circle in C. We provide a full technical description of all orthogonality preserving (non-necessarily continuous nor bijective) linear maps between commutative JB*-triples. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB*-triples is automatically continuous and bi-orthogonality preserving.Junta de Andalucia FQM375 PY20_00255MCIN/AEI/FEDER 'Una manera de hacer Europa' Ministerio de Ciencia, Innovacion y Universidades PGC2018-093332-B-I00 PID2021-122126NB-C31IMAG-Maria de Maeztu grant CEX2020-001105

Every commutative JB∗‑triple satisfies the complex Mazur–Ulam property

We prove that every commutative JB*-triple, represented as a space of continuous functions C-0(T)(L), satisfies the complex Mazur-Ulam property, that is, every surjective isometry from the unit sphere of C-0(T)(L) onto the unit sphere of any complex Banach space admits an extension to a surjective real linear isometry between the spaces.Universidad de Granada/CBU

Exploring new solutions to Tingley's problem for function algebras

In this note we present two new positive answers to Tingley's problem in certain subspaces of function algebras. In the first result we prove that every surjective isometry between the unit spheres, S(A) and S(B), of two uniformly closed function algebras A and B on locally compact Hausdorff spaces can be extended to a surjective real linear isometry from A onto B. In a second part we study surjective isometrics between the unit spheres of two abelian JB*-triples represented as spaces of continuous functions of the form C-0(T)(X) := { a is an element of C-0(X) : a(lambda t) = lambda a(t) for every (lambda,t) is an element of T x X}, where X is a (locally compact Hausdorff) principal T-bundle and T denotes the unit sphere of C. We establish that every surjective isometry Delta : S(C-0(T) (X)) -> -S(C-0(T)(Y)) admits an extension to a surjective real linear isometry between these two abelian JB*-triples.UK Research & Innovation (UKRI)Engineering & Physical Sciences Research Council (EPSRC) EP/R044228/1Spanish GovernmentEuropean Commission PGC2018-093332-B-I00Junta de Andalucia FQM375 A-FQM-242-UGR18 PY20 00255 CEX2020-001105-M/AEI/10.13039/501100011033Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT) Japan Society for the Promotion of ScienceGrants-in-Aid for Scientific Research (KAKENHI) PGC2018-093332-B-I00 JP 20K03650 MCIN/AEI/1

Can one identify two unital JB*-algebras by the metric spaces determined by their sets of unitaries?

First author supported by EPSRC (UK) project `Jordan Algebras, Finsler Geometry and Dynamics' ref. no. EP/R044228/1 and by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, and Consejeria de Economia, Innovacion, Ciencia y Empleo, Junta de Andalucia grants FQM375 and A-FQM-242-UGR18. Second author supported by MCIN/AEI/10.13039/501100011033/FEDER `Una manera de hacer Europa' project no. PGC2018-093332-B-I00, Consejeria de Economia, Innovacion, Ciencia y Empleo, Junta de Andalucia grants FQM375, A-FQM-242-UGR18 and PY20_00255, and by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033.Let M and N be two unital JB*-algebras and let U(M) and U(N) denote the sets of all unitaries in M and N, respectively. We prove that the following statements are equivalent: (a) M and N are isometrically isomorphic as (complex) Banach spaces; (b) M and N are isometrically isomorphic as real Banach spaces; (c) there exists a surjective isometry Delta : U(M) -> U(N). We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry Delta : U(M) -> U(N), we can find a surjective real linear isometry Psi : M -> N which coincides with Delta on the subset e(iMsa). If we assume that M and N are JBW*-algebras, then every surjective isometry Delta : U(M) -> U(N) admits a (unique) extension to a surjective real linear isometry from M onto N. This is an extension of the Hatori-Molnar theorem to the setting of JB*-algebras.EPSRC (UK) project `Jordan Algebras, Finsler Geometry and Dynamics' EP/R044228/1Spanish Ministry of Science, Innovation and Universities (MICINN)European Commission PGC2018-093332-B-I00Junta de Andalucia FQM375 A-FQM-242-UGR18 PY20_00255IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033 MCIN/AEI/10.13039/501100011033/FEDE

The Daugavet equation for polynomials on C⁎-algebras and JB⁎-triples

We prove that every JB∗-triple E (in particular, every C∗- algebra) satisfying the Daugavet property also satisfies the stronger polynomial Daugavet property, that is, every weakly compact polynomial P : E −→ E satisfies the Daugavet equation IdX +P = 1 + P . The analogous conclusion also holds for the alternative Daugavet propert

Orthogonally additive polynomials on C*-Algebras

We show that for every orthogonally additive scalar n-homogeneous polynomial P on a C*-algebra A there exists phi in A* satisfying P(x) = phi(x(n)), for each element x in A. The vector-valued analogue follows as a corollary