Additive mappings on C*-algebras preserving absolute values

AF-embeddings into [formula omitted]-algebras of real rank zero

We obtain an order sharp estimate for the distance from a given bounded operator A A on a Hilbert space to the set of normal operators in terms of ‖ [ A , A ∗ ] ‖ \|[A,A^*]\| and the distance to the set of invertible operators. A slightly modified estimate holds in a general C ∗ C^* -algebra of real rank zero.

We prove a number of fundamental facts about the canonical order on projections in C*-algebras of real rank zero. Specifically, we show that this order is separative and that arbitrary countable collections have equivalent (in terms of their lower bounds) decreasing sequences. Under the further assumption that the order is countably downwards closed, we show how to characterize greatest lower bounds of finite collections of projections, and their existence, using the norm and spectrum of simple product expressions of the projections in question. We also characterize the points at which the canonical homomorphism to the Calkin algebra preserves least upper bounds of countable collections of projections, namely that this occurs precisely when the span of the corresponding subspaces is closed.

Let Ext ( C ( X ), A ) be the set of unitarily equivalence classes of essential C *-algebra extensions of the following form: 0 → A → E → C ( X ) → 0, where A is a nonunital separable simple C *-algebra of real rank zero, stable rank one with unique normalized trace and X is a finite CW complex. We show that there is a bijection [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]: Ext ( C ( X ), A ) → KK ( C ( X ), M ( A )/ A ), where M ( A ) is the multiplier algebra of A . In particular, we determine when an extension is actually splitting. We also, in a more general setting, give a condition when an essential extension is quasidiagonal.

Semilinear operators on a complex Hilbert space are studied in a part of a program that aims to develop the theories of additive operators on complex and quaternionic Hilbert spaces for application to problems in mathematical physics. The more notable among the new results proved on the eigenvalue problem for semilinear operators are the following: (i) if α is an eigenvalue of a semilinear operator then so also is any complex number which has the same modulus as α; (ii) if a normal semilinear operator has two eigenvectors belonging to different eigenvalues, then either the two eigenvectors are orthogonal or two eigenvalues have the same moduli; and (iii) a normal semilinear operator has a complete set of eigenvectors if and only if it is self-adjoint. Further, it is shown that there exists a norm-preserving semilinear isomorphism between the spaces of bounded linear and semilinear operators on a complex Hilbert space. Finally it is demonstrated how the theory of semilinear operators can be exploited to solve the problems of finding three involutive mutually anticommuting self-adjoint two-by-two matrices and four four-by-four matrices with the same properties: the unusual and remarkably easy solution of this old familiar exercise establishes the relevance of the theory being developed here to physics.

Almost Commuting Unitaries and Classification of Purely Infinite SimpleC*-Algebras

On the Range of Certain ASH Algebras of Real Rank Zero

Classification of Homomorphisms from C (X) to Simple C *-Algebras of Real Rank Zero

A unital C*-algebra A is said to have cancellation of projections if the semigroup D(A) of Murray–von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies cancellation for any A, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras. 2000 Mathematics Subject Classification 46L80 (primary), 46L85 (secondary).

On the Classification of C ∗ -Algebras of Real Rank Zero, II

Distance between unitary orbits of unitary elements in $$C^*$$-algebras of real rank zero

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

Let H be a real or complex Hilbert space and B ( H ) the algebra of all bounded linear operators on H. Recall that a map δ : B ( H ) → B ( H ) is called an inner Jordan ∗ -derivation if there exists some T ∈ B ( H ) such that δ ( A ) = AT − T A ∗ for all A ∈ B ( H ) . In this paper, it is proved that inner Jordan ∗ -derivations are the only additive maps δ of B ( H ) with the property that δ ( P ) = δ ( P ) P ∗ + Pδ ( P ) for all idempotent operators P ∈ B ( H ) if dim ⁡ H = ∞ , which is satisfied by additive local Jordan ∗ -derivations. For the finite dimensional case, additional conditions are required for δ to be an inner Jordan ∗ -derivation. As applications, it is shown that, for any given C , D ∈ B ( H ) , δ satisfies δ ( A ) B ∗ + Bδ ( A ) + δ ( B ) A ∗ + Aδ ( B ) = D for all A , B ∈ B ( H ) with AB + BA = C if and only if δ is an inner Jordan ∗ -derivation and D = δ ( C ) . Also, several known results are generalized.

Let H and K be complex Hilbert spaces with dimensions greater than 2, and ξ ∈ C. Assume that Φ : B(H) → B(K) is an additive surjective map satisfying that, for any A, B ∈ B(H), AB = ξBA*Φ(A)Φ(B) = ξΦ(B)Φ(A)*. We show that,(1) if ξ = 1, then there exist a unitary or an anti-unitary operator U : H → K and a nonzero real number c such that Φ(A) = c U AU*for all A ∈ B(H);(2) if ξ ∈ R {1} and Φ is unital, then there is a unitary or an anti-unitary operator U : H → K such that Φ(A) = U AU*for all A ∈ B(H);(3) if ξ ∈ C R and Φ is unital, then there is a unitary operator U : H → K such that Φ(A) = U AU*for all A ∈ B(H).