Russo-Dye Theorem Research Articles

Linear Preserves of BP-quasi invertible elements in JB*-algebras

In this note, we study one of the main outcomes of the Russo-Dye Theorem of JB*-algebra: a linear operator that preserves Brown-Pedersen-quasi invertible elements between two JB*-algebras is characterized by a Jordan ∗-homomorphism. Earlier, in C*-setting of algebras, Russo and Dye gave a characterization of any linear operator that maps unitary elements into unitary elements; namely a Jordan ∗-homomorphism. Special sorts of linear preservers between C*-algebras and between JB*-triples were introduced by Burgos et al. As a result, if G is a linear operator between two JB*-algebras having non-empty sets of extreme points of the closed unit sphere that preserves extreme points, then there exists a Jordan ∗-homomorphism Φ which also preserves extreme points and characterizes the linear operator G. We also explore the connection between linear operators that strongly preserve Brown-Pedersen-quasi nvertible elements between two JB*-triples and the λ-property of both JB*-triples. Other geometric properties, such as extremally richness and the Bade property of two JB*-algebras or triples under linear preservers, are to be elaborated on in forthcoming research.

Positive linear maps on Hilbert space operators and noncommutative Lp spaces

We extend some inequalities for normal matrices and positive linear maps related to the Russo-Dye theorem. The results cover the case of some positive linear maps on a von Neumann algebra mapping any nonzero operator to an unbounded operator.

On the Russo-Dye Theorem for positive linear maps

We revisit a classical result, the Russo-Dye Theorem, stating that every positive linear map attains its norm at the identity.

The Russo–Dye theorem in nest subalgebras of factors

Let N be a nest in a factor R. In this paper, we introduce the concept of the admissible nest for R and prove that the weak operator topology closure of the convex hull of all the unitary elements in the nest subalgebra of the factor is the unit ball if and only if N is an admissible nest in R.

On the Geometry of the Unit Ball of aJB*-Triple

We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting ofC*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in aJB*-triple; this indicates their structural richness. We initiate a study of the unit ball of aJB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. SomeC*-algebra andJB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended toJB*-triples.

The Russo-Dye Theorem in a weakly closed 𝒯(𝒩)–module

Suppose that N \mathcal {N} is admissible. It is shown that the convex hull of unitary elements of a weakly closed T ( N ) \mathcal {T(N)} –module U \mathcal {U} contains the whole unit ball of U \mathcal {U} if and only if I ~ = I \widetilde {I}=I and for any N > 0 N>0 , N ~ > 0 \widetilde N>0 .

BANACH SPACES WHOSE ALGEBRAS OF OPERATORS ARE UNITARY: A HOLOMORPHIC APPROACH

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies ∥u∥ = ∥u−1∥ = 1. The norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A. If X is a complex Hilbert space, then the algebra BL(X) of all bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) BL(X) is unitary and, for Y equal to X, X* or X** there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y. 2000 Mathematics Subject Classification 46B04, 46B10, 46B20.

The Russo-Dye theorem in nest algebras

It is shown that the convex hull of the unitary elements of a nest algebra contains the whole unit ball if and only if both 0 + 0_+ and I − ⊥ I_-^\perp are either zero or infinite rank.

A non-selfadjoint Russo-Dye Theorem

One of the well-known results in the theory of C* algebras is the Russo-Dye Theorem [19]: given a C* algebra .~r the closed convex hull of the unitary elements in ~r equals the closed unit ball of ~r This result was later refined by Gardner and reached its final form by Kadison and Pedersen; today it is known that every operator in a C* algebra ~r whose norm is less than 1, is the average of unitaries from A. The Russo-Dye Theorem initiated the theory of unitary rank in selfadjoint operator algebras. If ~r is an operator algebra, the unitary rank of an element A E ~r is defined as the smallest number for which there is a convex combination of unitaries from ~r of length u(A) and equaling A. If no such decomposition exists (in particular if liAII > 1) we define u(A) = oo. The literature on unitary rank is vast. The earliest result is due to Murray and yon Neumann who proved that any selfadjoint operator of norm I or less is the mean of two unitary operators ([12] p. 239, 1937). The first systematic study was given by R. Kadison and G. Pedersen [8] in 1984 (previous work in the field included contributions by Popa [15], Robertson [17], Gardner [6] and others). In 1986, C. Olsen and G. Pedersen [14] characterized all elements in a factor von Neumarm algebra with finite unitary rank. In the general case of a C*-algebra, a characterization was obtained by Rordam in his important paper [18]. For more details and further information on the theory of unitary rank we refer to the excellent articles of U. Haagerup [7] and M. Rordam [ 18]. In the first section of the present paper, we prove a Russo-Dye type Theorem for infinite multiplicity nest algebras. The techniques employed in the proof of our result are different from that of Gardner and Kadison-Pedersen. To our knowledge, this is the first result of this type, for non-selfadjoint operator algebras and clearly initiates the unitary rank theory for such algebras.

Shorter Notes: An Elementary Proof of the Russo-Dye Theorem

Shorter Notes: An Elementary Proof of the Russo-Dye Theorem

An elementary proof of the Russo-Dye theorem

PROOF. It suffices to show that if x E AI, the open unit ball of A, then x E co(U), the closed convex hull of U. It is easy to reduce this to showing that for every u E U, y = (x + u)/2 E co(U): For then U C 2 co(U) x, which is closed and convex, so co(U) C 2 co(U)-x, or (x +co(U))/2 Cco(U); if x0 E U, and x n += (x + xn)/2, xn co(U) and xn -* x. But note thaty = ((xu-1 + 1)/2)u, so (since l1xu-111 = lixii < 1) IIYl < 1, and y is invertible. Thus y = v Iy I, with v unitary and (y*y)l/2 =y I= (W + w*)/2, where W = YI + i(l IY 12)1/2 is also unitary. This concludes the proof and indeed proves the stronger result: AI U C U + U. NOTE ADDED IN PROOF. C. K. Fong has pointed out that essentially the same argument proves also that Al C co(U): Extend slightly the segment from u through x to x' E Al, and apply the given argument to x' instead of x; then x' C co(U), and for large n, x lies between x' and u.

On the Russo-Dye theorem.

On the Russo-Dye theorem.