On the spectral theory of Rickart Ordered*-algebras

Collection of Full …

Spectral Theory on Commutative Kreın C*-algebras 39 The main result presented here can actually be also obtained in at least a few other ways that we briefly describe here below: • For a symmetric imprimitivity commutative unital Kreın C*-algebra A, the even part A+, as a commutative ...

We prove theorems of Perron-Frobenius type for positive elements in partially ordered topological algebras satisfying certain hypotheses. We show how some of our results relate to known results on Banach algebras. We give examples and state some open questions.

Proc Amer Math Soc, 1978

Let A be a closed linear subspace of a C*-algebra B. Adjoin, if necessary, the identity 1 to B. Then A is a C*-subalgebra if and only if, for each x in A, the elements x* and |jc| + 1-| \x\-1| are in A. If 1 is in A, then A is a C*-subalgebra if and only if [jc| is in A for each x in A. Here |jc| denotes the unique positive square root of x*x in B. This note is entirely devoted to the proof of and a discussion on the following result. Theorem. Let A be a closed linear subspace of a complex C*-algebra B. A is a C*-subalgebra of B if and only if, for each x in A, the elements x* and \x\ + 1-| |jc|-1| are in A.

Journal of Mathematical Analysis and Applications, 2010

In this paper we establish some results relating star, left-star, right-star, minus ordering and the reverse order law under certain conditions on Moore-Penrose invertible elements of C *-algebras.

Journal of the London Mathematical Society, 1971

We establish a theorem for which a number of absolute-value identities and inequalities in the framework of Banach * -algebras can be generated. To do this job, we construct an order-preserving linear map from the vector space of 2-by-2 hermitian matrices to a hermitian Banach * -algebras. is map can convert any suitable matrix ordering to a number of identities and inequalities in Banach * -algebras. Hence, we obtain a number of analogues of the well-known results in a framework of hermitian Banach * -algebras.

arXiv (Cornell University), 2020

In this paper we consider shift operators, self-adjoint, unitary and normal operators on the standard module over a unital C *-algebra A. We define various generalized spectra in A of these operators, give description of such spectra of these operators and investigate their further properties.

Journal of Functional Analysis, 2000

Studia Mathematica, 2002

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties. If A is an ordered Banach algebra with a normal algebra cone C, then an important problem is that of providing conditions under which certain spectral properties of a positive element b will be inherited by positive elements dominated by b. We are particularly interested in the property of b being an element of the radical of A. Some interesting answers can be obtained by the use of subharmonic analysis and Cartan's theorem.

A Banach involutive algebra (A, ∗) is called a Krein C*-algebra if there is a fundamental symmetry of (A, ∗), i.e., α ∈ Aut (A, ∗) such that α^2 = Id_A and ||α(x∗ )x|| = ||x||^2 for all x ∈ A. Using α, we can decompose A = A_+ ⊕ A_−, where A+ = {x ∈ A | α(x) = x} and A− = {x ∈ A | α(x) = −x}. The even part A_+ is a C*-algebra and the odd part A_− is a Hilbert C*-bimodule over the even part A+ . The ultimate goal is to develop a spectral theory for commutative Krein C*-algebras when the odd part is an imprimitivity bimodule over the commutative even part.