Classification and GNS-construction for general universal products

2015 ◽  
Vol 18 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Malte Gerhold ◽  
Stephanie Lachs
Keyword(s):  
Natural Products ◽  
Linear Functionals ◽  
Positive Linear Functionals ◽  
Boolean Product ◽  
Gns Construction ◽  
Two Parameter ◽  
Parameter Deformation

It is known that there are exactly five natural products, which are universal products fulfilling two normalization conditions simultaneously. We classify universal products without these extra conditions. We find a two-parameter deformation of the Boolean product, which we call (r, s)-products. Our main result states that, besides degnerate cases, these are the only new universal products. Furthermore, we introduce a GNS-construction for not necessarily positive linear functionals on algebras and study the GNS-construction for (r, s)-product functionals.

Interval oscillation criteria for self-adjoint matrix Hamiltonian systems

2005 ◽  
Vol 135 (5) ◽  
pp. 1085-1108 ◽  
Author(s):  
Qigui Yang ◽  
Yun Tang
Keyword(s):  
Hamiltonian Systems ◽  
Half Line ◽  
Linear Functionals ◽  
Matrix Space ◽  
Differential Systems ◽  
Adjoint Matrix ◽  
Oscillation Criteria ◽  
Positive Linear Functionals ◽  
Interval Oscillation ◽  
New Criteria

By using a monotonic functional on a suitable matrix space, some new oscillation criteria for self-adjoint matrix Hamiltonian systems are obtained. They are different from most known results in the sense that the results of this paper are based on information only for a sequence of subintervals of [t0, ∞), rather than for the whole half-line. We develop new criteria for oscillations involving monotonic functionals instead of positive linear functionals or the largest eigenvalue. The results are new, even for the particular case of self-adjoint second-differential systems which can be applied to extreme cases such as

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

INEQUALITIES OF JENSEN’S TYPE FOR POSITIVE LINEAR FUNCTIONALS ON HERMITIAN UNITAL BANACH -ALGEBRAS

2020 ◽  
Vol 102 (2) ◽  
pp. 308-318
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Convex Functions ◽  
Banach Algebras ◽  
General Setting ◽  
Linear Functionals ◽  
Positive Linear Functionals ◽  
Hermitian Unital

We establish inequalities of Jensen’s and Slater’s type in the general setting of a Hermitian unital Banach $\ast$-algebra, analytic convex functions and positive normalised linear functionals.

A mean-value theorem for positive linear functionals

2019 ◽  
Vol 189 (4) ◽  
pp. 675-681
Author(s):  
Mircea Ivan ◽  
Vicuta Neagos ◽  
Andra-Gabriela Silaghi
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Linear Functionals ◽  
Positive Linear Functionals
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