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Banach Algebra of Bounded Complex-Valued Functionals
In this article, we describe some basic properties of the Banach algebra which is constructed from all bounded complex-valued functionals.Kanazashi Katuhiko - Shizuoka High School, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanJózef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Banach algebra of bounded complex linear operators. Formalized Mathematics, 12(3):237-242, 2004.Noboru Endou. Complex linear space and complex normed space. Formalized Mathematics, 12(2):93-102, 2004.Noboru Endou. Complex valued functions space. Formalized Mathematics, 12(3):231-235, 2004.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Property of complex functions. Formalized Mathematics, 9(1):179-184, 2001.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Yasunari Shidama, Hikofumi Suzuki, and Noboru Endou. Banach algebra of bounded functionals. Formalized Mathematics, 16(2):115-122, 2008, doi:10.2478/v10037-008-0017-z.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990
The interplay between representable functionals and derivations on Banach quasi *-algebras
This note aims to highlight the link between representable functionals and
derivations on a Banach quasi *-algebra, i.e. a mathematical structure that can
be seen as the completion of a normed *-algebra in the case the multiplication
is only separately continuous. Representable functionals and derivations have
been investigated in previous papers for their importance concerning the study
of the structure properties of a Banach quasi *-algebra and applications to
quantum models.Comment: Contribution Proceedings of International Conference on Topological
Algebras and Applications 201
Representable and continuous functionals on Banach quasi *-algebras
In the study of locally convex quasi *-algebras an important role is played
by representable linear functionals; i.e., functionals which allow a
GNS-construction. This paper is mainly devoted to the study of the continuity
of representable functionals in Banach and Hilbert quasi *-algebras. Some other
concepts related to representable functionals (full-representability,
*-semisimplicity, etc) are revisited in these special cases. In particular, in
the case of Hilbert quasi *-algebras, which are shown to be fully
representable, the existence of a 1-1 correspondence between positive, bounded
elements (defined in an appropriate way) and continuous representable
functionals is proved
Manifolds of classical probability distributions and quantum density operators in infinite dimensions
The manifold structure of subsets of classical probability distributions and
quantum density operators in infinite dimensions is investigated in the context
of -algebras and actions of Banach-Lie groups. Specificaly, classical
probability distributions and quantum density operators may be both described
as states (in the functional analytic sense) on a given -algebra
which is Abelian for Classical states, and non-Abelian for
Quantum states. In this contribution, the space of states of a
possibly infinite-dimensional, unital -algebra is
partitioned into the disjoint union of the orbits of an action of the group
of invertible elements of . Then, we prove that the
orbits through density operators on an infinite-dimensional, separable Hilbert
space are smooth, homogeneous Banach manifolds of
, and, when admits a
faithful tracial state like it happens in the Classical case when we
consider probability distributions with full support, we prove that the orbit
through is a smooth, homogeneous Banach manifold for .Comment: 35 pages. Revised version in which some imprecise statements have
been amended. Comments are welcome
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