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It is stronger than the weak operator topology. The weak operator topology (WOT) or weak topology is defined by the seminorms |(x(h 1), h 2)| for h 1, h 2 ∈ H. (Warning: the weak Banach space topology, the weak operator topology, and the ultraweak topology are all sometimes called the weak topology, but they are different.)
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators , and consideration may be given to nonlinear operators .
Let M be an algebra consisting of bounded operators on a Hilbert space H, containing the identity operator, and closed under taking adjoints. Then the closures of M in the weak operator topology and the strong operator topology are equal, and are in turn equal to the bicommutant M′′ of M. This algebra is called the von Neumann algebra ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
III (2): General topology, from chapter 4 onwards; IV: Functions of a Real Variable; V: Topological Vector Spaces; VI: Integration [b] Thus the six books are also "logically ordered", with the caveat that some material presented in the later chapters of Algebra, the second book, invokes results from the early chapters of General Topology, the ...
In functional analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.. The prototypical example of an abelian von Neumann algebra is the algebra L ∞ (X, μ) for μ a σ-finite measure on X realized as an algebra of operators on the Hilbert space L 2 (X, μ) as follows: Each f ∈ L ∞ (X, μ ...
The predual of B(H) is the trace class operators C 1 (H), and it generates the w*-topology on B(H), called the weak-star operator topology or σ-weak topology. The weak-operator and σ-weak topologies agree on norm-bounded sets in B(H). A net {T α} ⊂ B(H) converges to T in WOT if and only Tr(T α F) converges to Tr(TF) for all finite-rank ...
The weak topology on a JW algebra M is define by the seminorms |f(a)| where f is a normal state; the strong topology is defined by the seminorms |f(a 2)| 1/2. The quadratic representation and Jordan product operators L(a)b = a ∘ b are continuous operators on M for both the weak and strong topology. An idempotent p in a JBW algebra M is called ...