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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
Double equal may refer to:
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Double equal may refer to: Double-equal operator, part of JavaScript syntax Double Equals, an album by Raw Radio War
In algebra, the bicommutant of a subset S of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written S ′ ′ {\displaystyle S^{\prime \prime }} . The bicommutant is particularly useful in operator theory, due to the von Neumann double commutant theorem, which relates the algebraic and analytic structures of operator algebras. Specifically, it shows that if M is a unital, self-adjoint operator algebra in the C*-algebra B(H), for some Hilbert space H, then the weak closure, strong closure and bicommutant of M are equal. This tells us that a unital C*-subalgebra M of B(H) is a von Neumann algebra if, and only if, M = M ′ ′ {\displaystyle M=M^{\prime \prime }} , and that if not, the von Neumann algebra it generates is M ′ ′ {\displaystyle M^{\prime \prime }} . The bicommutant of S always contains S. So S ′ ′ ′ = ( S ′ ′ ) ′ ⊆ S ′ {\displaystyle S^{\prime \prime \prime }=\left(S^{\prime \prime }\right)^{\prime }\subseteq S^{\prime }} . On the other hand, S ′ ⊆ ( S ′ ) ′ ′ = S ′ ′ ′ {\displaystyle S^{\prime }\subseteq \left(S^{\prime }\right)^{\prime \prime }=S^{\prime \prime \prime }} . So S ′ = S ′ ′ ′ {\displaystyle S^{\prime }=S^{\prime \prime \prime }} , i.e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have: S ′ = S ′ ′ ′ = S ′ ′ ′ ′ ′ = ⋯ = S 2 n − 1 = ⋯ {\displaystyle S^{\prime }=S^{\prime \prime \prime }=S^{\prime \prime \prime \prime \prime }=\cdots =S^{2n-1}=\cdots } and S ⊆ S ′ ′ = S ′ ′ ′ ′ = S ′ ′ ′ ′ ′ ′ = ⋯ = S 2 n = ⋯ {\displaystyle S\subseteq S^{\prime \prime }=S^{\prime \prime \prime \prime }=S^{\prime \prime \prime \prime \prime \prime }=\cdots =S^{2n}=\cdots } for n > 1. It is clear that, if S1 and S2 are subsets of a semigroup, ( S 1 ∪ S 2 ) ′ = S 1 ′ ∩ S 2 ′ . {\displaystyle \left(S_{1}\cup S_{2}\right)'=S_{1}'\cap S_{2}'.} If it is assumed that S 1 = S 1 ″ {\displaystyle S_{1}=S_{1}''\,} and S 2 = S 2 ″ {\displaystyle S_{2}=S_{2}''\,} (this is the case, for instance, for von Neumann algebras), then the above equality gives ( S 1 ′ ∪ S 2 ′ ) ″ = ( S 1 ″ ∩ S 2 ″ ) ′ = ( S 1 ∩ S 2 ) ′ . {\displaystyle \left(S_{1}'\cup S_{2}'\right)''=\left(S_{1}''\cap S_{2}''\right)'=\left(S_{1}\cap S_{2}\right)'.}
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator theory. The formal statement of the theorem is as follows: Von Neumann bicommutant theorem. 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 generated by M. There are several other topologies on the space of bounded operators, and one can ask what are the *-algebras closed in these topologies. If M is closed in the norm topology then it is a C*-algebra, but not necessarily a von Neumann algebra. One such example is the C*-algebra of compact operators (on an infinite dimensional Hilbert space). For most other common topologies the closed *-algebras containing 1 are von Neumann algebras; this applies in particular to the weak operator, strong operator, *-strong operator, ultraweak, ultrastrong, and *-ultrastrong topologies. It is related to the Jacobson density theorem.
In computer programming, operator overloading, sometimes termed operator ad hoc polymorphism, is a specific case of polymorphism, where different operators have different implementations depending on their arguments. Operator overloading is generally defined by a programming language, a programmer, or both.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Brown v. Board of Education in 1954.