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.

So, i. e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:

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.

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 the von Neumann algebra generated by M.

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.

*-A commutant is its own bicommutant.

*-A bicommutant is its own bicommutant.

# the effect of the operations is to yield an interesting bicommutant theory.

In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors ; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level.

Angular momenta of a classical object .< p > Left: intrinsic " spin " angular momentum S is really orbital angular momentum of the object at every point ,</ p >< p > right: extrinsic orbital angular momentum L about an axis ,</ p >< p > top: the moment of inertia tensor | moment of inertia tensor I and angular velocity ω ( L is not always parallel to ω )</ p >< p > bottom: momentum p and it's radial position r from the axis .</ p > The total angular momentum ( spin + orbital ) is J.

Most of the U. S. federal courts of appeal have adopted a rule under which, in the event of any conflict in decisions of panels ( most of the courts of appeal almost always sit in panels of three ), the earlier panel decision is controlling, and a panel decision may only be overruled by the court of appeals sitting en banc ( that is, all active judges of the court ) or by a higher court.

The 21 consonant letters in the English alphabet are B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, X, Z, and usually W and Y: The letter Y stands for the consonant in " yoke ", the vowel in " myth " and the vowel in " funny ", and " yummy " for both consonant and vowel, for examples ; W almost always represents a consonant except in rare words ( mostly loanwords from Welsh ) like " crwth " " cwm ".

Because of the similar conditions, both reactions in the E1 or E1cb elimination always compete with the S < sub > N </ sub > 1 substitution.

Because of the similar conditions and reagents, the E2 elimination is always in competition with the S < sub > N </ sub > 2-substitution.

For locations south of 34 ° S, Crux is circumpolar and thus always visible in the night sky.

Due to concerns about widespread and well-documented corruption in Chiang's government throughout his rule ( though not always with his knowledge ), the U. S. government limited aid to Chiang for much of the period of 1946 to 1948, in the midst of fighting against the People's Liberation Army led by Mao Zedong.

If there is a change in the potential energy of a system ; for example μ < sub > 1 </ sub >> μ < sub > 2 </ sub > ( μ is Chemical potential ) an energy flow will occur from S < sub > 1 </ sub > to S < sub > 2 </ sub >, because nature always prefers low energy and maximum entropy.

As always, Hawks was an average student and college friend Ray S. Ashbury remembered him as spending more of his time playing craps and drinking alcohol than studying, although Hawks was also known to be a voracious reader of popular American and English novels in college.

The greatest influence on use of the harp has always been the availability of fine harps and skilled players, and the great increase of them in the U. S. of the 20th century resulted in its spread into popular music.

In the U. S., hydrocodone is always found in combination with other drugs such as paracetamol ( also called acetaminophen ), aspirin, an NSAID, ibuprofen, an antihistamine, an expectorant, or homatropine methylbromide due to compounding regulations.

" Bureaucratic procedures are not always transparent and red tape often makes processing unnecessarily long ," reports the U. S. Department of Commerce.

An experiment done by C. S. Wu at Columbia University showed that neutrinos always have left-handed chirality.

If the elements of S are pairwise orthogonal, the answer is always yes: for any such set there exists a measurement which will ascertain the exact state of the system without disturbing it, and once we know the state we can prepare another system in the same state.

Laws mandating the reporting of domestic violence are now in place in all of the states of the U. S. Discussions of domestic violence are nearly always of a feminist construct, largely due to statistics that show women as having a higher rate of victimization.

The United States is almost always the most dysfunctional of the developed democracies, sometimes spectacularly so ... The view of the U. S. as a " shining city on the hill " to the rest of the world is falsified when it comes to basic measures of societal health.

Practicing lawyers who can afford them almost always use an annotated version of the U. S. Code from a private company.

This process of legislation and conversion is known as metrication, and in the U. S. is most evident in labeling requirements on food products, where SI units are almost always presented alongside customary units.

Traditionally, and due to tacit agreement between the United States and Europe, the U. S. has always chosen the President of the World Bank.

Made from 1942 to 1945, the seven films range from 40 to 76 minutes in length, and all are available on DVD or online at no cost, since they have always been public domain films produced by the U. S. government.

Consider, also, that all English speakers often pronounce ' Z ' where ' S ' is spelled, almost always when a noun ending in a voiced consonant or a liquid is pluralized, for example " seasons ", " beams ", " examples ", etc.

It was therefore obligatory to always add an N ( for north ) to Kristiansund ( Kristiansund N ) and an S ( for south ) to Kristiansand ( Kristiansand S ).

The unpredictable North Korean government has tested ( or, more likely, partially fizzled ) a nuclear device, and has historically threatened to turn Seoul into a " sea of fire ", or most recently, " ashes ", in response to unspecified, but always imminent, U. S. or South Korean " aggression " against it.

Hunter S. Thompson's Fear and Loathing in America: The Brutal Odyssey of an Outlaw Journalist, which contains hundreds of private letters written by Thompson over the years, contains a letter in which he uses A Modest Proposals satire technique against the Vietnam War.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

A Wyoming ( U. S .) rock with an intermediate claystone layer that contains 1000 times more iridium than the upper and lower layers.

Each of these strings p < sub > i </ sub > determines a subset S < sub > i </ sub > of Cantor space ; the set S < sub > i </ sub > contains all sequences in cantor space that begin with p < sub > i </ sub >.

The album contains a number of fast / short songs interspersed with jazz (" D. M. S. O.

According to S. N. Goenka, a teacher of Vipassana meditation, the original meaning of dhamma is " dharayati iti dharmaH ", or " one that contains, supports or upholds " and dharma in the Buddhist scriptures has a variety of meanings, including " phenomenon " and " nature " or " characteristic ".

The town of Harwich contains several smaller census-designated places ( CDPs ) for which the U. S. Census reports more focused geographic and demographic information.

However, it also contains many other fictional elements: countries such as Wakanda and Latveria ( very small nations ), and organizations like the espionage agency S. H. I. E. L. D.

If S contains two elements that are not pairwise orthogonal ( in particular, the set of all quantum states includes such pairs ) then an argument like that given above shows that the answer is no.

Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, S ( 0 ), S ( S ( 0 )), and so on ; i. e., it is informally known that

The shipyard also contains a portion of the United States Navy reserve fleet, a large collection of inactive U. S. Navy vessels, including four aircraft carriers ; the USS Independence, the USS Constellation, the USS Ranger and the USS Kitty Hawk.

Triangulum Australe contains several cepheid variables, all of which are too faint to be seen with the naked eye: R Trianguli Australis which ranges from apparent magnitude 6. 4 to 6. 9 over a period of 3. 389 days, S Trianguli Australis ranges from apparent magnitude 6. 1 to 6. 8 over a period of 6. 323 days, and U Trianguli Australis varies from 7. 5 to 8. 3 over 2. 568 days.

( For the example of x + y − z, the subgroup H in S < sub > 3 </ sub > contains the identity and the transposition ( xy ).

The genus Sarcophilus contains two other species, known only from Pleistocene fossils: S. laniarius and S. moomaensis.

As of 2005, the only childhood vaccine in the U. S. that contains thiomersal in greater than trace amounts is the influenza vaccine, which is currently recommended only for children with certain risk factors.

# S is locally finite and contains a copy of every finite group.

If S contains a least element, then that element is the infimum ; otherwise, the infimum does not belong to S ( or does not exist ).

For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S ( this point may be x itself ).