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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.
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bicommutant and is
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.
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.
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.
So, i. e. the commutant of the bicommutant of S is equal to the commutant of S. By induction, we have:
*-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.
bicommutant and von
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.
* von Neumann bicommutant theorem
# REDIRECT von Neumann bicommutant theorem
bicommutant and .
The bicommutant of S always contains S. So.
is and particularly
That is particularly true of sovereignty when it is applied to democratic societies, in which `` popular '' sovereignty is said to exist, and in federal nations, in which the jobs of government are split.
it is particularly the artist ( Tonio Kroger, Aschenbach, Leverkuhn ) who suffers from it.
It is different with his volume The Swedes And Their Chieftains ( Svenskarna och deras Hovdingar ), a history intended for the general reader and particularly suited for high school students.
But what a super-Herculean task it is to winnow anything of value from the mud-beplastered arguments used so freely, particularly since such common use is made of cliches and stereotypes, in themselves declarations of intellectual bankruptcy.
And it is this, particularly the establishment of archaeology and place-name studies on a scientific basis, which are immediately pertinent to the Saxon Shore.
In light of the scholarly reappraisals engendered by the higher criticism this is a most remarkable statement, particularly coming from one who was well known for his antifundamentalist views.
The country about Cambridge is flat and not particularly spectacular in its scenery, though it offers easy going to the foot traveler.
While it is hazardous to project the trend of history, it seems clear that a genuine community is painfully emerging in the Western world, particularly among the countries of Western Europe.
But competent observers believe he is making progress, particularly toward what Sen. Jackson lists as the primary need -- `` a clearer understanding of where our vital national interests lie and what we must do to promote them ''.
The headline is offensive, particularly in view of the total inaccuracy of the editorial.
`` Argiento, this is senseless '', he complained, not liking to work on the wet floors, particularly in cold weather.
These are particularly suitable for regions where water or rock is close to the surface.
A second fundamental principle is that involved particularly in the present proceeding -- the difference between nighttime and daytime propagation conditions with respect to the standard broadcast frequencies.
This is particularly true in site selection.
But it is the wooden sculpture from Bali, the one representing two men with their heads bent backward and their bodies interlaced by a fish, that I particularly call to your attention.
This is particularly true of highly charged ions, especially those ions which fall into the class of polyelectrolytes.
It is quite likely that an even greater area was covered, particularly downwind.
There is some evidence that naturally occurring goitrogens may play a role in the development of goitre, particularly in Tasmania and Australia ( Clements and Wishart, 1956 ).
We pointed out that emotional excitement may lead to psychosomatic disorders and neurotic symptoms, particularly in certain types of personality, but it is also known that the reliving of a strong emotion ( `` abreaction '' ) may cure a battle neurosis.
There is some indication from a limited number of interviews with members of the population that the element of power, primarily the voluntary influence of non-authoritative power, has been exerted on actors in the system, particularly in regard to mate selection.
Confused and divided though this tradition may be, it is an important part of the social and cultural heritage of the group, and acts as a means of socialization, particularly for members of the rural community.
is and useful
Or is it relevant because it teaches us something useful to know about ourselves??
We may further grant to those of her ( Poetry's ) defenders who are lovers of poetry and yet not poets, the permission to speak in prose on her behalf: let them show not only that she is pleasant but also useful to States and to human life, and we will listen in a kindly spirit ; ;
RCA Victor has an ambitious and useful project in a stereo series called `` Adventures In Music '', which is an instructional record library for elementary schools.
The adjustable fly cutter is very useful for cutting large diameter holes and can be used to cut exact-size discs by reversing the cutter blade.
This is true because of savings in utility lines and the fact that your buildings have a useful radius equal in all directions.
Such an instrument is expected to be especially useful if it could be used to measure the elasticity of heavy pastes such as printing inks, paints, adhesives, molten plastics, and bread dough, for the elasticity is related to those various properties termed `` length '', `` shortness '', `` spinnability '', etc., which are usually judged by subjective methods at present.
The latter is useful for modifying information about some or all forms of a word, hence reducing the work required to improve dictionary contents.
It is of course useful to have a sovereign cause on one's social criticism, for it makes diagnosis and prescription much easier than they might otherwise be.
In Coriolanus the agnomen of Marcius is used deliberately and pointedly, but the Homeric epithets and the Anglo-Saxon kennings are used casually and recall to the hearer `` a familiar story or situation or a useful or pleasant quality of the referent ''.
He knew instinctively that next to voice and face an actor's hands are his most useful possession -- that in fiction as in the theatre, gesture is an indispensable shorthand for individualizing character and dramatizing action and response.
A concurrent effort is needed to make oceanographic data useful on the spot.
The aim is to collect a very broad range of physical, chemical, morphological, and structural data for crystals on an encyclopedic scale and to seek all possible useful and revealing correlations of properties with internal structure.
A useful by-product of this system is that the information necessary to set the gyro drift biases is available from the currents necessary to hold the system in level.
In spite of the shading of one type of course into another, I believe it is useful to talk about vocational courses as apart from academic courses.
In practice, a bidirectional reflectance distribution function ( BRDF ) may be required to characterize the scattering properties of a surface accurately, although the albedo is a very useful first approximation.
ANOVA " is probably the most useful technique in the field of
Regression is often useful.
A new computer program is used to create the most comfortable and useful prosthetics.
While the Arrhenius concept is useful for describing many reactions, it is also quite limited in its scope.
" Good ", for example, can mean " useful " or " functional " ( That's a good hammer ), " exemplary " ( She's a good student ), " pleasing " ( This is good soup ), " moral ( a good person versus the lesson to be learned from a story ), " righteous ", etc.
Straw is useful in binding the brick together and allowing the brick to dry evenly.
Astronomy is sometimes promoted as one of the few remaining sciences for which amateurs can still contribute useful data.
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