Projective Unitary Group - The Topology of PU(H) - PU(H) Is A Classifying Space For Circle Bundles

PU(H) Is A Classifying Space For Circle Bundles

The same construction may be applied to matrices acting on an infinite-dimensional Hilbert space .

Let U(H) denote the space of unitary operators on an infinite-dimensional Hilbert space. When f: X → U(H) is a continuous mapping of a compact space X into the unitary group, one can use a finite dimensional approximation of its image and a simple K-theoretic tric

to show that it is actually homotopic to the trivial map onto a single point. This means that U(H) is weakly contractible, and an additional argument shows that it is actually contractible. Note that this is a purely infinite dimensional phenomenon, in contrast to the finite-dimensional cousins U(n) and their limit U(∞) under the inclusion maps which are not contractible admitting homotopically nontrivial continuous mappings onto U(1) given by the determinant of matrices.

The center of the infinite-dimensional unitary group U is, as in the finite dimensional case, U(1), which again acts on the unitary group via multiplication by a phase. As the unitary group does not contain the zero matrix, this action is free. Thus U is a contractible space with a U(1) action, which identifies it as EU(1) and the space of U(1) orbits as BU(1), the classifying space for U(1).

Read more about this topic:  Projective Unitary Group, The Topology of PU(H)

Famous quotes containing the words circle, space and/or bundles:

    Change begets change. Nothing propagates so fast. If a man habituated to a narrow circle of cares and pleasures, out of which he seldom travels, step beyond it, though for never so brief a space, his departure from the monotonous scene on which he has been an actor of importance would seem to be the signal for instant confusion.... The mine which Time has slowly dug beneath familiar objects is sprung in an instant; and what was rock before, becomes but sand and dust.
    Charles Dickens (1812–1870)

    No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.
    Isaac Newton (1642–1727)

    He bundles every forkful in its place,
    And tags and numbers it for future reference,
    So he can find and easily dislodge it
    In the unloading. Silas does that well.
    He takes it out in bunches like birds’ nests.
    Robert Frost (1874–1963)