Representations of *-Algebras, Locally Compact Groups, and by J. M.G. Fell, R. S. Doran

By J. M.G. Fell, R. S. Doran

This is often an all-encompassing and exhaustive exposition of the idea of infinite-dimensional Unitary Representations of in the community Compact teams and its generalization to representations of Banach algebras. The presentation is certain, available, and self-contained (except for a few basic wisdom in algebra, topology, and summary degree theory). within the later chapters the reader is dropped at the frontiers of present-day wisdom within the zone of Mackey common subgroup analysisand its generalization to the context of Banach *-Algebraic Bundles.

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Additional info for Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles: Basic Representation Theory of Groups and Algebras: 001 ... of Algebras, Locally Compact Groups, & Banac)

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It becomes meaningful only when we point to a particular observer, equipped with a particular coordinate system, and ask what is the velocity with respect to this coordinate system. Indeed, no observer can be intrinsically distinguished from any other observer on the basis of the absolute speed or direction of his motion. Quite generally, then, by a frame of rejkrence, or coordinate system, we shall mean a particular material system, attached to an observer, in terms of which we suppose that the observer can describe and measure exhaustively (in terms of numbers or other impersonal symbols) any object or event in the universe.

A physical example of a tensor of rank 2 is the state of “stress” at a point within a solid medium. If we were to multiply the right side of (33) by det(g), we would obtain the definition of a pseudo-tensor of rank r. The physical applicability of these examples leads us, from a purely physical motivation, to wonder how many essentially different real-linear representations of O(3) actually exist. That is, what types of real-linear quantities are there with respect to this group? Interestingly enough, it turns out that the quantities which we have already constructed in a certain sense exhaust all the possibilities.

These two possibilities, at first sight qualitatively so different, become unified when described in terms of W-particles. In both cases a Wparticle, initially in a proton state, collides with the nucleus and rebounds from it in an altered state-in the first case a proton state with altered velocity, in the second a neutron state. This approach suggests a group-representational basis for classifying and unifying the multitudes of different so-called elementary particles observed in high-energy physics laboratories.

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