By Garrett P.
Read or Download Unitary representations of topological groups PDF
Similar differential geometry books
Minimal surfaces and Teichmuller theory
The notes from a suite of lectures writer brought at nationwide Tsing-Hua college in Hsinchu, Taiwan, within the spring of 1992. This notes is the a part of booklet "Thing Hua Lectures on Geometry and Analisys".
Complex, contact and symmetric manifolds: In honor of L. Vanhecke
This ebook is targeted at the interrelations among the curvature and the geometry of Riemannian manifolds. It includes study and survey articles in response to the most talks introduced on the foreign Congress
Differential Geometry and the Calculus of Variations
During this e-book, we research theoretical and functional points of computing equipment for mathematical modelling of nonlinear structures. a few computing options are thought of, resembling equipment of operator approximation with any given accuracy; operator interpolation suggestions together with a non-Lagrange interpolation; equipment of approach illustration topic to constraints linked to innovations of causality, reminiscence and stationarity; equipment of approach illustration with an accuracy that's the top inside a given category of versions; tools of covariance matrix estimation;methods for low-rank matrix approximations; hybrid equipment in keeping with a mixture of iterative methods and top operator approximation; andmethods for info compression and filtering below clear out version should still fulfill regulations linked to causality and forms of reminiscence.
- Geometry VI: Riemannian Geometry
- Algorithmic and Computer Methods for Three-Manifolds
- Surveys in differential geometry: essays on Einstein manifolds, Vol 6
- Characteristic classes
- Comprehensive Introduction to Differential Geometry
Additional resources for Unitary representations of topological groups
Sample text
That is, X = L2 (ZΓ\G, ω) is the completion of a direct sum of irreducibles. To see that a π-isotype is of finite multiplicity, again use compactness of the operators from Cco (G). With ϕ = ϕ∗ , R(ϕ) acts by scalar λ on X π (λ). Non-zero scalar operators are compact only for finite-dimensional spaces, so X π (λ) is finite-dimensional for λ = 0. Given π, take ϕ = ϕ∗ such that R(ϕ) is non-zero on the representation space Vπ of π, invoking the existence of approximate identities. Then Vπ (λ) = 0 for some non-zero λ.
Let π be unitary. Then 0≤| ci π(gi )v|2 = ij i ci cj π(gj−1 gi )vi , v ci cj π(gj )∗ π(gi )vi , v = ci cj π(gi )vi , π(gj )v = ij ij by unitariness. On the other hand, suppose f is positive definite. Let ci f (ggi ) E(g) = dj f (ghj ) F (g) = j i Using f (g −1 ) = f (g), ci dj f (h−1 j gi ) = E, F = dj E(h−1 j )= j ij ci F (gi−1 ) i Thus, the inner product E, F is independent of the expressions for E and F , but only depends on the functions. That is, the inner product is well-defined. That the inner product has the positive semi-definiteness property F, F ≥ 0 is immediate from the definition of positive-definiteness.
Traces, characters, central functions for Z ÒG compact Take Z\G compact with Z a closed subgroup of the center of G (so G is unimodular). From above, every irreducible unitary of G is finite dimensional, with some unitary central ω on Z. 1] Definition: Let π, V be an irreducible (finite-dimensional) of G. The character χπ of π is the function on G defined by χπ (g) = trace π(g) = π(g)ei , ei i for any orthonormal basis ei of π. 2] Remark: If π were not finite-dimensional, the character χπ of π could not be defined as a pointwise-evaluatable function, but only as some more general sort of entity.