Topological group: multiplication and inversion are continuous

Depends on

• Group and abelian group
• Continuity of a map of topological spaces at a point and globally
• The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space

Used by

• Compact and discrete transforms are the two extreme Plancherel cases Corollary
• Dualisation is a contravariant involution Corollary
• Folner sequences for second countable compactly generated groups Corollary
• Normalized positive type and pointed cyclic unitary representations Corollary
• A bounded continuous normalized function that is not of positive type Counterexample
• Forgetting the compact-open topology destroys Pontryagin duality Counterexample
• LCA Fourier inversion is not an everywhere statement for arbitrary L¹ functions Counterexample
• No normalized translation-invariant Haar probability on the real line Counterexample
• The integers do not have property (T) Counterexample
• The unitary dual need not be Hausdorff Counterexample
• Topological abelian groups are additive but not abelian Counterexample
• A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups Definition
• Almost invariant vectors for a unitary representation Definition
• Compact-group isotypic projection Definition
• Compactly generated locally compact groups Definition
• Continuous positive-type functions and normalization Definition
• Kazhdan pairs, Kazhdan sets and Kazhdan constants Definition
• Kazhdan's property (T) Definition
• Left-invariant means on L^∞ of a locally compact group Definition
• Principal g bundle and associated fiber bundle Definition
• Relative property (T) for a pair and relative Kazhdan pairs Definition
• Representative functions on a compact group Definition
• States, tracial states and faithful normal traces on a von Neumann algebra Definition
• Strongly continuous unitary representations, invariant linear subspaces and intertwiners Definition
• Systems of imprimitivity for a Borel G-space Definition
• The annihilator of a subgroup Definition
• The Heisenberg group of a finite-dimensional symplectic vector space Definition
• The left and right uniformities of a topological group Definition
• The Pontryagin dual with the compact-open topology Definition
• The real projective line and the action of SL2(R) Definition
• The transformation (covariance) algebra C_c(G× X) Definition
• Transitive systems of imprimitivity and their normalized measure class Definition
• A circle representation with an averaged orthogonal weight form Example
• A compact group with no faithful continuous finite-dimensional representation Example
• A Kazhdan pair for a compact group via Haar averaging Example
• Følner sets in ℝⁿ Example
• GNS representation of a continuous unitary character Example
• Isotypic Haar projections specialize to finite character sums Example
• Peter-Weyl for a profinite group Example
• Positive type on a discrete group: the identity mass, characters, and the regular GNS model Example

…and 74 more results.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)