scholarly journals The ~ -representations of symmetric homogeneous algebras

1986 ◽  
Vol 40 (2) ◽  
pp. 267-275
Author(s):  
J. A. Ward
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Representation Space ◽  
Continuous Unitary Representation ◽  
Degenerate Representation ◽  
Homogeneous Algebras ◽  
Image Position

AbstractIn 1947 I. E. Segal proved that to each non-degenerate ~ -representation R of L1 (= L1 (G) for a compact group G) with representation space , there corresponds a continuous unitary representation W of G, also with representation space , which satisfiesfor each fL1 and hk . This was extended to Lp,1 p < , in 1970 by E. Hewitt and K. A. Ross. We now generalize this result to any symmetric homogeneous convolution Banach alebra of pseudomeasures on G. Further we prove that the correspondence preserves irreduibility.

On a question of R. Godement about the spectrum of positive, positive definite functions

1991 ◽  
Vol 110 (1) ◽  
pp. 137-142
Author(s):  
Mohammed B. Bekka
Keyword(s):  
Compact Group ◽  
Uniform Convergence ◽  
Unitary Representation ◽  
Convex Set ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Image Position ◽  
Compact Subsets

Let G be a locally compact group, and let P(G) be the convex set of all continuous, positive definite functions ø on G normalized by ø(e) = 1, where e denotes the group unit of G. For ø∈P(G) the spectrum spø of ø is defined as the set of all indecomposable ψ∈P(G) which are limits, for the topology of uniform convergence on compact subsets of G, of functions of the form(see [5], p. 43). Denoting by πø the cyclic unitary representation of G associated with ø, it is clear that sp ø consists of all ψ∈P(G) for which πψ is irreducible and weakly contained in πø (see [3], chapter 18).

scholarly journals Rudin-Shapiro sequences for arbitrary compact groups

1976 ◽  
Vol 22 (4) ◽  
pp. 421-430 ◽  
Author(s):  
J. R. McMullen ◽  
J. F. Price
Keyword(s):  
Compact Group ◽  
Trigonometric Polynomials ◽  
Compact Groups ◽  
Group A ◽  
Image Position

AbstractLet G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

scholarly journals Extension problems and non-Abelian duality for C*-algebras

2007 ◽  
Vol 75 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Astrid an Huef ◽  
S. Kaliszewski ◽  
Iain Raeburn
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Test Question ◽  
Locally Compact ◽  
Dual Representation ◽  
C Algebras

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

Counterexample to a Conjecture of Greenleaf

1970 ◽  
Vol 13 (4) ◽  
pp. 497-499 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Image Position

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that

Random Fourier Series on Compact Noncommutative Groups

1975 ◽  
Vol 27 (6) ◽  
pp. 1400-1407 ◽  
Author(s):  
Massimo A. Picardello
Keyword(s):  
Fourier Series ◽  
Compact Group ◽  
Irreducible Representations ◽  
Countable Subset ◽  
Dual Object ◽  
Image Position ◽  
Random Fourier Series

1. Let G be a compact group, let I b e a subset of its dual object 𝚪, which, without loss of generality, will be assumed to be a countable subset. Let Di, i ∈ I , be irreducible representations of G of degree di. The Fourier series of a function F in L1(G) is denned bywhere

scholarly journals A Stone-Weierstrass theorem for group representations

1978 ◽  
Vol 1 (2) ◽  
pp. 235-244 ◽  
Author(s):  
Joe Repka
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Regular Representation ◽  
Discrete Series ◽  
Series Representation ◽  
Tensor Products ◽  
Weil Representation ◽  
Weierstrass Theorem ◽  
Group Representations ◽  
Discrete Series Representation

It is well known that ifGis a compact group andπa faithful (unitary) representation, then each irreducible representation ofGoccurs in the tensor product of some number of copies ofπand its contragredient. We generalize this result to a separable typeIlocally compact groupGas follows: letπbe a faithful unitary representation whose matrix coefficient functions vanish at infinity and satisfy an appropriate integrabillty condition. Then, up to isomorphism, the regular representation ofGis contained in the direct sum of all tensor products of finitely many copies ofπand its contragredient.We apply this result to a symplectic group and the Weil representation associated to a quadratic form. As the tensor products of such a representation are also Weil representations (associated to different forms), we see that any discrete series representation can be realized as a subrepresentation of a Weil representation.

scholarly journals Some examples of nonmeasurable sets

1974 ◽  
Vol 18 (2) ◽  
pp. 236-238 ◽  
Author(s):  
Edwin Hewitt ◽  
Karl Stromberg
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Outer Measure ◽  
Recent Issue ◽  
Locally Compact ◽  
Image Position ◽  
Nonmeasurable Sets

In a recent issue of this Journal, Pu [3] has given an interesting construction of a nonmeasurable subset A of R such that for all intervals I in R. [Throughout this note, the symbol λ denotes Lebesgue outer measure on R or Haar outer measure on a general locally compact group.] This solves a problem stated in [2], p. 295, Exercise (18.30).

scholarly journals Integral operators in the theory of induced Banach representation II. The bundle approach

1981 ◽  
Vol 4 (4) ◽  
pp. 625-640 ◽  
Author(s):  
I. E. Schochetman
Keyword(s):  
Compact Group ◽  
Cross Sections ◽  
Closed Subgroup ◽  
Integral Operators ◽  
Locally Compact Group ◽  
Representation Space ◽  
Locally Compact

LetGbe a locally compact group,Ha closed subgroup andLa Banach representation ofH. SupposeUis a Banach representation ofGwhich is induced byL. Here, we continue our program of showing that certain operators of the integrated form ofUcan be written as integral operators with continuous kernels. Specifically, we show that: (1) the representation space of a Banach bundle; (2) the above operators become integral operators on this space with kernels which are continuous cross-sections of an associated kernel bundle.

scholarly journals A Note on the Kawada-into Theorem

1963 ◽  
Vol 13 (4) ◽  
pp. 295-296 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
Image Position ◽  
Complex Valued

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

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