Embedding Theorems in Group C*-Algebras†

1983 ◽  
Vol 26 (2) ◽  
pp. 157-166 ◽  
Author(s):  
Tan-Yu Lee
Keyword(s):  
Open Subgroup ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Embedding Theorems ◽  
Group Representations ◽  
Locally Compact ◽  
C Algebras ◽  
Reduced Group ◽  
Natural Way

AbstractLet G be a locally compact group and H an open subgroup of G. The embeddings of group C*-algebras associated with H into the group C*-algebras associated with G are studied. Three conditions for the embeddings given in terms of C*-norms of the group algebras, group representations and positive definite functions are shown to be equivalent. As corollary, we prove that the full C*-algebra of H can be embedded into the full C*-algebra of G in a natural way as well as the case for the reduced group C*-algebras. We also show that the embeddings hold for their duals and double duals.

HYPERREFLEXIVITY CONSTANTS OF THE BOUNDED -COCYCLE SPACES OF GROUP ALGEBRAS AND C*-ALGEBRAS

2019 ◽  
Vol 109 (1) ◽  
pp. 112-130
Author(s):  
EBRAHIM SAMEI ◽  
JAFAR SOLTANI FARSANI
Keyword(s):  
Polynomial Growth ◽  
Banach Algebras ◽  
Amenable Group ◽  
Open Subgroup ◽  
Group Algebras ◽  
Convolution Operators ◽  
Locally Compact ◽  
C Algebra ◽  
C Algebras ◽  
Strong Property

We introduce the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288\unicode[STIX]{x1D70B}(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of ${\mathcal{Z}}^{n}(A,X)$, the space of bounded $n$-cocycles from $A$ into $X$, where $A$ is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule for which ${\mathcal{H}}^{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1<p<\infty$, the space $CV_{P}(G)$ of convolution operators on $L^{p}(G)$ is hyperreflexive with a constant given by $384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$. This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, Math. Scand. 50(1) (1982), 111–122] for $p=2$.

scholarly journals A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

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.

scholarly journals META-CENTRALIZERS OF NON-LOCALLY COMPACT GROUP ALGEBRAS

2014 ◽  
Vol 57 (2) ◽  
pp. 349-364 ◽  
Author(s):  
S. V. LUDKOVSKY
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact

AbstractMeta-centralizers of non-locally compact group algebras are studied. Theorems about their representations with the help of families of generalized measures are proved. Isomorphisms of group algebras are investigated in relation with meta-centralizers.

scholarly journals Equivariant KK-theory for inverse limits of G-C*-algebras

1994 ◽  
Vol 56 (2) ◽  
pp. 183-211 ◽  
Author(s):  
Claude Schochet
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Inverse Limits ◽  
Locally Compact ◽  
C Algebras

AbstractThe Kasparov groups are extended to the setting of inverse limits of G-C*-algebras, where G is assumed to be a locally compact group. The K K-product and other important features of the theory are generalized to this setting.

scholarly journals Crossed products of Hilbert pro-C-bimodules and associated pro-C-algebras

2016 ◽  
Vol 32 (2) ◽  
pp. 195-201
Author(s):  
MARIA JOITA ◽  
◽  
RADU-B. MUNTEANU ◽  
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Locally Compact ◽  
C Algebras

An action (γ, α) of a locally compact group G on a Hilbert pro-C∗-bimodule (X, A) induces an action γ × α of G on A ×X Z the crossed product of A by X. We show that if (γ, α) is an inverse limit action, then the crossed product of A ×α G by X ×γ G respectively of A ×α,r G by X ×γ,r G is isomorphic to the full crossed product of A ×X Z by γ × α respectively the reduced crossed product of A ×X Z by γ × α.

On various types of convergence of positive definite functions on foundation semigroups

1992 ◽  
Vol 111 (2) ◽  
pp. 325-330 ◽  
Author(s):  
M. Lashkarizadeh-Bami
Keyword(s):  
Compact Group ◽  
Pointwise Convergence ◽  
Positive Definite ◽  
Limit Problem ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Interesting Discussion ◽  
Central Limit Problem ◽  
The Relationship

As is known, on a locally compact group G, the mere assumption of pointwise convergence of a sequence (n) of continuous positive definite functions implies uniform convergence of (n) to on compact subsets of G. This result was first proved in 1947 by Raikov8 (and independently by Yoshizawa9). An interesting discussion of the relationship between such theorems and various Cramr-Lvy theorems of the 1920s and 1930s, concerning the Central Limit Problem of probability, is given by McKennon(7, p. 62).

scholarly journals Multinorms and Approximate Amenability of Weighted Group Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Saman Ghaderkhani
Keyword(s):  
Weight Function ◽  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Group Algebras ◽  
Locally Compact ◽  
Approximate Amenability

Let G be a locally compact group, and take p,q with 1≤p,q<∞. We prove that, for any left (p,q)-multiinvariant functional on L∞(G) and for any weight function ω≥1 on G, the approximate amenability of the Banach algebra L1(G,ω) implies the left (p,q)-amenability of G, but in general the opposite is not true. Our proof uses the notion of multinorms. We also investigate the approximate amenability of M(G,ω).

scholarly journals On the stable rank and real rank of group $C^*$-algebras of nilpotent locally compact groups

2005 ◽  
Vol 97 (1) ◽  
pp. 89
Author(s):  
Robert J. Archbold ◽  
Eberhard Kaniuth
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Stable Rank ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Rank ◽  
Locally Compact ◽  
C Algebras ◽  
Necessary And Sufficient

It is shown that if $G$ is an almost connected nilpotent group then the stable rank of $C^*(G)$ is equal to the rank of the abelian group $G/[G,G]$. For a general nilpotent locally compact group $G$, it is shown that finiteness of the rank of $G/[G,G]$ is necessary and sufficient for the finiteness of the stable rank of $C^*(G)$ and also for the finiteness of the real rank of $C^*(G)$.

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