48 research outputs found
Weak amenability and 2-weak amenability of Beurling algebras
Let L^1_\om(G) be a Beurling algebra on a locally compact abelian group
. We look for general conditions on the weight which allows the vanishing of
continuous derivations of L^1_\om(G). This leads us to introducing
vector-valued Beurling algebras and considering the translation of operators on
them. This is then used to connect the augmentation ideal to the behavior of
derivation space. We apply these results to give examples of various classes of
Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be
even 2-weakly amenable.Comment: 25 page
On local properties of Hochschild cohomology of a C- algebra
Let be a C-algebra, and let be a Banach -bimodule. B. E.
Johnson showed that local derivations from into are derivations. We
extend this concept of locality to the higher cohomology of a -algebra
%for -cocycles from into and show that, for every ,
bounded local -cocycles from into are -cocycles.Comment: 13 page
Exotic C*-algebras of geometric groups
We consider a new class of potentially exotic group C*-algebras
for a locally compact group , and its connection with the
class of potentially exotic group C*-algebras introduced by
Brown and Guentner. Surprisingly, these two classes of C*-algebras are
intimately related. By exploiting this connection, we show
for , and the C*-algebras
are pairwise distinct for when belongs to
a large class of nonamenable groups possessing the Haagerup property and either
the rapid decay property or Kunze-Stein phenomenon by characterizing the
positive definite functions that extend to positive linear functionals of
and . This greatly generalizes earlier results
of Okayasu and the second author on the pairwise distinctness of
for when is either a noncommutative free group or the group
, respectively.
As a byproduct of our techniques, we present two applications to the theory
of unitary representations of a locally compact group . Firstly, we give a
short proof of the well-known Cowling-Haagerup-Howe Theorem which presents
sufficient condition implying the weak containment of a cyclic unitary
representation of in the left regular representation of . Also we give a
near solution to a 1978 conjecture of Cowling. This conjecture of Cowling
states if is a Kunze-Stein group and is a unitary representation of
with cyclic vector such that the map belongs to for some , then
. We show for every
(recall )
Quotients of Fourier algebras, and representations which are not completely bounded
We observe that for a large class of non-amenable groups , one can find
bounded representations of on Hilbert space which are not completely
bounded. We also consider restriction algebras obtained from , equipped
with the natural operator space structure, and ask whether such algebras can be
completely isomorphic to operator algebras; partial results are obtained, using
a modified notion of Helson set which takes account of operator space
structure. In particular, we show that if is virtually abelian, then the
restriction algebra is completely isomorphic to an operator algebra if
and only if is finite.Comment: v3: 10 pages, minor edits and slight change to title from v2. Final
version, to appear in Proc. Amer. Math. So
Weighted discrete hypergroups
Weighted group algebras have been studied extensively in Abstract Harmonic
Analysis where complete characterizations have been found for some important
properties of weighted group algebras, namely amenability and Arens regularity.
One of the generalizations of weighted group algebras is weighted hypergroup
algebras. Defining weighted hypergroups, analogous to weighted groups, we study
Arens regularity and isomorphism to operator algebras for them. We also examine
our results on three classes of discrete weighted hypergroups constructed by
conjugacy classes of FC groups, the dual space of compact groups, and
hypergroup structure defined by orthogonal polynomials. We observe some
unexpected examples regarding Arens regularity and operator isomorphisms of
weighted hypergroup algebras.Comment: 27 pages. This version is shorter but still covers all the main
results of the previous on
LOCAL PROPERTIES OF THE HOCHSCHILD COHOMOLOGY OF C*-ALGEBRAS
Let A be a C*-algebra, and let X be a Banach A-bimodule. Johnson [B.E.Johnson, ‘Local derivations on C*-algebras are derivations', Trans. Amer. Math. Soc. 353 (2000), 313-325] showed that local derivations from A into X are derivations. We extend this concept of locality to the higher cohomology of a C*-algebra and show that, for every , bounded local n-cocycles from A(n) into X are n-cocycle