Representations of $p$-convolution algebras on $L^q$-spaces

@article{Gardella2018RepresentationsO,
  title={Representations of \$p\$-convolution algebras on \$L^q\$-spaces},
  author={Eusebio Gardella and Hannes Thiel},
  journal={Transactions of the American Mathematical Society},
  year={2018}
}
For a nontrivial locally compact group $G$, and $p\in [1,\infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1,\infty)$, these Banach algebras can be represented on an $L^q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\frac 1p - \frac 12|=|\frac… 
14 Citations

Isomorphisms of Algebras of Convolution Operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered

Rigidity results for $L^p$-operator algebras and applications

For $p\in [1,\infty)$, we show that every unital $L^p$-operator algebra contains a unique maximal $C^*$-subalgebra, which we call the $C^*$-core. When $p\neq 2$, the $C^*$-core of an $L^p$-operator

Extending representations of Banach algebras to their biduals

We show that a representation of a Banach algebra A on a Banach space X can be extended to a canonical representation of $$A^{**}$$ A ∗ ∗ on X if and only if certain orbit maps $$A\rightarrow X$$ A →

Dynamic asymptotic dimension and $K$-theory of Banach crossed product algebras

We apply controlled (or quantitative) $K$-theory to prove that a certain $L_p$ assembly map is an isomorphism for $p\in(1,\infty)$ when a countable discrete group $\Gamma$ acts with finite dynamic

A modern look at algebras of operators on Lp-spaces

Open Cones and $K$-theory for $\ell^p$ Roe Algebras

In this paper, we verify the l coarse Baum-Connes conjecture for open cones and show that the K-theory for l Roe algebras of open cones are independent of p ∈ [1,∞). Combined with the result of T.

Weighted homomorphisms between C*-algebras

. We show that a bounded, linear map between C ∗ -algebras is a weighted ∗ -homomorphism (the central compression of a ∗ -homomorphism) if and only if it preserves zero-products, range-orthogonality,

Dynamical complexity and K-theory of Lp operator crossed products

We apply quantitative (or controlled) [Formula: see text]-theory to prove that a certain [Formula: see text] assembly map is an isomorphism for [Formula: see text] when an action of a countable

Preduals and complementation of spaces of bounded linear operators

For Banach spaces [Formula: see text] and [Formula: see text], we establish a natural bijection between preduals of [Formula: see text] and preduals of [Formula: see text] that respect the right

Expanders are counterexamples to the $\ell^p$ coarse Baum-Connes conjecture

A BSTRACT . We consider an ℓ p coarse Baum-Connes assembly map for 1 < p < ∞ , and show that it is not surjective for expanders arising from residually finite hyperbolic groups.

References

SHOWING 1-10 OF 31 REFERENCES

Isomorphisms of Algebras of Convolution Operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered

Group Algebras Acting on $$L^p$$Lp-Spaces

For $$p\in [1,\infty )$$p∈[1,∞) we study representations of a locally compact group $$G$$G on $$L^p$$Lp-spaces and $$\textit{QSL}^p$$QSLp-spaces. The universal completions $$F^p(G)$$Fp(G) and

Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces

For $p \in [1, \infty),$ we define and study full and reduced crossed products of algebras of operators on $\sigma$-finite $L^p$ spaces by isometric actions of second countable locally compact

Functoriality of group algebras acting on $L^p$-spaces

We continue our study of group algebras acting on $L^p$-spaces, particularly of algebras of $p$-pseudofunctions of locally compact groups. We focus on the functoriality properties of these objects.

CONVOLUTION OPERATORS AND HOMOMORPHISMS OF LOCALLY COMPACT GROUPS

Abstract Let $1{\TL }p{\TL }\infty $, let G and H be locally compact groups and let ω be a continuous homomorphism of G into H. We prove, if G is amenable, the existence of a linear contraction of

Contractive projections in Cp

We show that the range of a norm one projection on a commutative C*-algebra has a ternary product structure (Theorem 2). We describe and char- acterize all such projections in terms of extreme points

REPRESENTATIONS OF LOCALLY COMPACT GROUPS ON QSLP-SPACES AND A P-ANALOG OF THE FOURIER-STIELTJES ALGEBRA

For a locally compact group G and p ∈ (1, oo), we define Bp(G) to be the space of all coefficient functions of isometric representations of G on quotients of subspaces of L p spaces. For p = 2, this

Representations of C*-algebras in dual and right dual Banach algebras

The range of a contractive algebra morphism from a C*-algebra to a Banach algebra is closed, and the morphism is a C*-morphism onto its range. When the codomain is a dual Banach algebra, or only a

Banach algebras and automatic continuity

Banach algebras combine algebraic and analytical aspects: it is the interplay of these structures that gives the subject its fascination. This volume expounds the general theory of Banach algebras,