Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Monfared

2010

Self Cite

Section: Introductionmentioning

confidence: 84%

Section: Preliminariesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On ideals in the bidual of the Fourier algebra and related algebras

Filali

Monfared

2010

Self Cite

“…Secondly, on the one hand, for many amenable groups G, the Fourier algebra A(G) is SAI (cf. [10,17,18,32,33]). On the other hand, as shown by Losert [35,36], both A(F 2 ) and A(SU (3) where λ m and ρ m are the maps on A * * given by n −→ m n and n −→ n ♦ m, respectively (cf.…”

Section: Proof (I) Due To Proposition 24 and Corollarymentioning

confidence: 97%

Module maps on duals of Banach algebras and topological centre problems

Ruan

2011

We study various spaces of module maps on the dual of a Banach algebra A, and relate them to topological centres. We introduce an auxiliary topological centre Z t ( A * A * ) ♦ for the left quotient Banach algebra A * A * of A * * . Our results indicate that Z t ( A * A * ) ♦ is indispensable for investigating properties of module maps over A * and for understanding some asymmetry phenomena in topological centre problems as well as the interrelationships between different Arens irregularity properties. For the class of Banach algebras of type (M) introduced recently by the authors, we show that strong Arens irregularity can be expressed both in terms of automatic normality of A * * -module maps on A * and through certain commutation relations. This in particular generalizes the earlier work on group algebras by Ghahramani and McClure (1992) [13] and by Ghahramani and Lau (1997) [12]. We link a module map property over A * to the space WAP(A) of weakly almost periodic functionals on A, generalizing a result by Lau and Ülger (1996) [34] for Banach algebras with a bounded approximate identity. We also show that for a locally compact quantum group G, the quotient strong Arens irregularity of L 1 (G) can be obtained from that of M(G) and can be characterized via the canonical C 0 (G)-module structure on LUC(G) * .

Geometry of C⁎-algebras, and the bidual of their projective tensor product

2020

Let A be a C * -algebra, and consider the Banach algebra A⊗ γ A, where ⊗ γ denotes the projective Banach space tensor product; if A is commutative, this is the Varopoulos algebra V A . It has been an open problem for more than 35 years to determine precisely when A ⊗ γ A is Arens regular; cf. [26], [40], [41]. Even the situation for commutative A, in particular the case A = ℓ ∞ , has remained unsolved. We solve this classical question for arbitrary C * -algebras by using von Neumann algebra and operator space methods, mainly relying on versions of the (commutative and non-commutative) Grothendieck Theorem, and the structure of completely bounded module maps. Establishing these links allows us to show that A ⊗ γ A is Arens regular if and only if A has the Phillips property; equivalently, A is scattered and has the Dunford-Pettis Property. A further equivalent condition is that A * has the Schur property, or, again equivalently, the enveloping von Neumann algebra A * * is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A ⊗ γ A is encoded in the geometry of the C * -algebra A. In case A is a von Neumann algebra, we conclude that A ⊗ γ A is Arens regular (if and) only if A is finite-dimensional. We also show that this does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C * -algebras A, we obtain that V A is Arens regular if and only if A is scattered. In fact, we determine precisely the centre of the bidual, namely, Z(V A * * ) is Banach algebra isomorphic to A * * ⊗ eh A * * , where ⊗ eh denotes the extended Haagerup tensor product. We deduce that V A is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, V A is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered (as mentioned above, this is new even for the case A = ℓ ∞ ).