On the size of quotients of function spaces on a topological group

Abstract

For a non-precompact topological group G, we consider the space C(G) of bounded, continuous, scalar-valued functions on G with the supremum norm, together with the subspace LMC (G) of left multiplicatively continuous functions, the subspace LUC (G) of left norm continuous functions, and the subspace WAP (G) of weakly almost periodic functions.We establish that the quotient space LUC (G)/WAP (G) contains a linear isometric copy of ∞, and that the quotient space C(G)/LMC (G) (and a fortiori C(G)/LUC (G)) contains a linear isometric copy of ∞ when G is a normal non-P -group. When G is not a P -group but not necessarily normal we prove that the quotient is non-separable. For nondiscrete P -groups, the quotient may sometimes be trivial and sometimes non-separable. When G is locally compact, we show that the quotient space LUC (G)/WAP (G) contains a linear isometric copy of ∞(κ(G)), where κ(G) is the minimal number of compact sets needed to cover G. This leads to the extreme non-Arens regularity of the group algebra L 1 (G) when in addition either κ(G) is greater than or equal to the smallest cardinality of an open base at the identity e of G, or G is metrizable. These results are improvements and generalizations of theorems proved by various authors along the last 35 years and until very recently.

Extreme non-Arens regularity of the group algebra

“…So far, the Banach algebras known to be enAr are: the group algebra L 1 (G) for any infinite locally compact group [12] (some particular, but important, cases were previously proved in [4,22]), ℓ 1 (S) for any infinite discrete cancellative semigroup [15] (the proof in [4] is given for locally compact groups but can also be applied also in this case) and the Fourier algebra A(G) for any locally compact group satisfying χ(G) ≥ κ(G), where χ(G) is the smallest cardinality of an open basis of the identity of G and κ(G) is the smallest cardinality of a compact covering of G ( [26], [22] when χ(G) = ω).…”

Interpolation sets and the size of quotients of function spaces on a locally compact group

“…We prove, in Section 5, that there is a linear isometric copy of ℓ ∞ (κ) in CB(G)/LUC(G), where as before κ is the compact covering G, if and only if G is a neither compact nor discrete. This leads again to a linear isometric copy of ℓ ∞ (κ) into the quotient L ∞ (G)/WAP(G), and of course may be used to deduce again the extreme non-Arens regularity of of L 1 (G) when κ(G) ≥ w(G) ≥ ω as in [6,Theorem 4.4].…”

“…It was also proved in [6,Section 4] that the quotient L ∞ (G)/CB(G) always contains a linear isometric copy of ℓ ∞ , yielding extreme non-Arens regularity for the group algebra of compact metrizable groups. Due to a result by Rosenthal proved in [48,Proposition 4.7,Theorem 4.8], larger copies of ℓ ∞ cannot be expected in L ∞ (G) when G is compact.…”

“…So this theorem improved actually also Dzinotyiweyi's result for locally compact groups. For non-discrete, P -groups, the quotient CB(G)/LUC(G) was seen to be trivial in the case when for instance G is a Lindelöf P -group (see [28,Theorem 5.1]), but may also contain a linear isometric copy of ℓ ∞ for some other P -groups (see [6,Theorem 3.3]). In [7, Theorem 3.1], using a technique due to Alas (see [1]), the quotient space CB(G)/LUC(G) was also seen to contain a linear isometric copy of ℓ ∞ whenever G is a non-SIN topological group.…”

The bidual of a radical operator algebra can be semisimple

“…Let H be a Hilbert space with orthonormal basis (e i ) i∈N . Let T 0 : H → H be the operator with (1) T 0 e i = e i+1 , if i is odd; 0, if i is even.…”