Quotient algebras of Banach operator ideals related to non-classical approximation properties

Abstract

We investigate the quotient algebra A I X := I(X)/F(X)||•|| I for Banach operator contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace X ⊂ c0 such that the compact-by-approximable algebra AX = K(X)/A(X) contains two incomparable countably infinite chains of nilpotent closed ideals.2010 Mathematics Subject Classification. 46B28, 47L20, 47B10. Key words and phrases. Quotient algebra, Banach operator ideals, approximation properties, quasi p-nuclear operators, Sinha-Karn p-compact operators.This work was supported by the Magnus Ehrnrooth Foundation and the Swedish Cultural Foundation in Finland.

On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators

“…We point out that Sinha-Karn p-compact operators were introduced by Sinha and Karn [42] as p-compact operators and with the notation (K p , κ p ) for the corresponding Banach operator ideal. However, we have decided to adopt the terminology and notation from [44,45] in order to obtain a clear distinction from the historically earlier class of (classical) p-compact operators [15,35], which are denoted by K p in this paper (see below).…”

“…However, our approach is based on a factorisation argument of Reinov [40,Lemma 1.1] in contrast to the approach of Choi and Kim in [4]. For other similar constructions based on [40, Lemma 1.1] we refer to [44, Theorem 3.9] and [45,Proposition 4.3].…”

“…[9,32,27,24,25,26] and their references. Moreover, results about special instances of the I-AP have been used in the recent studies [44,45] of radical quotient algebras of Banach operator ideals.…”