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

We investigate the quotient algebra U-X(I) := I(X)/(F(X))over bar||middot||(I) for Banach operator ideals I contained in the ideal of the compact operators, where X is a Banach space that fails the I-approximation property. The main results concern the nilpotent quotient algebras U-X(QNp) and U-X(SKp) for the quasi p-nuclear operators QN(p) and the Sinha-Karn p-compact operators SKp. The results include the following: (i) if X has cotype 2, then U-X(QNp) = {0} for every p >= 1; (ii) if X & lowast; has cotype 2, then U-X(SKp) = {0} for every p >= 1; (iii) the exact upper bound of the index of nilpotency of U-X(QNp) and U-X(SKp) for p &NOTEQUexpressionL; 2 is max{2, inverted right perpendicularp/2inverted left perpendicular}, where inverted right perpendicularp/2inverted left perpendicular denotes the smallest n is an element of N such that n >= p/2; (iv) for every p > 2 there is a closed subspace X subset of c(0) such that both U-X(QNp) and U-X(SKp) contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace X subset of c(0) such that the compact-by-approximable algebra U-X = K(X)/A(X) contains two incomparable countably infinite chains of nilpotent closed ideals. (C) 2022 The Author(s). Published by Elsevier Inc.