Numerical Index and Daugavet Property of Operator Ideals and Tensor Products

Maximal ideals of generalized summing linear operators

“…Following the long tradition of the interplay between operator ideals and tensor norms, which comes from Grothendieck's seminal works and stands to the day (see recent developments in [1,2,13,17,18,21,23,24,25,26]), we prove our main results defining, developing and applying a tensor quasi-norm α X;Y determined by the sequence classes X and Y . The tensor quasi-norms α X;Y can be regarded as generalizations of the classical Chevet-Saphar tensor norms (see [14,31]).…”

Estimations of the numerical index of a JB$^*$-triple

“…It should be noted that, in most of cases, the von Neumann tensor product A⊗C is out of scope for previous results computing the numerical index for C(K, X) spaces [63, Theorem 5], L ∞ (µ, X) spaces for a σ-finite measure µ [64], and projective and injective tensor products of Banach spaces [61].…”

“…The numerical index of each function algebra is 1 [73]. The numerical index of the projective and the injective tensor product of two Banach spaces is less than or equal to the minimum of the numerical indexes of both factors [61]. Many other recent publications have been devoted to the study of the numerical index (see, for example, [20,21,52,60,65], there are over 1600 papers published under the MSC item 47A12 "Numerical range, numerical radius").…”