Generalized induced norms

Abstract

Let . be a norm on the algebra M n of all n × n matrices over C. An interesting problem in matrix theory is that "are there two norms . 1 and . 2 on C n such that A = max{ Ax 2 : x 1 = 1} for all A ∈ M n . We will investigate this problem and its various aspects and will discuss under which conditions . 1 = . 2 . PreliminariesThroughout the paper M n denotes the complex algebra of all n × n matrices A = [a ij ] with entries in C together with the usual matrix operations. Denote by {e 1 , e 2 , · · · e n } the standard basis for C n , where e i has 1 as its ith entry and 0 elsewhere. We denote by E ij the n × n matrix with 1 in the (i, j) entry and 0 elsewhere.For 1 ≤ p ≤ ∞ the norm ℓ p on C n is defined as follows:A norm . on C n is said to be unitarily invariant if x = Ux for all unitaries U and all x ∈C n . * 2000 Mathematics Subject Classification 15A60 (Primary) 47A30, 46B99 (Secondary).Keywords and phrases. induced norm, generalized induced norm, algebra norm, the full matrix algebra, unitarily invariant, generalized induced congruent.

On the nonexistence of nontrivial involutive $n$-homomorphisms of $C^{\star }$-algebras

“…Furthermore, every homomorphism is clearly also an n-homomorphism for all n ≥ 2, but the converse is false, in general. The concept of n-homomorphism was studied for complex algebras by Hejazian, Mirzavaziri, and Moslehian [7]. This concept also makes sense for rings and (semi)groups.…”

“…The nonunital case is more subtle. For example, if A and B are nonunital (Banach) algebras such that A n = B n = {0}, then every linear map L : A → B (bounded or unbounded) is, trivially, an n-homomorphism (see Examples 2.5 and 4.3 of [7]). …”

“…Every homomorphism is an n-homomorphism for all n ≥ 2, but the converse is false, in general. Hejazian et al (2005) ask: Is every * -preserving n-homomorphism between C -algebras continuous? We answer their question in the affirmative, but the even and odd n arguments are surprisingly disjoint.…”

AUTOMATIC CONTINUITY OF n-HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS

“…For certain properties of 3-homomorphisms one may refer to [1]. If A is unital with the unit element e A and θ : A → B is an n-homomorphism then by [7,Proposition 2.2], there exists a homomorphism ϕ : A → B such that θ (a) = θ (e A )ϕ(a) for all a ∈ A. Furthermore, a 2-homomorphism is then just a homomorphism, in the usual sense.…”

“…Thus we may assume in the following that n ≥ 3. The concept of n-homomorphism was studied for complex algebras by Hejazian et al in [7]. Fragoulopoulou [4,5] in 1991 and 1993, and then Honary and Najafi [8] in 2008, obtained some results on the automatic continuity of homomorphisms between topological Q-algebras.…”