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.
In this paper, we consider three classes of bounded linear operators on a topological vector space with respect to three different topologies which are introduced by Troitsky. We obtain some properties for the spectral radii of a linear operator on a topological vector space. We find some sufficient conditions for the completeness of these classes of operators. Finally, as a special application, we deduce some sufficient conditions for invertibility of a bounded linear operator.
Given p ∈ (1, ∞), let G be a countable Powers group, and let (G, A, α) be a separable nondegenerately representable isometric G-L p operator algebra. We show that if A is G-simple and unital then the reduced L p operator crossed product of A by G, F p r (G, A, α), is simple. This generalizes special cases of some results due to de la Harpe and Skanadalis in the C * -algebra context. We will also show that the result is not true for p = 1. Moreover, we prove that traces on F p r (G, A, α) are in natural bijection with G-invariant traces on A via the standard conditional expectation. As a consequence, for a countable powers group G the reduced L p operator group algebra, F p r (G), is simple and has a unique normalized trace.2010 Mathematics Subject Classification. Primary 46H05; Secondary 46H35, 47L10.
Let A and B be two Banach algebras and let M be a Banach B-bimodule. Suppose that σ : A → B is a linear mapping and d : A → M is a σ-derivation. We prove several results about automatic continuity of σderivations on Banach algebras. In addition, we define a notion for m-weakly continuous linear mapping and show that, under certain conditions, d and σ are m-weakly continuous. Moreover, we prove that if A is commutative and σ : A → A is a continuous homomorphism such that σ 2 = σ then σdσ(A) ⊆ σ(Q(A)) ⊆ rad(A).
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