✦ LIBER ✦

On invertibility preserving linear mappings, simultaneous triangularization and Property L

✍ Scribed by Erik Christensen

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 133 KB
Volume: 301
Category: Article
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(99)00195-0

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✦ Synopsis

Let A be a unital matrix algebra, ϕ : A → M n (C) a unital linear mapping and B the algebra generated by ϕ(A). The mapping ϕ is a homomorphism modulo the Jacobson radical in B if and only if for k = dim(B)dim(ϕ(A)) + 3 the mapping ϕ ⊗ id : A ⊗ M k (C) → B ⊗ M k (C) preserves invertibility. This result is closely related to the question on the relationship between Property L and simultaneous triangularization. We introduce a Property kL and give criteria in terms of conditions on matrix pencils with coefficients in M k (C) rather than in C which ensure simultaneous triangularization.