By I. Kaplansky
An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and simple effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings maintaining invertible parts Nonassociativity Bibliography
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Extra info for Algebraic and analytic aspects of operator algebras
Further, T R must satisfy I = QQ−1 = QPT. As QP is a square matrix, we have R T = (QP)−1 . Thus, every χ-inverse must be of the form P(QP)−1 P−1 L . −1 −1 −1 −1 Conversely, A(P(QP) PL )A = PQP(QP) PL PQ = PQ = A −1 −1 and C(P(QP)−1 P−1 PL −1 L ) = C(P) = C(A), for each PL . So, P(QP) is a χ-inverse of A. Proof of (iii) is similar to proof of (ii). (iv) follows form (ii) and (iii). 5. Let A be an n × n matrix such that ρ(A) = ρ(A2 ). Let A = Pdiag(C, 0) P−1 , where P and C are non-singular. 28 Matrix Partial Orders, Shorted Operators and Applications (i) The class of all A− χ is given by P C−1 L P−1 ; where L is arbitrary.
N ). Then the spectral decomposition A = UΛU of A can also be written as n λi Ui Ui . 42. The above decomposition is in general not unique (unless all eigen-values are distinct). However, if µ1 , . . , µs (s ≤ n) are distinct eigen-vectors of A, then by appropriate pooling of the eigen-values we can write s µi Vi Vi , where Vi Vi = I , Vi Vj = 0 and I = U= Vi Vi . i=1 This decomposition is unique in the sense that the orthogonal projectors Vi Vi are unique. 43. (Spectral Decomposition of a Hermitian Matrix) Let A be an n×n matrix over C.
Then A is rangehermitian if and only if there exists a unitary matrix U such that A = Udiag(T, 0)U , where T is non-singular. 30. A matrix A over C is said to be simple if all its eigenvalues are distinct. A is said to be semi-simple, if the algebraic multiplicity for each of its distinct eigen-values equals its geometric multiplicity. 31. (Spectral Decomposition of a Semi-Simple Matrix) Let A be an n×n matrix over C. Then A is semi-simple if and only if there exist matrices E1 , . . , Es of order n × n and distinct complex numbers λ1 , .
Algebraic and analytic aspects of operator algebras by I. Kaplansky