RANGE PROJECTIONS OF IDEMPOTENTS IN C*-ALGEBRAS
By Jerry J. Koliha. 2001
52 citations recorded by the source corpus.
In this paper we study range projections of idempotents in C*-algebras, and use them to obtain a Schur type decomposition that leads to simple proofs of results on Moore-Penrose inverse and norms of idempotents. We analyze the continuity of range projections, obtain a general result on their approximation, and recover a result of Vidav on two projections in a Hilbert space. Several representations of range projections are given. 1. Range projections Basic facts about C*-algebras needed in this paper can be found, for example, in Davidson's monograph [3]. In this paper, 21 is a unital C*algebra with unit I . By 2 l 1 we denote the set of all invertible elements of 21. We recall that I + A*A € 21" for all A € 21 and that \\A*A\\ = \\A\\ (the C*-identity). An element A € 21 is polar if 0 is at most a pole of the resolvent of A, and quasipolar if 0 is an isolated singularity of the resolvent of A. We will need the following characterization of quasipolar elements of 21. LEMMA 1.1 [9, Theorem 4.2]. An element A G 21 is quasipolar if and only if there exists an idempotent P € 21 commuting with A such that AP is quasinilpotent and A + P E 2 l 1 . Such idempotent is unique, and is called the spectral idempotent of A corresponding to 0, written A. From the preceding lemma it follows that A is polar if and only it is quasipolar and AA = 0 for some integer k. In particular, A is simply polar if and only if A is quasipolar and AA = 0. The word 'projection' will be reserved for an element Q of a C*-algebra 21 which is self-adjoint and idempotent, that is, Q* = Q = Q. A motivation for 1991 Mathematics Subject Classification: 46L05, 46H30, 47A60.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/01f76c6efa4575f50f4a0e63c6e44a6a1ee960a7
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In Algebra and Number Theory
- Dragan S. Djordjevi'c
- Vladimir Rakocevic
- Dragana S. Cvetković
- Xavier Mary
- Jerry J. Koliha
- Ilya M. Spitkovsky
- Albrecht Böttcher
- Pedro Patrício
In Computational Theory and Mathematics
- Dragan S. Djordjevi'c
- Vladimir Rakocevic
- Dragana S. Cvetković
- Xavier Mary
- Jerry J. Koliha
- Chunyuan Deng
- Pedro Patrício
- Yimin Wei
In Mathematical Physics
- Dragan S. Djordjevi'c
- Vladimir Rakocevic
- Dragana S. Cvetković
- Xavier Mary
- Chunyuan Deng
- Dragana S. Cvetkovic-Ilic
Citations in the corpus
- On generalized inverses and Green's relations (2009, background): “Combining theorem 4 and theorem 9, we then get directly the following existence criteria and commuting relations for the classical inverses [6], [3], [7], [8], [9], [10], [13]:”
- Moore–Penrose inverse in rings with involution (2007, methodology): “strong emphasis on algebraic methods, appeared [16, 17 , 18].”
- Invertibility of the Difference of Idempotents (2003, methodology): “The question of the invertibility of P Q, where P, Q are idempotent hermitian matrices or, more generally, hermitian idempotent operators on a Hilbert space H, is of great interest in operator theory as it is connected with the question of when the space H is the direct sum H ¼ RðPÞ RðQÞ of the rang”
- The difference and sum of projectors (2004, background): “Conditions equivalent to H = R(P ) ⊕R (Q), where P, Q are orthogonal projectors on a Hilbert space H, were investigated in [6,8,11], and in a number of recent papers [1,2, 4 ,5,9,12,13].”