TY - JOUR

T1 - On C*-algebras generated by idempotents

AU - Krupnik, Naum

AU - Roch, Steffen

AU - Silbermann, Bernd

PY - 1996/5/1

Y1 - 1996/5/1

N2 - The topic of the present paper is concrete Banach and C*-algebras which are generated by a finite number of idempotents. Our first result is that, for each finitely generated Banach algebra script capital A sign, there is a number n0 so that the algebra script capital A signn×n of all n × n matrices with entries in script capital A sign is generated by three idempotents whenever n ≥n 0, and that script capital A signn×n is generated by two idempotents if and only if n = 2 and if script capital A sign is singly generated. As an application we find that the algebra Cn×n(K) of all continuous ℂn×n-matrix-valued functions on a compact K ⊂ ℂ with connected complement but without interior points, is generated by 2 or 3 idempotents in case n = 2 or n > 2, respectively. This result is used to construct examples of C*-algebras which are generated by 2 idempotents but not 2 projections. For these algebras, the standard 2 × 2 matrix symbol fails to be symmetric. We finally show that each C*-algebra satisfying a polynomial identity (in particular, each C*-algebra generated by two idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one.

AB - The topic of the present paper is concrete Banach and C*-algebras which are generated by a finite number of idempotents. Our first result is that, for each finitely generated Banach algebra script capital A sign, there is a number n0 so that the algebra script capital A signn×n of all n × n matrices with entries in script capital A sign is generated by three idempotents whenever n ≥n 0, and that script capital A signn×n is generated by two idempotents if and only if n = 2 and if script capital A sign is singly generated. As an application we find that the algebra Cn×n(K) of all continuous ℂn×n-matrix-valued functions on a compact K ⊂ ℂ with connected complement but without interior points, is generated by 2 or 3 idempotents in case n = 2 or n > 2, respectively. This result is used to construct examples of C*-algebras which are generated by 2 idempotents but not 2 projections. For these algebras, the standard 2 × 2 matrix symbol fails to be symmetric. We finally show that each C*-algebra satisfying a polynomial identity (in particular, each C*-algebra generated by two idempotents) possesses a symmetric matrix valued symbol and, hence, the standard symbol can always be replaced by a symmetric one.

UR - http://www.scopus.com/inward/record.url?scp=0030138835&partnerID=8YFLogxK

U2 - 10.1006/jfan.1996.0048

DO - 10.1006/jfan.1996.0048

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AN - SCOPUS:0030138835

SN - 0022-1236

VL - 137

SP - 303

EP - 319

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -