C*-Algebras with the Approximate Positive Factorization Property

Authors:
G. J. Murphy and N. C. Phillips

Journal:
Trans. Amer. Math. Soc. **348** (1996), 2291-2306

MSC (1991):
Primary 46L05, 46L10

DOI:
https://doi.org/10.1090/S0002-9947-96-01657-1

MathSciNet review:
1357402

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Abstract: We say that a unital -algebra has the approximate positive factorization property (APFP) if every element of is a norm limit of products of positive elements of . (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of -algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial , and stable rank 1. In the unital case, the group separates the tracial states. The APFP passes to matrix algebras, and if is an ideal in such that and have the APFP, then so does . We also give some new examples of -algebras with the APFP, including type factors and infinite-dimensional simple unital direct limits of homogeneous -algebras with slow dimension growth, real rank zero, and trivial group. Simple direct limits of homogeneous -algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a -algebra that may be of interest in the future.

**[1]**C. S. Ballantine,*Products of positive definite matrices, IV*, Linear Alg. Appl.**3**(1970), 79--114. MR**41:1766****[2]**B. Blackadar, M. D\v{a}d\v{a}rlat, and M. Rørdam,*The real rank of inductive limit C*-algebras*, Math. Scand.**69**(1991), 211--216. MR**93e:46067****[3]**L. G. Brown and G. K. Pedersen,*C*-algebras of real rank zero*, J. Funct. Anal.**99**(1991), 131--149. MR**92m:46086****[4]**H. Choda,*An extremal property of the polar decomposition in von Neumann algebras*, Proc. Japan Acad.**46**(1970), 341--344. MR**50:8098****[5]**M.-D. Choi,*A simple C*-algebra generated by two finite-order unitaries*, Canadian J. Math.**31**(1979), 867-880. MR**80j:46092****[6]**J. Cuntz,*-theoretic amenability for discrete groups*, J. Reine Ang. Math.**344**(1983), 180-195. MR**86e:46064****[7]**J. Cuntz and G. K. Pedersen,*Equivalence and traces on C*-algebras*, J. Funct. Anal.**33**(1979), 135--164. MR**80m:46053****[8]**P. de la Harpe and G. Skandalis,*Déterminant associé à une trace sur une algèbre de Banach*, Ann. Inst. Fourier**43-1**(1984), 241--260. MR**87i:46146a****[9]**K. R. Goodearl,*Notes on a class of simple C*-algebras with real rank zero*, Publ. Sec. Mat. Univ. Autónoma Barcelona**36**(1992), 637--654. MR**94f:46092****[10]**K. R. Goodearl,*Riesz decomposition in inductive limit -algebras*, Rocky Mountain J. Math.**24**(1994), 1405--1430. CMP**95:9****[11]**U. Haagerup,*Quasitraces on exact C*-algebras are traces*, (1991) (handwritten manuscript).**[12]**P. R. Halmos,*A Hilbert Space Problem Book*, Springer, New York 1982. MR**84e:47001****[13]**G. G. Kasparov,*The operator K-functor and extensions of C*-algebras*, Izv. Akad. Nauk SSSR Ser. Mat.**44**(1980), 571--636; English transl. , Math. USSR Izv.**16**(1981), 513--572. MR**81m:58075****[14]**M. Khalkhali, C. Laurie, B. Mathes, and H. Radjavi,*Approximation by products of positive operators*, J. Operator Theory**29**(1993), 237--247. CMP**94:15****[15]**M. Leen,*Factorization in the Invertible Group of a -Algebra*, Ph.D. thesis, University of Oregon, Eugene, OR, 1994.**[16]**H. Lin,*Exponential rank of C*-algebras with real rank zero and Brown-Pedersen's conjecture*, J. Funct. Anal.**114**(1993), 1--11. MR**95a:46079****[17]**G. J. Murphy,*The analytic rank of a C*-algebra*, Proc. Amer. Math. Soc.**115**(1992), 741--746. MR**92i:46085****[18]**C. Pearcy and D. M. Topping,*Sums of small numbers of idempotents*, Michigan J. Math.**14**(1967), 453--465. MR**36:2006****[19]**N. C. Phillips,*Simple C*-algebras with the property weak (FU)*, Math. Scand.**69**(1991), 127--151. MR**93d:46121****[20]**N. C. Phillips,*Exponential length and traces*, Proc. Royal Soc. Edinburgh, Sec. A**125**(1995), 13--29. CMP**95:8****[21]**N. C. Phillips,*Factorization problems in the invertible group of a homogeneous C*-algebra*, (preprint) University of Oregon, 1993.**[22]**T. Quinn,*Ideals in AF-algebras and approximation by products of positive operators*, preprint, Trinity College, Dublin, 1993.**[23]**H. Radjavi,*Products of Hermitian matrices and symmetries*, Proc. Amer. Math. Soc.**21**(1969), 369--372; , vol. 26, 1970, pp. 701. MR**39:1470****[24]**M. A. Rieffel,*Dimension and stable rank in the K-theory of C*-algebras*, Proc. London Math. Soc.**46**((3)) (1983), 301--333. MR**84g:46085****[25]**M. A. Rieffel,*The homotopy groups of the unitary groups of noncommutative tori*, J. Operator Theory**17**(1987), 237--254. MR**88f:22018****[26]**M. Rørdam,*On the structure of simple C*-algebras tensored with a UHF algebra*, J. Funct. Anal.**100**(1991), 1--17. MR**92m:46091****[27]**M. Rørdam,*On the structure of simple C*-algebras tensored with a UHF algebra, II*, J. Funct. Anal.**107**(1992), 255--269. MR**93f:46094****[28]**J. Rosenberg and C. Schochet,*The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor*, Duke Math. J.**55**(1987), 431--474. MR**88i:46091****[29]**P. Y. Wu,*Products of normal operators*, Canadian J. Math.**40**(1988), 1322--1330. MR**90d:47039**

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Additional Information

**G. J. Murphy**

Affiliation:
Department of Mathematics, University College, Cork, Ireland

Email:
gjm@ucc.ie

**N. C. Phillips**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Email:
phillips@bright.uoregon.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01657-1

Received by editor(s):
April 21, 1994

Received by editor(s) in revised form:
December 19, 1994

Additional Notes:
Partially supported by NSF grant DMS-9106285.

Article copyright:
© Copyright 1996
American Mathematical Society