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Journal articles on the topic 'Unitary rank'

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Published: 9 March 2023
Last updated: 10 March 2023

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1

Delvaux, Steven, and Marc Van Barel. "Unitary rank structured matrices." Journal of Computational and Applied Mathematics 215, no. 1 (May 2008): 49–78. http://dx.doi.org/10.1016/j.cam.2007.03.020.

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2

Hanzer, Marcela. "Rank one reducibility for unitary groups." Glasnik matematicki 46, no. 1 (June 12, 2011): 121–48. http://dx.doi.org/10.3336/gm.46.1.12.

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3

Elliott, George A., and Zhichao Liu. "Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero." Journal of Operator Theory 86, no. 2 (November 15, 2021): 299–316. http://dx.doi.org/10.7900/jot.2020apr21.2306.

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Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.
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4

Dudas, Olivier, and Gunter Malle. "Decomposition matrices for low-rank unitary groups." Proceedings of the London Mathematical Society 110, no. 6 (April 6, 2015): 1517–57. http://dx.doi.org/10.1112/plms/pdv008.

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5

Bevilacqua, Roberto, Enrico Bozzo, and Gianna M. Del Corso. "Transformations to rank structures by unitary similarity." Linear Algebra and its Applications 402 (June 2005): 126–34. http://dx.doi.org/10.1016/j.laa.2004.12.029.

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6

Jiang, Yunjiang, Lon H. Mitchell, and Sivaram K. Narayan. "Unitary matrix digraphs and minimum semidefinite rank." Linear Algebra and its Applications 428, no. 7 (April 2008): 1685–95. http://dx.doi.org/10.1016/j.laa.2007.10.031.

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7

Delvaux, Steven, and Marc Van Barel. "Eigenvalue computation for unitary rank structured matrices." Journal of Computational and Applied Mathematics 213, no. 1 (March 2008): 268–87. http://dx.doi.org/10.1016/j.cam.2007.01.006.

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8

Barnet-Lamb, Thomas, Toby Gee, and David Geraghty. "Serre weights for rank two unitary groups." Mathematische Annalen 356, no. 4 (January 9, 2013): 1551–98. http://dx.doi.org/10.1007/s00208-012-0893-y.

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9

Flaschka, Hermann, and John Millson. "Bending Flows for Sums of Rank One Matrices." Canadian Journal of Mathematics 57, no. 1 (February 1, 2005): 114–58. http://dx.doi.org/10.4153/cjm-2005-006-3.

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AbstractWe study certain symplectic quotients of n-fold products of complex projective m-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group.
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10

PARIS, MATTEO G. A. "UNITARY LOCAL INVARIANCE." International Journal of Quantum Information 03, no. 04 (December 2005): 655–59. http://dx.doi.org/10.1142/s0219749905001523.

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We address unitary local (UL) invariance of bipartite pure states. Given a bipartite state |Ψ〉〉 = ∑ij ψij |i〉1 ⊗ |j〉2 the complete characterization of the class of local unitaries U1 ⊗ U2 for which U1 ⊗ U2|Ψ〉〉 = |Ψ〉〉 is obtained. The two relevant parameters are the rank of the matrix Ψ, [Ψ]ij = ψij, and the number of its equal singular values, i.e. the degeneracy of the eigenvalues of the partial traces of |Ψ〉〉.
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11

Putnam, Ian F., and Mikael Rordam. "The maximum unitary rank of some $C^*$-algebras." MATHEMATICA SCANDINAVICA 63 (June 1, 1988): 297. http://dx.doi.org/10.7146/math.scand.a-12242.

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12

Douglas, Ronald G., and Constanze Liaw. "A geometric approach to finite rank unitary perturbations." Indiana University Mathematics Journal 62, no. 1 (2013): 333–54. http://dx.doi.org/10.1512/iumj.2013.62.5028.

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13

Wong, W. J. "Rank 1 preservers on the unitary Lie ring." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 3 (December 1990): 399–417. http://dx.doi.org/10.1017/s1446788700032419.

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AbstractThe surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.
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14

Baranov, Anton, Vladimir Kapustin, and Andrei Lishanskii. "On hypercyclic rank one perturbations of unitary operators." Mathematische Nachrichten 292, no. 5 (November 28, 2018): 961–68. http://dx.doi.org/10.1002/mana.201800242.

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15

Ringrose, J. R. "Exponential length and exponential rank in C*-algebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 121, no. 1-2 (1992): 55–71. http://dx.doi.org/10.1017/s0308210500014141.

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SynopsisIn an operator algebra, the general element of the connected component of the unitary group can beexpressed as a finite product of exponential unitary elements. The recently introduced concept of exponential rank is defined in terms of the number of exponentials required for this purpose. The present paper is concerned with a concept of exponential length, determined not by the number of exponentials but by the sum of the norms of their self-adjoint logarithms. Knowledge of the exponential length of an algebra provides an upper bound for its exponential rank (but not conversely). This is used to estimate the exponential rank of certain algebras of operator-valued continuous functions.
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16

Ng, P. W., and P. Skoufranis. "Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras." Canadian Journal of Mathematics 69, no. 5 (October 1, 2017): 1109–42. http://dx.doi.org/10.4153/cjm-2016-045-5.

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AbstractIn this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.
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17

Voronetsky, Egor. "Injective stability for odd unitary K1." Journal of Group Theory 23, no. 5 (September 1, 2020): 781–800. http://dx.doi.org/10.1515/jgth-2020-0013.

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AbstractWe give a new purely algebraic approach to odd unitary groups using odd form rings. Using these objects, we give a self-contained proof of injective stability for the odd unitary {K_{1}}-functor under the stable rank condition.
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18

Choi, Man-Duen, John A. Holbrook, David W. Kribs, and Karol Życzkowski. "Higher-rank numerical ranges of unitary and normal matrices." Operators and Matrices, no. 3 (2007): 409–26. http://dx.doi.org/10.7153/oam-01-24.

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19

Blok, Rieuwert J., and Bruce N. Cooperstein. "The generating rank of the unitary and symplectic Grassmannians." Journal of Combinatorial Theory, Series A 119, no. 1 (January 2012): 1–13. http://dx.doi.org/10.1016/j.jcta.2011.07.002.

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20

Shkarin, Stanislav. "A hypercyclic finite rank perturbation of a unitary operator." Mathematische Annalen 348, no. 2 (January 28, 2010): 379–93. http://dx.doi.org/10.1007/s00208-010-0479-5.

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21

Bevilacqua, Roberto, Gianna M. Del Corso, and Luca Gemignani. "Fast QR iterations for unitary plus low rank matrices." Numerische Mathematik 144, no. 1 (October 23, 2019): 23–53. http://dx.doi.org/10.1007/s00211-019-01080-4.

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22

Grivaux, Sophie. "A hypercyclic rank one perturbation of a unitary operator." Mathematische Nachrichten 285, no. 5-6 (January 10, 2012): 533–44. http://dx.doi.org/10.1002/mana.201000112.

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23

Hu, Mengyao, Lin Chen, and Yize Sun. "Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2235 (March 2020): 20190754. http://dx.doi.org/10.1098/rspa.2019.0754.

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Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2 ⊗ V and I 2 ⊗ W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.
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24

Poguntke, Detlev. "Unitary Representations of Lie Groups and Operators of Finite Rank." Annals of Mathematics 140, no. 3 (November 1994): 503. http://dx.doi.org/10.2307/2118617.

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25

Riva, Enea, and Lidia Stoppino. "The slope of fibred surfaces: Unitary rank and Clifford index." Proceedings of the London Mathematical Society 124, no. 1 (January 2022): 83–105. http://dx.doi.org/10.1112/plms.12424.

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26

Rordam, Mikael. "Advances in the Theory of Unitary Rank and Regular Approximation." Annals of Mathematics 128, no. 1 (July 1988): 153. http://dx.doi.org/10.2307/1971465.

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27

Cao, Huiqin, Jianbei An, and Jiwen Zeng. "The essential 2-rank of general linear and unitary groups." Journal of Pure and Applied Algebra 225, no. 3 (March 2021): 106518. http://dx.doi.org/10.1016/j.jpaa.2020.106518.

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28

Zhu, Fu-hai, and Ke Liang. "Dirac cohomology of unitary representations of equal rank exceptional groups." Science in China Series A: Mathematics 50, no. 4 (April 2007): 515–20. http://dx.doi.org/10.1007/s11425-007-2081-1.

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29

Larson, Hannah K. "Pseudo-unitary non-self-dual fusion categories of rank 4." Journal of Algebra 415 (October 2014): 184–213. http://dx.doi.org/10.1016/j.jalgebra.2014.05.032.

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30

Chen, Ke, Xin Lu, and Kang Zuo. "The Oort Conjecture for Shimura Curves of Small Unitary Rank." Communications in Mathematics and Statistics 6, no. 3 (August 25, 2018): 249–68. http://dx.doi.org/10.1007/s40304-018-0155-8.

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31

Chen, Lin, and Li Yu. "On the Schmidt-rank-three bipartite and multipartite unitary operator." Annals of Physics 351 (December 2014): 682–703. http://dx.doi.org/10.1016/j.aop.2014.09.026.

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32

Henn, Andreas, Michael Hentschel, Aloys Krieg, and Gabriele Nebe. "On the Classification of Lattices Over Which Are Even Unimodular -Lattices of Rank 32." International Journal of Mathematics and Mathematical Sciences 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/837080.

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33

Cantier, Laurent. "A unitary Cuntz semigroup for C⁎-algebras of stable rank one." Journal of Functional Analysis 281, no. 9 (November 2021): 109175. http://dx.doi.org/10.1016/j.jfa.2021.109175.

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34

Chen, Shangdi, Qin Xu, and Junmei Zhang. "Constructions of rank metric codes under actions of the unitary groups." Linear Algebra and its Applications 645 (July 2022): 293–306. http://dx.doi.org/10.1016/j.laa.2022.03.031.

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35

Boyadzhiev, Khristo N. "A trace formula for two unitary operators with rank one commutator." Proceedings of the American Mathematical Society 113, no. 1 (January 1, 1991): 157. http://dx.doi.org/10.1090/s0002-9939-1991-1057950-x.

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36

Scaramuzzi, Roberto. "A notion of rank for unitary representations of general linear groups." Transactions of the American Mathematical Society 319, no. 1 (January 1, 1990): 349–79. http://dx.doi.org/10.1090/s0002-9947-1990-0958900-8.

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37

Lin, Huaxin. "Approximate unitary equivalence in simple $C^{*}$-algebras of tracial rank one." Transactions of the American Mathematical Society 364, no. 4 (April 1, 2012): 2021–86. http://dx.doi.org/10.1090/s0002-9947-2011-05431-0.

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38

Grbac, Neven, and Joachim Schwermer. "Eisenstein series for rank one unitary groups and some cohomological applications." Advances in Mathematics 376 (January 2021): 107438. http://dx.doi.org/10.1016/j.aim.2020.107438.

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39

Kozioł, Karol, and Stefano Morra. "Serre weight conjectures for p-adic unitary groups of rank 2." Algebra & Number Theory 16, no. 9 (December 19, 2022): 2005–97. http://dx.doi.org/10.2140/ant.2022.16.2005.

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40

LIN, HUAXIN. "UNITARIES IN A SIMPLE C*-ALGEBRA OF TRACIAL RANK ONE." International Journal of Mathematics 21, no. 10 (October 2010): 1267–81. http://dx.doi.org/10.1142/s0129167x10006446.

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Let A be a unital separable simple infinite dimensional C*-algebra with tracial rank not more than one and with the tracial state space T(A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ϵ > 0, there exists a self-adjoint element h ∈ As.a such that [Formula: see text] We also study the problem when u can be approximated by unitaries in A with finite spectrum. Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A) ⊂ U0(A). Denote by [Formula: see text] the affine function on T(A) defined by [Formula: see text]. We show that u can be approximated by unitaries in A with finite spectrum if and only if u ∈ CU(A) and [Formula: see text] for all n ≥ 1. Examples are given for which there are unitaries in CU(A) which cannot be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.
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41

Nien, Chufeng. "Klyachko Models for General Linear Groups of Rank 5 over a p-Adic Field." Canadian Journal of Mathematics 61, no. 1 (February 1, 2009): 222–40. http://dx.doi.org/10.4153/cjm-2009-011-2.

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Abstract. This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of GL5 (ℱ), where ℱ is a p-adic field. It is an extension of the work of Heumos and Rallis on GL4(ℱ).
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42

González-Alonso, Víctor, Lidia Stoppino, and Sara Torelli. "On the rank of the flat unitary summand of the Hodge bundle." Transactions of the American Mathematical Society 372, no. 12 (July 8, 2019): 8663–77. http://dx.doi.org/10.1090/tran/7868.

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43

Bevilacqua, Roberto, Gianna M. Del Corso, and Luca Gemignani. "Efficient Reduction of Compressed Unitary Plus Low Rank Matrices to Hessenberg Form." SIAM Journal on Matrix Analysis and Applications 41, no. 3 (January 2020): 984–1003. http://dx.doi.org/10.1137/19m1280363.

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44

Hanzer, Marcela. "The unitary dual of the hermitian quaternionic group of split rank 2." Pacific Journal of Mathematics 226, no. 2 (August 1, 2006): 353–88. http://dx.doi.org/10.2140/pjm.2006.226.353.

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45

Di, YaoMin, and Li Liu. "Entanglement capacity of two-qubit unitary operator for rank two mixed states." Science in China Series G: Physics, Mechanics and Astronomy 50, no. 6 (December 2007): 691–97. http://dx.doi.org/10.1007/s11433-007-0075-1.

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46

Zhang, Pingping, Hu Yang, and Hanyu Li. "Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/219025.

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Some new perturbation bounds for both weighted unitary polar factors and generalized nonnegative polar factors of the weighted polar decompositions are presented without the restriction thatAand its perturbed matrixA˜have the same rank. These bounds improve the corresponding recent results.
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47

Hurley, Ted. "Unique builders for classes of matrices." Special Matrices 9, no. 1 (January 1, 2021): 52–65. http://dx.doi.org/10.1515/spma-2020-0122.

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Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.
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48

Bamberg, John, Cheryl E. Praeger, and Binzhou Xia. "The covering radii of the 2-transitive unitary, Suzuki, and Ree groups." Journal of Group Theory 22, no. 1 (January 1, 2019): 103–17. http://dx.doi.org/10.1515/jgth-2018-0039.

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49

Tan, Victor. "Poles of Siegel Eisenstein Series on U(n, n)." Canadian Journal of Mathematics 51, no. 1 (February 1, 1999): 164–75. http://dx.doi.org/10.4153/cjm-1999-010-4.

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AbstractLet U(n, n) be the rank n quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of U(n, n) has at most simple poles at the integers or half integers in certain strip of the complex plane.
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50

Christopher Phillips, N. "Reduction of Exponential Rank in Direct Limits of C*-Algebras." Canadian Journal of Mathematics 46, no. 4 (August 1, 1994): 818–53. http://dx.doi.org/10.4153/cjm-1994-047-7.

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AbstractWe prove the following result. Let A be a direct limit of direct sums of C*-algebras of the form C(X) ⊗ Mn, with the spaces X being compact metric. Suppose that there is a finite upper bound on the dimensions of the spaces involved, and suppose that A is simple. Then the C* exponential rank of A is at most 1 + ε, that is, every element of the identity component of the unitary group of A is a limit of exponentials. This is true regardless of whether the real rank of A is 0 or 1.
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