On the continuity of the group inverse in $C^*$-algebras

J. Benítez; D. Cvetković-Ilić; X. Liu

doi:10.15352/bjma/1396640064

2014 On the continuity of the group inverse in $C^*$-algebras

J. Benítez, D. Cvetković-Ilić, X. Liu

Banach J. Math. Anal. 8(2): 204-213 (2014). DOI: 10.15352/bjma/1396640064

Abstract

Let $\{a_n\}_{n=1}^\infty$ be a sequence of group invertible elements of a unital $C^*$ algebra $\mathcal{A}$ that converges to $a$. We present some equivalent conditions for the group invertibility of $a$ and for the convergence of $\{a_n^\#\}_{n=1}^\infty$ to $a^\#$.

References

1.

J. Benítez and V. Rakočević, Invertibility of the commutator of an element in a $C^*$-algebra and its Moore–Penrose inverse, Studia Math. 200 (2010), 163–174. J. Benítez and V. Rakočević, Invertibility of the commutator of an element in a $C^*$-algebra and its Moore–Penrose inverse, Studia Math. 200 (2010), 163–174.

2.

J. Benítez and D. Cvetković-Ilić, On the elements $aa^\dagger$ and $a^\dag a$ in a ring, Appl. Math. Comput. 222 (2013), 478–489. J. Benítez and D. Cvetković-Ilić, On the elements $aa^\dagger$ and $a^\dag a$ in a ring, Appl. Math. Comput. 222 (2013), 478–489.

3.

D. Buckholtz, Hilbert space idempotents and involutions, Proc. Amer. Math. Soc. 128 (2000), 1415–1418. MR1653425 10.1090/S0002-9939-99-05233-8 D. Buckholtz, Hilbert space idempotents and involutions, Proc. Amer. Math. Soc. 128 (2000), 1415–1418. MR1653425 10.1090/S0002-9939-99-05233-8

4.

D. Buckholtz, Inverting the difference of Hilbert space projections, Amer. Math. Monthly 104 (1997), 60–61. MR1426419 10.2307/2974825 D. Buckholtz, Inverting the difference of Hilbert space projections, Amer. Math. Monthly 104 (1997), 60–61. MR1426419 10.2307/2974825

5.

S.L. Campbell and C.D. Meyer, Jr., Continuity Properties of the Drazin Pseudoinverse, Linear Algebra Appl. 10 (1975), 77–83. MR364283 10.1016/0024-3795(75)90097-X S.L. Campbell and C.D. Meyer, Jr., Continuity Properties of the Drazin Pseudoinverse, Linear Algebra Appl. 10 (1975), 77–83. MR364283 10.1016/0024-3795(75)90097-X

6.

M.P. Drazin, Pseudoinverse in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506–514. MR98762 10.2307/2308576 M.P. Drazin, Pseudoinverse in associative rings and semigroups, Amer. Math. Monthly 65 (1958), 506–514. MR98762 10.2307/2308576

7.

A. Galántai, Subspaces, angles and pairs of orthogonal projections, Linear Multilinear Algebra 56 (2008), 227–260. MR2384652 10.1080/03081080600743338 A. Galántai, Subspaces, angles and pairs of orthogonal projections, Linear Multilinear Algebra 56 (2008), 227–260. MR2384652 10.1080/03081080600743338

8.

R. Harte and M. Mbekhta, On generalized inverses in $C^*$-algebras II, Studia Math. 106 (1993), 129–138. MR1240309 R. Harte and M. Mbekhta, On generalized inverses in $C^*$-algebras II, Studia Math. 106 (1993), 129–138. MR1240309

9.

Q. Huang, J. Yu and L. Zhu, Some new perturbation results for generalized inverses of closed linear operators in Banach spaces, Banach J. Math. Anal. 6 (2012), 58–68. MR2945988 euclid.bjma/1342210160 Q. Huang, J. Yu and L. Zhu, Some new perturbation results for generalized inverses of closed linear operators in Banach spaces, Banach J. Math. Anal. 6 (2012), 58–68. MR2945988 euclid.bjma/1342210160

10.

I.C.F. Ipsen and C.D. Meyer, The angle between complementary subspaces, Amer. Math. Monthly 102 (1995), 904–911. MR1366052 10.2307/2975268 I.C.F. Ipsen and C.D. Meyer, The angle between complementary subspaces, Amer. Math. Monthly 102 (1995), 904–911. MR1366052 10.2307/2975268

11.

S. Izumino, Convergence of generalized inverses and spline projectors, J. Approx. Theory 38 (1983), 269–278. MR705545 10.1016/0021-9045(83)90133-8 S. Izumino, Convergence of generalized inverses and spline projectors, J. Approx. Theory 38 (1983), 269–278. MR705545 10.1016/0021-9045(83)90133-8

12.

J.J. Koliha and V. Rakočević, Continuity of the Drazin inverse II, Studia Math. 131 (1998), 167–177. MR1636348 J.J. Koliha and V. Rakočević, Continuity of the Drazin inverse II, Studia Math. 131 (1998), 167–177. MR1636348

13.

J.J. Koliha, Range projections of idempotents in $C^*$-algebras, Demonstratio Math. 34 (2001), 91–103. MR1823088 J.J. Koliha, Range projections of idempotents in $C^*$-algebras, Demonstratio Math. 34 (2001), 91–103. MR1823088

14.

J.J. Koliha and V. Rakočević, On the norm of idempotents in $C^*$-algebras, Rocky Mountain J. Math., 34 (2004), 685–697. MR2072801 10.1216/rmjm/1181069874 euclid.rmjm/1181069874 J.J. Koliha and V. Rakočević, On the norm of idempotents in $C^*$-algebras, Rocky Mountain J. Math., 34 (2004), 685–697. MR2072801 10.1216/rmjm/1181069874 euclid.rmjm/1181069874

15.

V.E. Ljance, Certain properties of idempotent operators (Russian) Tr. Inst. Prikl. Mat. Mekh. 1 (1958), 16–22. MR196493 V.E. Ljance, Certain properties of idempotent operators (Russian) Tr. Inst. Prikl. Mat. Mekh. 1 (1958), 16–22. MR196493

16.

S. Maeda, On the distance between two projections in $C^*$-algebras, Math. Japon. 22 (1977), 61–65. MR450983 S. Maeda, On the distance between two projections in $C^*$-algebras, Math. Japon. 22 (1977), 61–65. MR450983

17.

M. Mbekhta, Conorme et inverse généralisé dans les $C^*$-algèbres, Canad. Math. Bull. 35 (1992), 515–522. MR1191512 10.4153/CMB-1992-068-8 M. Mbekhta, Conorme et inverse généralisé dans les $C^*$-algèbres, Canad. Math. Bull. 35 (1992), 515–522. MR1191512 10.4153/CMB-1992-068-8

18.

V. Rakočević, Continuity of the Drazin inverse, J. Operator Theory, 41 (1999), 55–68. MR1675243 V. Rakočević, Continuity of the Drazin inverse, J. Operator Theory, 41 (1999), 55–68. MR1675243

19.

V. Rakočević, On the continuity of the Moore–Penrose inverse in $C^*$-algebras, Math. Montisnigri 2 (1993), 89–-92. MR1284899 V. Rakočević, On the continuity of the Moore–Penrose inverse in $C^*$-algebras, Math. Montisnigri 2 (1993), 89–-92. MR1284899

20.

G.W. Stewart, On the continuity of the generalized inverse, SIAM J. Appl. Math. 17 (1969), 33–45. MR245583 10.1137/0117004 G.W. Stewart, On the continuity of the generalized inverse, SIAM J. Appl. Math. 17 (1969), 33–45. MR245583 10.1137/0117004

Citation Download Citation

J. Benítez, D. Cvetković-Ilić, and X. Liu "On the continuity of the group inverse in $C^*$-algebras," Banach Journal of Mathematical Analysis 8(2), 204-213, (2014). https://doi.org/10.15352/bjma/1396640064

Published: 2014

Access the abstract

JOURNAL ARTICLE
10 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY