##
**Some \(C^\ast\)-algebras which are coronas of non-\(C^\ast\)-Banach algebras.**
*(English)*
Zbl 1357.47087

Let \(A\) be a separable, unital \(C^\ast\)-algebra and let \(\pi_1,\pi_2: A\to\mathcal{B}(H)\) be two faithful, unital representations of \(A\) on a separable Hilbert space \(H\) satisfying \(\pi_k(A)\cap\mathcal{K}(H)=\{0\}\) for \(k=1,2\), where \(\mathcal{K}(H)\) is the closed two-sided ideal of compact operators on \(H\).

The author’s famous “noncommutative Weyl-von Neumann theorem” states that \(\pi_1\) and \(\pi_2\) are unitarily equivalent modulo \(\mathcal{K}(H)\), that is, there exists a unitary \(U\in\mathcal{B}(H)\) such that \(U^\ast\pi_1(a)U-\pi_2(a)\in\mathcal{K}(H)\) for all \(a\in A\) [the author, Rev. Roum. Math. Pures Appl. 21, 97–113 (1976; Zbl 0335.46039)].

If \(J\subseteq\mathcal{K}(H)\) is a smaller two-sided ideal, one may ask whether \(\pi_1\) and \(\pi_2\) are unitarily equivalent modulo \(J\). This problem has been considered by the author in [J. Oper. Theory 2, 3–37 (1979; Zbl 0446.47003)] in the case that \(J\) is a normed ideal. By definition, a normed ideal is a two-sided ideal \(J\) of \(\mathcal{B}(H)\) containing all finite-rank operators together with a norm \(| \cdot|_J\) on \(J\) making it a Banach space, and such that the following compatibility properties are satisfied: We have \(| AXB|_J\leq \| A\| | X|_J \| B\|\) for all \(A,B\in\mathcal{B}(H)\) and \(X\in J\), and \(| X|_J=\| X\|\) for every rank-one operator \(X\). The most important examples of normed ideals are the Schatten \(p\)-class operators for \(p\in[1,\infty)\).

Assume that \(A\) is generated (as a \(C^\ast\)-algebra) by the selfadjoint elements \(X_1\), \(X_2,\dots,X_n\). To obtain unitary equivalence of \(\pi_1\) and \(\pi_2\) modulo \(J\), one has to additionally require that the invariant \(k_J\) vanishes on the tuples \((\pi_k(X_1),\dots,\pi_k(X_n))\) for \(k=1,2\). The definition of \(k_J\) is to involved to be stated here.

In Section 3, the main objects of the paper are introduced. Given a tuple \(\tau=(T_1,\dots,T_n)\) of selfadjoint operators in \(\mathcal{B}(H)\) and a normed ideal \(J\), the algebra \(\mathcal{E}(\tau,J)\) is defined as the collection of operators \(X\in\mathcal{B}(H)\) that commute with each \(T_k\) up to \(J\). Given \(X\) in \(\mathcal{E}(\tau,J)\), the norm \(\| X\| + | X|_J\) makes \(\mathcal{E}(\tau,J)\) into a Banach \(\ast\)-algebra with isometric involution.

We set \(\mathcal{K}(\tau,J):=\mathcal{K}(H)\cap\mathcal{E}(\tau,J)\), which is a closed, two-sided \(\ast\)-invariant ideal in \(\mathcal{E}(\tau,J)\). In every nontrivial case, \(\mathcal{K}(\tau,J)\) and \(\mathcal{E}(\tau,J)\) are not \(C^\ast\)-algebras. Nevertheless, it is shown that, in certain cases, the quotient Banach \(\ast\)-algebra \(\mathcal{E}(\tau,J)/\mathcal{K}(\tau,J)\) is a \(C^\ast\)-algebra and that there is a natural identification of \(\mathcal{E}(\tau,J)\) with the algebra of bounded multipliers on \(\mathcal{K}(\tau,J)\). Thus, in these cases the corona algebra of the non-\(C^\ast\)-Banach algebra \(\mathcal{K}(\tau,J)\) is a \(C^\ast\)-algebra.

The paper is mostly surveying and describing interesting connections between known results from other papers, and suggesting problems for future research. In particular, the proofs of the main results are contained in [the author, Groups Geom. Dyn. 8, No. 3, 985–1006 (2014; Zbl 1315.46062)].

The author’s famous “noncommutative Weyl-von Neumann theorem” states that \(\pi_1\) and \(\pi_2\) are unitarily equivalent modulo \(\mathcal{K}(H)\), that is, there exists a unitary \(U\in\mathcal{B}(H)\) such that \(U^\ast\pi_1(a)U-\pi_2(a)\in\mathcal{K}(H)\) for all \(a\in A\) [the author, Rev. Roum. Math. Pures Appl. 21, 97–113 (1976; Zbl 0335.46039)].

If \(J\subseteq\mathcal{K}(H)\) is a smaller two-sided ideal, one may ask whether \(\pi_1\) and \(\pi_2\) are unitarily equivalent modulo \(J\). This problem has been considered by the author in [J. Oper. Theory 2, 3–37 (1979; Zbl 0446.47003)] in the case that \(J\) is a normed ideal. By definition, a normed ideal is a two-sided ideal \(J\) of \(\mathcal{B}(H)\) containing all finite-rank operators together with a norm \(| \cdot|_J\) on \(J\) making it a Banach space, and such that the following compatibility properties are satisfied: We have \(| AXB|_J\leq \| A\| | X|_J \| B\|\) for all \(A,B\in\mathcal{B}(H)\) and \(X\in J\), and \(| X|_J=\| X\|\) for every rank-one operator \(X\). The most important examples of normed ideals are the Schatten \(p\)-class operators for \(p\in[1,\infty)\).

Assume that \(A\) is generated (as a \(C^\ast\)-algebra) by the selfadjoint elements \(X_1\), \(X_2,\dots,X_n\). To obtain unitary equivalence of \(\pi_1\) and \(\pi_2\) modulo \(J\), one has to additionally require that the invariant \(k_J\) vanishes on the tuples \((\pi_k(X_1),\dots,\pi_k(X_n))\) for \(k=1,2\). The definition of \(k_J\) is to involved to be stated here.

In Section 3, the main objects of the paper are introduced. Given a tuple \(\tau=(T_1,\dots,T_n)\) of selfadjoint operators in \(\mathcal{B}(H)\) and a normed ideal \(J\), the algebra \(\mathcal{E}(\tau,J)\) is defined as the collection of operators \(X\in\mathcal{B}(H)\) that commute with each \(T_k\) up to \(J\). Given \(X\) in \(\mathcal{E}(\tau,J)\), the norm \(\| X\| + | X|_J\) makes \(\mathcal{E}(\tau,J)\) into a Banach \(\ast\)-algebra with isometric involution.

We set \(\mathcal{K}(\tau,J):=\mathcal{K}(H)\cap\mathcal{E}(\tau,J)\), which is a closed, two-sided \(\ast\)-invariant ideal in \(\mathcal{E}(\tau,J)\). In every nontrivial case, \(\mathcal{K}(\tau,J)\) and \(\mathcal{E}(\tau,J)\) are not \(C^\ast\)-algebras. Nevertheless, it is shown that, in certain cases, the quotient Banach \(\ast\)-algebra \(\mathcal{E}(\tau,J)/\mathcal{K}(\tau,J)\) is a \(C^\ast\)-algebra and that there is a natural identification of \(\mathcal{E}(\tau,J)\) with the algebra of bounded multipliers on \(\mathcal{K}(\tau,J)\). Thus, in these cases the corona algebra of the non-\(C^\ast\)-Banach algebra \(\mathcal{K}(\tau,J)\) is a \(C^\ast\)-algebra.

The paper is mostly surveying and describing interesting connections between known results from other papers, and suggesting problems for future research. In particular, the proofs of the main results are contained in [the author, Groups Geom. Dyn. 8, No. 3, 985–1006 (2014; Zbl 1315.46062)].

Reviewer: Hannes Thiel (Münster)

### MSC:

47L30 | Abstract operator algebras on Hilbert spaces |

47L20 | Operator ideals |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

47B20 | Subnormal operators, hyponormal operators, etc. |

### Keywords:

commutant modulo a normed ideal operator; \(K\)-theory; Calkin algebra; almost normal operator; degree \(-1\) saturation
Full Text:
DOI

### References:

[1] | Voiculescu, D. V., Almost normal operators mod Hilbert-Schmidt and the \(K\)-theory of the algebras \(E \Lambda(\Omega)\), J. Non-commutat. Geom., 8, 4, 1123-1145 (2014) · Zbl 1325.46074 |

[3] | Voiculescu, D. V., Countable degree \(- 1\) saturation of certain \(C^\ast \)-algebras which are coronas of Banach algebras, Groups Geom. Dyn., 8, 985-1006 (2014) · Zbl 1315.46062 |

[4] | Voiculescu, D. V., Some results on norm-ideal perturbations of Hilbert space operators, I, J. Oper. Theory, 2, 3-37 (1979) · Zbl 0446.47003 |

[5] | Voiculescu, D. V., Some results on norm-ideal perturbations of Hilbert space operators, II, J. Oper. Theory, 5, 77-100 (1981) · Zbl 0483.46036 |

[6] | Voiculescu, D. V., On the existence of quasicentral approximate units relative to normed ideals, I, J. Funct. Anal., 91, 1, 1-36 (1990) · Zbl 0762.46051 |

[7] | Voiculescu, D. V., Perturbations of operators, connections with singular integrals, hyperbolicity and entropy, (Picardello, M. A., Harmonic Analysis and Discrete Potential Theory. Harmonic Analysis and Discrete Potential Theory, Proceedings of the International Meeting held in Frascati, July 1-10, 1991 (1992), Plenum Press), 181-191 |

[8] | Voiculescu, D. V., A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl., 21, 97-113 (1976) · Zbl 0335.46039 |

[9] | Voiculescu, D. V., Almost normal operators mod \(\gamma_p\), (Havin, V. P.; Hruscev, S. V.; Nikolski, N. K., Linear and Complex Analysis Problem Book. Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043 (1984), Springer Verlag), 227-230 |

[10] | Brown, L. G.; Douglas, R. G.; Fillmore, P. A., Unitary equivalence modulo the compact operators and extensions of \(C^\ast \)-algebras, (Proc. Conf. on Operator Theory. Proc. Conf. on Operator Theory, Lecture Notes in Math., vol. 345 (1973), Springer Verlag), 58-127 · Zbl 0277.46053 |

[11] | Carey, R. W.; Pincus, J. D., Commutators, symbols and determining functions, J. Funct. Anal., 19, 50-80 (1975) · Zbl 0309.47026 |

[12] | Voiculescu, D. V., The analogues of entropy and of Fisher’s information measure in free probability theory, V, Non-commutative Hilbert transforms, Inventi. Math., 132, 182-227 (1998) · Zbl 0930.46053 |

[13] | Connes, A., On the spectral characterization of manifolds, J. Noncommut. Geom., 7, 1, 1-82 (2013) · Zbl 1287.58004 |

[14] | Cuntz, J.; Thom, A., Algebraic \(K\)-theory and locally convex algebras, Math. Ann., 334, 339-371 (2006) · Zbl 1095.19003 |

[15] | Gohberg, I. T.; Krein, M. G., Introduction to the Theory of Linear Non-Self-Adjoint Operators (1965), Nauka: Nauka Moscow, Translated from the Russian by A. Feinstein, American Mathematical Society, Providence, RI, 1969 |

[16] | Simon, B., (Trace Ideals and Their Applications. Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120 (2005), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1074.47001 |

[17] | Bercovici, H.; Voiculescu, D. V., The analogue of Kuroda’s theorem for \(n\)-tuples, (Operator Theory: Advances and Applications, vol. 41 (1989), Birkhauser: Birkhauser Basel) · Zbl 0681.47018 |

[18] | Farah, I.; Hart, B., Countable saturation of corona algebras, C. R. Math. Rep. Acad. Sci., Canada, 35, 2, 35-56 (2013) · Zbl 1300.46047 |

[22] | Paschke, W. L., \(K\)-theory for commutants in the Calkin algebra, Pacific J. Math., 95, 2, 427-434 (1981) · Zbl 0478.46056 |

[23] | Martin, M.; Putinar, M., Lectures on hyponormal operators, (Operator Theory: Advances and Applications, vol. 39 (1989), Birkhauser) · Zbl 0684.47018 |

[24] | Clancey, K. F., Seminormal operators, (Lecture Notes in Math., vol. 742 (1979), Springer Verlag) · Zbl 0435.47032 |

[25] | Xia, D., Spectral theory of hyponormal operators, (Operator Theory: Advances and Applications, vol. 10 (1983)) · Zbl 0523.47012 |

[26] | Helton, J. W.; Howe, R., Integral operators, commutator traces, index and homology, (Proceedings of a Conference on Operator Theory. Proceedings of a Conference on Operator Theory, Lecture Notes in Math., vol. 345 (1973), Springer Verlag), 141-209 |

[27] | Connes, A., Non-commutative differential geometry, (Inst. Hautes Etudes Sci. Publ. Math., vol. 62 (1985)), 257-360 · Zbl 0564.58002 |

[28] | Brown, L. G., Operator algebras and algebraic \(K\)-theory, Bull. Amer. Math. Soc., 1119-1121 (1975) · Zbl 0332.46038 |

[30] | Voiculescu, D. V., Remarks on Hilbert-Schmidt perturbations of almost-normal operators, (Topics in Modern Operator Theory (1981), Birkhauser), 311-318 |

[31] | Cipriani, F., Dirichlet forms on non-commutative spaces, (Quantum Potential Theory. Quantum Potential Theory, Lecture Notes in Math., vol. 1954 (2008), Springer Verlag), 161-276 · Zbl 1166.81028 |

[32] | Cipriani, F.; Sauvageot, J.-L., Derivations and square roots of Dirichlet forms, J. Funct. Anal., 13, 3, 521-545 (2003) |

[33] | Choi, M. D.; Effros, E. G., The completely positive lifting problem for \(C^\ast \)-algebras, Ann. of Math. (2), 104, 3, 585-609 (1976) · Zbl 0361.46067 |

[34] | Arveson, W. B., Notes on extensions of \(C^\ast \)-algebras, Duke Math. J., 44, 329-355 (1977) · Zbl 0368.46052 |

[35] | Pasnicu, C., Weighted shifts as direct summands mod \(C_2\) of normal operators, (Operator Theory Advances and Applications, vol. 11 (1983), Birkhaus), 275-281 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.