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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)].

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.
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