Homotopy of a pair of approximately commuting unitaries in a simple \(C^*\)-algebra
Let A be a \(K_{1}\)-simple real rank zero C*-algebra, as defined in the article. This class of algebras includes unital purely infinite simple C*-algebras [\textit{J. Cuntz}, Ann. Math., II. Ser. 113, 181-197 (1981; Zbl 0446.46052)], unital real rank zero simple A{\textbf{T}}-algebras [\textit{G. A. Elliot}, J. Reine Angew. Math. 443, 179-219 (1993; Zbl 0809.46067)] and (not necessarily simple) unital AF-algebras. The main results are homotopy lemmas like the following: given \(\varepsilon > 0\) there exists a \(\delta > 0\) such that if u and v are unitaries in A, almost commuting in that \(\|vu - uv \|< \delta\), the spectrum of \(u\), except for a single gap larger than \(\delta\), is \(\delta\)-dense in \({\mathbb T}\), i.e. spectral projections corresponding to spectral intervals of length greater than or equal to \(\delta\) are non-zero, \([v]_{1} = 0\) and a certain \(K_{0}\)-valued obstruction, the isospectral obstruction, is zero, then \(v\) can be deformed to \(1\) along a path in \({\mathcal{U}}(A)\) of length less than \(5\pi + 1\). The authors prove also a `super-homotopy' lemma of a more symmetric form in that there are two pairs of almost commuting unitaries (u,v); one pair can be deformed into the other along a path of pairs of almost commuting unitaries in the algebra, the length of the path being bounded by a larger constant than for the homotopy lemma. The Bott obstruction for almost commuting unitaries in a unital C*-algebra seemed [in \textit{T. A. Loring}, Can. J. Math. 40, No. 1, 197-216 (1988; Zbl 0647.18007)] to come out of the blue. When \(A\) is a \(K_1\)-simple real rank zero C*-algebra and \(u\) has finite spectrum, or even if \(u \in {\mathcal{U}}_{0}(A)\), the authors define the more concrete isospectral obstruction. They generalize the role of \(v\) by considering a unital endomorphism \(\lambda\) of \(A\). Both \(Q\) and \(E\) denote spectral projections of intersecting half-open intervals with a large half-open gap. They assume \(\|\lambda(u) - u \|\) small so that \(\lambda(Q) \) approximately commutes with \(E\). When \(\lambda = Adv\), the isospectral obstruction is defined to be \([\lambda(E)]_{0} - [QE]_{0}\); more generally they need a quotient to get into \(K_{0}\). They show that the isospectral obstruction is independent of the choice of \(Q\) and \(E\) and that the isospectral obstructions can be identified with Bott obstructions for \(A\). The proof of the main section progresses in terms of various homotopy lemmas, sometimes overlapping, with differing upper bounds for Length \((u_{t})\). 1) When there is a spectral gap, large in the sense that it has a fixed lower bound, the logarithmic map is bi-continuous so the unitaries \(u\) may be replaced by self-adjoint operators \(h\) with finite spectrum. In this case there is no obstruction. To prove the homotopy result the authors assume first, for some \(N\), that \(h\) has no spectral gap longer than or equal to \(1 \over 2N\) in [0,1]; the spectral projections \(p_k\), over the intervals \([{{N -k +1} \over {N}},1]\) for \( k \in {\mathbb N_{0}} \cup {1 \over 2} {\mathbb N}\), \(k \leq N+1\), are then strictly increasing and the spectrum of \({1 \over N}\sum_{1}^{N}{p_{k}}\) is contained in \( \{ {j \over N} :j = 0,1,\dots,N \}\). Now using fully the assumption of \( K_{1}\)-simple real rank zero, they prove various spectral theoretical lemmas in order to construct a path of unitaries, of bounded length (dependent on \(N\)), connecting \(p_{2k}-p_{2k-1}\) with corresponding spectral projections of \(h\) deformed by \(v\). Taking away the first assumption (the word `every' on p. 493 is a misprint), some projections may be equal but the proof still can take the same lines as before. 2) When \(K_{1}(A) = 0\) and there is a single spectral gap, they prove a homotopy lemma where they do not need to even mention isospectral obstruction. 3) When there are no large gaps then the isospectral obstruction may be non-zero. They prove a homotopy lemma for when the isospectral obstruction is zero. 4) For the remaining case which is proved, that mentioned in the first paragraph above, there is no obstruction, one single large gap in the spectrum of \(u\), the unitary not to be deformed, and \([u]_{1} \neq 0\), i.e., they have dropped the condition, in (2) above. that \(K_{1}(A)\) is trivial.The authors also give some related results [cf. \textit{G. A. Elliott} and \textit{M. Rørdam}, Comment. Math. Helv. 70, No. 4, 615-638 (1995; Zbl 0864.46038); \textit{G. A. Elliot}'s `Normal elements of a simple \(C^*\)-algebra', in Algebraic Methods in Operator Theory, 109-123 (1994; Zbl 0807.46073)].
\(K_1\)-simple real rank zero \(C^*\)-algebra
scientific article; zbMATH DE number 1250656
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