On homotopy invariants of tensor products of Banach algebras (Q2176071)

Let \(\mathfrak A\) be unital (complex) Banach algebra and \(Z(\mathfrak A)\) its center. Consider unital subalgebras \(A\subseteq Z(\mathfrak A)\) and \(B\subseteq\mathfrak A\) and denote by \(A\hat{\otimes}_{\mathfrak A}B\) the closure in \(\mathfrak A\) of the algebra \[\langle A, B\rangle=\Bigl\{\sum_{k=1}^na_kb_k: a_k\in A,\ b_k\in B,\ n\in{\mathbb Z}^+\Bigr\}.\] Let \({\mathfrak M}(A)\) denote the set of all non-zero homomorphisms \(A\rightarrow\mathbb C\), equipped with the Gelfand topology, and \(C({\mathfrak M}(A), B)\) the Banach algebra of all continuous functions \(f:{\mathfrak M}(A)\rightarrow B\) with the norm \[||f||=\max_{\xi\in{\mathfrak M}(A)}||f(\xi)||_B.\] The authors state some conditions under which there exist homotopy equivalences between the sets of invertible/left invertible/idempotent elements of \begin{itemize} \item[(1)] \(A\hat{\otimes}_{\mathfrak A}B\) and \(C({\mathfrak M}(A), B)\);\par \item[(2)] \(A\hat{\otimes}_{\mathfrak A}B\) and \(B\). \end{itemize} Many examples are provided, but the proofs use a lot of results from other papers without stating the results. The proofs are often presented only schematically, leaving the details to be worked out by the reader.