For unital $$C^*$$-algebras A and B, we completely characterize the isometric ($$*$$-) automorphisms of their Banach space projective tensor product $$A\otimes ^\gamma B$$. This leads to the characterization of inner and outer isometric $$*$$-automorphisms of $$A\otimes ^\gamma B$$ as well. As an application, we provide a partial affirmative answer to a question posed by Kaijser and Sinclair, viz., we prove that for unital $$C^*$$-algebras A and B, the set of norm-one unitaries of $$A\otimes ^\gamma B$$ coincides with $$U(A) \otimes U(B)$$, where U(A) is the unitary group of A. We also establish the fact that the relative commutant of $$A\otimes ^\gamma {\mathbb {C}}1$$ in $$A \otimes ^\gamma B$$ is the same as $${\mathcal {Z}}(A) \otimes ^\gamma B$$, where B is a subhomogenous unital $$C^*$$-algebra, and A is any $$C^*$$-algebra.

中文翻译：

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$$C^*$$C∗-代数的巴拿赫空间投影张量积的自同构

对于单位 $$C^*$$-代数 A 和 B，我们完全刻画了它们的 Banach 空间投影张量积 $$A\otimes ^\gamma B$$ 的等距 ($$*$$-) 自同构。这也导致了 $$A\otimes ^\gamma B$$ 的内部和外部等距 $$*$$-自同构的表征。作为一个应用，我们对 Kaijser 和 Sinclair 提出的问题提供了部分肯定的答案，即，我们证明对于单位 $$C^*$$-代数 A 和 B，$$ 的范一幺正的集合A\otimes ^\gamma B$$ 与 $$U(A) \otimes U(B)$$ 重合，其中 U(A) 是 A 的酉群。我们还建立了 $$ 的相对换向符的事实A\otimes ^\gamma {\mathbb {C}}1$$ in $$A \otimes ^\gamma B$$ 与 $${\mathcal {Z}}(A) \otimes ^\gamma B$$ 相同$$，其中 B 是次齐次单位 $$C^*$$-代数，A 是任何 $$C^*$$-代数。