Abstract Given C ⁎ -algebras A and B , consider the Banach algebra A ⊗ γ B , where ⊗ γ denotes the projective Banach space tensor product. If A and B are commutative, this is the Varopoulos algebra V A , B ; we write V A for V A , A . It has been an open problem for almost 40 years to determine precisely when A ⊗ γ B is Arens regular; see, e.g., [33] , [48] , [49] . We solve this classical question for arbitrary C ⁎ -algebras. Indeed, we show that A ⊗ γ B is Arens regular if and only if A or B has the Phillips property; note that A has the latter property if and only if it is scattered and has the Dunford–Pettis Property. A further equivalent condition is that A ⁎ has the Schur property, or, again equivalently, the enveloping von Neumann algebra A ⁎ ⁎ is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A ⊗ γ B is entirely encoded in the geometry of the C ⁎ -algebras. In case A and B are von Neumann algebra, we conclude that A ⊗ γ B is Arens regular (if and) only if A or B is finite-dimensional. We also show that this characterization does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C ⁎ -algebras A and B , we obtain that V A , B is Arens regular if and only if A or B is scattered. We further describe the centre Z ( V A ⁎ ⁎ ) , showing that it is Banach algebra isomorphic to A ⁎ ⁎ ⊗ e h A ⁎ ⁎ , where ⊗ e h denotes the extended Haagerup tensor product. We deduce that V A is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, V A is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered; this is the case, e.g., for A = l ∞ .

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C⁎-代数的几何及其投影张量积的二元性

摘要 给定 C ⁎ -代数 A 和 B ，考虑 Banach 代数 A ⊗ γ B ，其中 ⊗ γ 表示射影 Banach 空间张量积。如果 A 和 B 是可交换的，则这是 Varopoulos 代数 VA , B ；我们将 VA 写为 VA , A 。近 40 年来，精确确定 A ⊗ γ B 何时为阿伦斯正则一直是一个悬而未决的问题。参见，例如，[33]、[48]、[49]。我们为任意 C ⁎ -代数解决了这个经典问题。事实上，我们证明 A ⊗ γ B 是阿伦斯正则的当且仅当 A 或 B 具有菲利普斯性质；请注意，当且仅当 A 是分散的并且具有 Dunford-Pettis 性质时，A 才具有后一个性质。另一个等价条件是 A ⁎ 具有 Schur 性质，或者，同样等价地，包络冯诺依曼代数 A ⁎ ⁎ 是有限原子的，即矩阵代数的直接和。因此，A ⊗ γ B 的阿伦斯正则性完全编码在 C ⁎ -代数的几何中。在 A 和 B 是冯诺依曼代数的情况下，我们得出结论，只有当 A 或 B 是有限维时，A ⊗ γ B 才是阿伦斯正则（当且）。我们还表明，这种表征不能推广到非自伴随对偶（甚至可交换）算子代数类。专门研究可交换 C ⁎ -代数 A 和 B ，我们得到 VA , B 是阿伦斯正则当且仅当 A 或 B 是分散的。我们进一步描述中心 Z ( VA ⁎ ⁎ ) ，表明它与 A ⁎ ⁎ ⊗ eh A ⁎ ⁎ 是 Banach 代数，其中 ⊗ eh 表示扩展的 Haagerup 张量积。仅当 A 是有限维时，我们推导出 VA 是强不规则的（当且）。因此，VA 既不是阿伦斯正则也不是强阿伦斯不规则，当且仅当 A 是非散射的；情况是这样的